Singularities of configuration and graph hypersurfaces

THE UNIVERSITY OF CHICAGO
SINGULARITIES OF CONFIGURATION AND GRAPH HYPERSURFACES
A DISSERTATION SUBMITTED TO
THE FACULTY OF THE DIVISION OF THE PHYSICAL SCIENCES
IN CANDIDACY FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
DEPARTMENT OF MATHEMATICS
BY
ERIC PATTERSON
CHICAGO, ILLINOIS
JUNE 2009
UMI Number: 3362457
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To my gardener
ABSTRACT
I show that the singularities of a configuration hypersurface correspond to the points
whose corank is at least two. The result generalizes to compute multiplicities at
higher corank loci in configuration hypersurfaces. Though the result is similar to the
case of a generic determinantal variety, the proof proceeds from a different perspective. The key is simple linear algebra: the rank of a quadratic form is determined
by its ranks when restricted to each hyperplane in a complete set of hyperplanes.
I review the construction of the first and second graph hypersurfaces as examples
of configuration hypersurfaces. In particular, the singularities of the first and second
graph hypersurfaces are corank at least two loci. As an application, the singularities
of the first graph hypersurface are contained in the second graph hypersurface, and
the singularities of the second graph hypersurface are contained in the first hypersurface.
I show that the map to which the graph hypersurface is dual is not an embedding
in general. Therefore, the incidence variety may not resolve the singularities of the
graph hypersurface.
I present a formula that describes the graph polynomial of a graph produced by
a specific gluing-deleting operation on a pair of graphs. The operation produces logdivergent graphs from log-divergent graphs, so it is useful for generating examples
relevant for physics. A motivic understanding of the operation is still missing.
m
ACKNOWLEDGMENTS
I am in great debt to my advisor, Spencer Bloch. He introduced me to the subject
of graph hypersurfaces and was a source of infinitely many ideas, questions, and suggestions during the research process. I greatly appreciate the direction and freedom
he gave me, and his encouragement was indispensable in the completion of this work.
Moreover, he has worked quickly and tirelessly to help me meet every deadline even
though I never left sufficient time. His role cannot be overstated.
I would like to thank Bob Kottwitz for sitting on my committee and for reading
the manuscript under extreme time constraints. His willing assistance during my
graduate education has been a boon of utmost importance.
I learned Theorem 4.4.8 from Laci Babai while working for his discrete math
course for the REU in 2004. At that time, I had no idea it would figure prominently
in this work. I am grateful to him for this contribution.
I would like to thank the Department of Mathematics at the University of Chicago
for its generous support and wonderful academic environment, which I often took
for granted but were instrumental in the completion of this work.
I must thank Gabe Kerr for his interest in this project, especially during its
infancy. While we were never successful in implementing his approaches, I learned
an enormous amount from his suggestions.
In the final months of this project, Adam Ginensky has provided many valuable
contributions. His interest and enthusiasm alone has been a great motivator. He
also pointed me to many useful resources and perspectives. These suggestions have
greatly improved this project. I regret that I have not been able to make a precise
connection between his work and mine in the present work. That will have to be the
topic of future work. His input on the writing process has prevented much anguish
and provided many improvements to the manuscript. I am grateful for all of these
contributions.
iv
V
Through all my years at the University of Chicago, Nick Longo has been a valued
colleague and friend. His door was always open for all matters mathematical and
otherwise.
Nick provided a key element: coffee.
Before Nick's influence, I failed
to recognize the efficacy of this wonderful drug, without which I doubt I would
have completed the present work. He also provided all of the necessary threats to
motivate me, and by having twice as much to do as me at all times, he provided a
great example to emulate.
There are two common features to almost all Ph.D.'s. First, there is the decision
to start the Ph.D. Second, as Spencer Bloch once put it, "The point of graduate
school is to graduate."
Regarding the first, I must thank everyone who inspired
me to begin this endeavor, especially Abhijit Champanerkar, Bob Friedman, Patrick
Gallagher, Ilya Kofman, Max Lipyanskiy, Henry Pinkham, Michael Prim, and Greg
Smith. Regarding the second, I must thank WH Trading whose light at the end of
the tunnel motivated me to get out of the tunnel.
Finally, I must thank and apologize to everyone who put up with my abuse and
neglect for the past six years, particularly my students and my family. My parents
have been more supportive and encouraging throughout graduate school and my life
than I could ever express. Lauren Duensing, has been by my side through everything.
Her patience is extraordinary. She is completely responsible for preserving my sanity,
for which I am eternally grateful. She has also done a tremendous job of preventing
extensive abuse of the English language by thoroughly proofreading and generally
providing grammatical advice. And Shelby has been my constant companion during
all of my writing and research—thank you, Shel.
INTRODUCTION
This dissertation essentially proves a few facts about some very specific degeneracy
loci. In simple terms, the motivating questions of this dissertation are:
1. Is the dual of a graph hypersurface smooth?
2. How are the first and second graph hypersurfaces related, and where do they
intersect?
I find partial answers to each question, but an important contribution of this dissertation is a result discovered while trying to answer the second question. Namely,
the singular locus of the graph hypersurface, and more generally any configuration
hypersurface, is the same as the corank at least 2 locus when thought of as quadratic
forms.
The motivation for both questions is that the defining polynomials of the first and
second graph hypersurfaces appear in parametric Feynman integrals [IZ80]. Broadhurst and Kreimer [BK95,BK97] numerically calculated values of these integrals and
discovered that they are (at least approximately) multizeta values. Their discovery
interested algebraic geometers who view the integrals as periods of the first graph
hypersurface. That these periods may be multizeta values suggests trying to apply
the techniques of motives, which already establish connections between geometry
and multizeta values [Bor77, Sou86, DG05]. Belkale and Brosnan [BB03] show that
the graph hypersurfaces are not mixed Tate in general. Therefore, whether the periods are multizeta values requires more delicate machinery to identify whether these
particular periods are multizeta values and/or which graphs have multizeta periods. Bloch, Esnault, and Kreimer [BEK06] verified that the motives of the wheeland-spokes graph hypersurfaces are consistent with the period being a zeta value.
Brown [Bro09] has developed algorithmic methods for evaluating these integrals and
determines a criterion to determine whether particular graphs evaluate to multizeta
vi
vii
values. The criterion is strong enough to find new results but is not sufficient for all
graphs. Aluffi and Marcolli have studied graph hypersurfaces from the point of view
of the Grothendieck group and Chern-Schwartz-MacPherson classes [AM08] and have
addressed the mixed Tate issue under combinatorial conditions on the graph [AM09].
The work of Brown, Aluffi, and Marcoli draws attention to working with the affine
hypersurface that is the cone over the graph hypersurface; my results translate to
the affine case, as well. Their work also suggests how the complexity of the topology
of the graph (as measured by its crossing number and its genus) may be influencing
the complexity of the multizeta values (as in multizeta values are more complex than
zeta values).
The first graph polynomial has received more attention than the second in the
algebraic geometry literature because a convergent parametric Feynman integral is
a period for the first graph hypersurface, not the second. However, a more complete
understanding of these periods will require a better understanding of the second
hypersurface, as well. A configuration hypersurface is a degeneracy locus naturally
associated with a nonzero subspace of a based vector space. The paper [BEK06]
shows that the first graph hypersurface is a configuration hypersurface; following the
same ideas, I show in Proposition 4.2.12 that the same is true for the second graph
hypersurface.
The motivation for the first question is that if the dual of the graph hypersurface were smooth, then the incidence variety defining the dual would resolve the
singularities of the graph hypersurface. The paper [BEK06] noted that the graph
hypersurface is naturally a dual variety when a specific quadratic linear series is
an embedding, and one could hope that the explicit nature of the incidence variety
would provide a useful resolution. In Theorem 4.4.1, I show that the quadratic linear
series is not an embedding in general, so the incidence variety may not resolve the
singularities. There are exceptional graphs for which the quadratic linear series is
an embedding, but they must satisfy strong conditions. Depending on the reader's
background, this result may require very little or very much new material. I present
this result last as it is not a prerequisite for the other geometric results of the paper.
viii
Readers interested only in the graph theory may skip to Chapter 4; the prerequisite
geometry for this result is in Section 2.2.
My partial answer to the second question, Corollary 4.2.15, is that the singular
loci of both the first and second graph hypersurfaces are contained in the intersection
of the two graph hypersurfaces.
I discovered this result by a careful analysis of
restrictions of configurations. I show in Section 4.2 that the singularities of the graph
hypersurfaces agree with the corank at least two locus. This result, interesting in its
own right, also provides a tool for comparing the singular loci of the hypersurfaces
of subconfigurations such as the first and second graph hypersurfaces. Moreover, I
show that the points of multiplicity k correspond to the corank k locus, and I show
this for configuration hypersurfaces, of which graph hypersurfaces are a special case.
These results are discussed in Chapter 3 and depend on the preliminaries presented
in the preceding chapters.
There are two other results I present that I would like to point out.
First,
Corollary 1.2.24 shows that the rank of a degenerate bilinear form will remain the
same on at least one hyperplane in a complete set of hyperplanes.
This is the
generic behavior for hyperplanes, and the complete condition implies that one of the
hyperplanes is generic in this sense. This result is not true for nonsymmetric bilinear
forms. This corollary and its generalization, Lemma 1.2.4, are essential in the proof
that the singular locus of a configuration hypersurface is the corank greater than one
locus. Because it is an interesting linear algebra result in its own right, including its
proof seems preferable to giving a reference.
Second, I present a graph operation that glues graphs along an edge and deletes
that edge. This operation is useful for the study of log-divergent graphs because
from two log-divergent graphs it produces another log-divergent graph. Therefore, it
is a useful operation for writing down high-loop, log-divergent examples. In Proposition 4.3.19, I derive the effect of this operation on the graph polynomial.
The
geometry of this formula, and in particular how the graph motive is affected, needs
further development. Physicists suggest that the Feynman amplitudes corresponding
to this operation will be a product, so one would hope to find a product structure
IX
reflected in the motive.
There have been many approaches to studying graph hypersurfaces in the literature. Many approaches use the combinatorics of the graphs to make conclusions
about the hypersurface; Theorem 4.4.1 falls into that category.
My approach to
Corollary 4.2.14 uses the linear algebra construction of the graph polynomial. This
construction can be traced back to Kirchhoff using graph Laplacians to study electrical networks. In fact, alternative names for the graph polynomials in the literature
are Kirchhoff
polynomials and Kirchhoff-Symanzik
polynomials (Symanzik applied
them in quantum field theory). The more recent literature [BEK06] emphasized this
approach, and my progress would have been impossible without the philosophy of
that source.
In Chapter 1, I introduce the bilinear form notation and terminology that I
will use, including the geometry of quadrics.
The key result of this chapter is
Lemma 1.2.4, which describes constraints on the rank of a bilinear form in terms of
its rank on its subspaces.
Chapter 2 discusses some general geometric constructions of families of quadratic
forms, namely degeneracy loci in Section 2.1 and quadratic linear series in Section 2.2.
The remaining chapters study specific cases of these constructions.
The goal of Chapter 3 is to apply Lemma 1.2.4 in the case of certain degeneracy
loci defined for configurations.
The results of this chapter are about rank loci of
specific families of symmetric matrices spanned by rank one matrices. I begin in
Section 3.1 by recalling the definition of this family and its degeneracy locus, the
configuration hypersurface. Its defining polynomial has a simple description in terms
of Pliicker coordinates, and I explain how this varies when restricting the configuration in Section 3.2. In Section 3.3, I exploit these descriptions to find that the
singularities of a configuration hypersurface and their order are determined by the
rank of these points as matrices. This connection between singularity and rank is
applied to subconfigurations in Section 3.4, which is used in Chapter 4 to compare
the first and second graph hypersurfaces.
X
In Chapter 4, I apply the preceding results to the case of graph hypersurfaces.
I start with a brief review of the necessary combinatorics and algebraic topology of
graphs in Section 4.1. I continue by describing two configurations that come from
graphs. Computing their configuration polynomials reveals the familiar combinatorial definitions of the graph polynomials in Section 4.2. I proceed to state the
singularity result for the graph hypersurfaces, which does not require anything new
by that point. In Section 4.3, I explain some of the ways in which the special nature of these polynomials reflects operations on graphs. These operations and their
effects on the polynomials are well-known and well used in the literature, but I have
included one formula (4.6), perhaps not as well-known, that is lacking a geometric interpretation. To the extent possible, my arguments emphasize linear algebra
coming from the algebraic topology of graphs rather than combinatorics. The final
Section 4.4 describes how the quadratic linear series of the graph configuration is
not an embedding for most graphs.
CONTENTS
ABSTRACT
iii
ACKNOWLEDGMENTS
iv
INTRODUCTION
vi
LIST OF FIGURES
xii
CHAPTER 1. DEGENERATE BILINEAR FORMS
1.1 Basics of Bilinear Forms
1.2
Restriction of Bilinear Forms and Rank Lemmas
CHAPTER 2.
FAMILIES OF BILINEAR FORMS
1
2
7
32
2.1
Degeneracy Loci
32
2.2
Quadratic Linear Series
38
CHAPTER 3.
CONFIGURATIONS
48
3.1
Configuration Hypersurfaces
49
3.2
Restrictions and Projections of Configurations
62
3.3
Singularities of Configuration Hypersurfaces
76
3.4
Subconfigurations
85
CHAPTER 4. GRAPH HYPERSURFACES
90
4.1
Graph Preliminaries
90
4.2
Configurations from Graphs
105
4.3
Polynomial Formulas for Some Graph Operations
123
4.4
Quadratic Linear Series Defined by Graphs
146
REFERENCES
161
XI
FIGURES
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
A typical graph
A circuit with 6 edges and vertices
A circuit with three edges
Spanning trees of the circuit with three edges
Two-trees of the circuit with three edges
Minimal cut sets of the circuit with three edges
A simple graph to demonstrate the first graph polynomial
A circuit with two edges; a banana
Two banana graph
Two bananas, one edge split
The graphs of Figures 4.9 and 4.10 glued along e, then with e removed to create a bubble on edge e^ of the two banana graph. . . .
Two banana graph, revisited
A wheel with three spokes
A complete bipartite graph, K33
xn
91
100
102
102
103
103
105
141
141
143
144
148
149
150
CHAPTER 1
D E G E N E R A T E BILINEAR FORMS
This is a chapter on the linear algebra of bilinear forms.
The theory of bilinear
forms is well-known, especially the classification of nondegenerate bilinear forms.
Chapter 4 of Serre's text [Ser73] provides an elegant, concise treatment. The reader
looking for a more complete treatment should consult [Bou07]. An algebro-geometric
perspective can be found in [Har92,Del73]. This chapter will gently introduce the
results necessary for what follows. The reader familiar with degenerate bilinear forms
may prefer to skip this chapter.
This chapter introduces the key definitions and properties of bilinear forms that I
will use. In the following chapters, I study specific families of bilinear forms and the
locus of forms in families that are degenerate. As preparation, I begin in Section 1.1
with the basics of bilinear forms and their radicals.
In Section 1.2, I prove Lemma 1.2.4 and Corollary 1.2.24, which relate the rank
of a degenerate bilinear form to its rank when restricted to subspaces. In the case of
a symmetric bilinear form B on a vector space V, Corollary 1.2.24 says that if B is
degenerate on V, then ranky B — r a n k # B for some hyperplane in a complete set of
hyperplanes in V. This equality is generically true for hyperplanes: the radical of B
will meet a generic H in the radical of B\JJ. However, the equality may not hold for a
specific hyperplane. The point of the corollary is that if one has enough hyperplanes
(a complete set), then the equality must hold for one of them. These corollaries
are the key to relating the singular and rank loci of configuration hypersurfaces in
Theorem 3.3.6. This result may be familiar to the reader, in which case Section 1.2
may be skipped.
1
1.1
Basics of Bilinear Forms
Let V be a finite dimensional vector space over a field K; denote its dimension by n.
Denote the dual space by V v = Hom^-(V, K). A bilinear form B on V is an element
of (V ®V)y
= Vy
®Vy.
Suppose that (5 = {b\,...,
bn} is a basis of V. Then the matrix Mp = (B(bi, bj))
represents B in this basis in the sense that B(v,w)
= v^Mpwp for all v,w 6 V,
where the right side is multiplication of matrices written in the coordinates of the
basis (6. Here we consider vp and wp as column vectors in the basis d, and v'o
denotes the transpose of the column vector vp. In other words, there are unique
v\,...,
v
ih (^^e coordinates of v in p = {b\,...,
vn G K such that v — YA=I
bn}),
which I express as the column vector
I v\ \
v2
V0
\vnJ
and similarly for w. By bilinearity of B and the definition of matrix multiplication,
both B(v,w) and VgMrjwp are equal to
]T
ViB(bi,bj) Wn
i,j=l
If we represent B in another basis 7 = { c i , . . . , c n } , then the matrix M 7 =
(-B(CJ,
Cj)) is related to Mp as follows. There exist unique scalars S^ in K such that
bj = X^fc=i Sikck- This change-of-basis matrix 5* = (So*) transforms column vectors
in the new basis to column vectors in the old basis:
vp = Sv1.
3
Therefore, for all v, w 6 V,
B(v,w)
=
VgMpwg
{Sv1)tMiiSw1
=
=
and M 7 = StMpS.
vt1StMfjSw1
Therefore, det M 7 = (det S)2 det Mg, so the determinant is
not a well-defined invariant of the bilinear form.
x
0 x K
x
/(K )i
; this element is the discriminant
It is a well-defined element of
of £?.
The bilinear form B defines two linear maps from V to V^, rB and £B:
0 -> r a d f l ( F , B)^V
rB(v)
^ V
=
v
B(-,v)
and
0 - ^ r a d L ( ^ , £ ) -^ V ^
£B(v)
=
Vw
B(v,-).
Definition 1.1.1 (Left and Right Radicals). The right radical of B in V, rad/,-(V, B),
is
rad i ? (V, B) = Ker r B = {v e V | B(w,v)
= 0 for all w
eV},
and the /e/t radical of B in V, rad£ / (l / , B), is
rad L (V, 5 ) = K e r ^ j = {v e V | B(t;,u>) = 0 for all w e
V).
These subspaces of V may be different if B is not symmetric, but they have the
same dimension because row and column rank always agree. I will often work with
one bilinear form B and restrict it to various subspaces W of V.
In that case, I
may suppress the B from the notation and write radjj V for rad^(V, B), r a d # W
4
for rad^W,
B\yy),
and similarly for the left radicals. When B is symmetric, I will
suppress the left and right subscripts from the notation.
The corank of B on V, c o r a n k B , is
corank i? = d i m r a d ^ V , B) = dimrad^>(V~, B) = dim V — rank B.
When I restrict B to a subspace W, I will denote rankB\yy and corankB\yy by
rankpy B and coranki^/ B. In particular, r a n k y B and coranky B are synonyms for
rank-B and corank B that include V in the notation. The following lemma is wellknown, and I will not include the proof. I make frequent use of the equivalence of
these conditions, often without comment.
L e m m a 1.1.2 (Characterization of Degenerate/Nondegenerate Bilinear Forms).
Suppose that B is a bilinear form on V and W is a nonzero subspace of V.
following conditions
on B\\y
1. B is nondegenerate
2. r&dL(W,B)
=0,
3. rndR(W,B)
= 0,
are
equivalent:
on W,
4- corankjy B = 0,
5. rankj^/ B = dim W, and
6. detMg
Similarly,
^ 0 for every matrix MQ representing B\\y in a basis /3 ofW.
the following are equivalent:
1. B is degenerate,
2. mdL(W,B)
3.
T^O,
radR(W,B)^0,
4- corankjy B > 0,
The
5
5. ranlqy B < dim VK, and
6. det MR = 0 for every matrix Ma representing B\\y
in a basis 13 of W.
More specifically, the following are equivalent:
1. corankjy B > dim W — k,
2. rank^/ B < k, and
3. all kxk
minors of MQ vanish for a matrix Mfj representing B\w
in some basis
(3 ofW.
A bilinear form B is symmetric
if B(v, w) = B(w, v) for all v, w e V . If a bilinear
form B is symmetric, then it defines a quadratic form Q : V —•> K by the formula
Q(v) = B(v,v).
If the characteristic of K is not 2, then Q determines B by the
formula
B(v, w) = ± (Q(v + w) - Q(v) -
Q(w)).
Because B is symmetric, rjj = £j] : V —> l / v . I denote this map by
Q :V ^
Vv.
I denote rad(y, B) by r a d Q . In the context of quadratic forms, r a d Q is also called
the kernel of Q and the vertex of Q.
Let F(V) be the projective space of lines in the vector space V. A quadric in
F(V) is a variety defined by Q(v) = 0 for a quadratic form Q. It is conventional
to use the notation Q for both the quadric and the quadratic form. The quadric is
degenerate if rad Q ^ 0 and nondegenerate otherwise. Every line v in rad Q is in this
quadric. In particular, the quadratic form Q : V/ rad Q —> K is well-defined, and
the quadratic form Q factors as
V/vadQ
For every short exact sequence of vector spaces
0 -» W -»• V -> V/W -> 0,
there is a birational map
TTW : P(V) —> P(V/W),
the projection from the linear subspace W. It is defined on F(V) — F(W) because
every line in V that is not a line in W maps to a line in V/W.
The fibers of this
projection are isomorphic to the affine space W. Using projection from the linear
subspace radQ, a degenerate quadric Q is a cone over Q, i.e., the following diagram
commutes and the vertical maps have fibers isomorphic to raclQ:
Q - P(radQ) ^
n
iadQ\Q
F(V) - P(radQ)
^radQ
-P(V/radQ).
The quadrics are hypersurfaces, so their singularities are determined by the vanishing
of the differential dQ — 2Q. Recall that the kernels of nonzero elements of Vv are
hyperplanes in V, and the elements of V v may be used to denote the corresponding
hyperplane in V. The tangent space to Q at a smooth point v is
TVQ = Q(v)
7
because Q(v) is a hyperplane in V tangent to Q at v.
Q(v)(v)
=
Q(v)=0.
Therefore, the smooth locus of Q is Q — P ( r a d Q ) . In particular, the nondegenerate
quadrics are smooth.
1.2
Restriction of Bilinear Forms and Rank Lemmas
This section studies how the rank of a bilinear form changes when restricted to subspaces with emphasis on degenerate bilinear forms. The main result is Lemma 1.2.4.
It is the key lemma in Theorem 3.3.6, which identifies the singularities of configuration hypersurfaces with a rank locus. The proof of Lemma 1.2.4 requires only the
linear algebra of bilinear forms, so all the prerequisites have already been covered
in Section 1.1. The specifics are quite distinct from the topics of Section 3.3 where
Theorem 3.3.6 is explained, which is why the proof of the lemma is provided in this
section.
The linear algebra can be somewhat dry, so I try to motivate it by preceding
the proofs with some examples and the statement of the main lemma. A few definitions are required before the statement of the lemma. The first is a mild symmetry
condition.
Definition 1.2.1 (Reflexive Bilinear Form). A bilinear form B on a vector space V
is reflexive if B(v\, i^) = 0 implies B(v2, t>i) = 0 for v\, t>2 £ V.
This condition is sufficient, to imply rad^> W = rad^ W for every subspace W,
which is a key feature that is needed in Lemma 1.2.4.
Definition 1.2.2 (Complete Set of Hyperplanes). A set of hyperplanes {Hi,..
in V is complete if Pq=lHi
V in this definition.
., H^}
= 0; note that k may be greater than the dimension of
Every hyperplane is the kernel of a linear form on V, and scaling the linear form
does not change the kernel. Therefore, the hyperplanes in V are identified with the
points of P(V V ). I mention this well-known fact so that the following proposition
makes sense.
Proposition 1.2.3 (Complete Equivalent to Spanning). A set of hyperplanes
{Hi,.
..,Hk}
spans V v if and only if it is complete.
This fact is well-known, and spanning is probably the preferred adjective. I prefer
the term complete to emphasize that I will use the intersection property.
Proof Let {Hi,...,
H^} be a set of hyperplanes in V. Consider the exact sequence
of vector spaces
k
k
O^^Hi^V^V/^Hi^O
i=l
i=l
and its dual
k
\
V
k
v
->y ->(H^:v
o-+\v/f)H{\
Note that the span of the Hi as linear forms is exactly ( V/ Hi=l Hi)
• Therefore,
spanning is equivalent to completeness.
•
Lemma 1.2.4 (Criterion to Bound the Rank). Suppose that B is a reflexive, degenerate bilinear form on V and k is a positive integer less than dim V. Let {Hi,...,
be a complete set of hyperplanes in V. For a subset J C. {!,...,£},
Hj =
njGjH3.
If B\JJ is degenerate for all subsets J with \J\ < n — k, then
ranky B < k.
define
H^}
9
I work in the generality of bilinear forms that are not necessarily symmetric, but
the bilinear forms to which I will apply these results will be symmetric. In particular,
the distinction between left and right that I make in this section can be safely ignored
for the applications I have in mind. I have included the generality of nonsymmetric
bilinear forms because it draws attention to some details of the linear algebra.
Let W be a subspace of the vector space V, and let B be a bilinear form on V.
The bilinear form restricts to W, and this section seeks to clarify how the rank of B
on V relates to the rank of B on W. In particular, the rank of each is determined by
the dimensions of the radicals rad^> V = rad#(V, B) and r a d ^ W = rad^>(W,
B\\y),
which fit into the following commutative diagram:
0
w
rad^H^
V{Blw)
. wy
n
(1.1)
0
rad/j> V
There is an analogous diagram for the left radicals. The rows are left exact by the
definition of the radical. There is no natural map between the radicals of V and W.
Note the difference between the notation of the horizontal map
W
r{Blw)
:
WW
which is the linear map induced by the bilinear form B\w
on W, and the diagonal
map
W
{rB)lW
:
VV,
which is the linear map TQ restricted to the subspace W.
Before I proceed to the necessary lemmas, I provide a sketch of the argument in
the symmetric case, which may be sufficient for some readers. Then I will give a few
examples that demonstrate the possibilities.
The fundamental idea of this section and the key to proving Lemma 1.2.4 is
10
Corollary 1.2.24. It says that if B is a symmetric degenerate bilinear form on V,
then its rank is generic in the sense that in every complete set of hyperplanes, there
is a hyperplane H for which ranky B = rank// B. In fact, the symmetric condition
can be weakened to rad/, V = rad/j V, but it cannot be entirely removed as shown
by Example 1.2.8. In general, the rank on the hyperplane has well-known bounds
r a n k y B > rank// B > ranky B — 2,
which I derive in Corollaries 1.2.13 and 1.2.15. These ranks are determined by r a d i i
and rad V, which have no natural map between them. The possibilities are:
1. rank// B — ranky B — 2 when dim rad H = dim rad V + 1, which occurs when
r a d V $1 rad if;
2. rank// B = ranky B — 1 when dim rad H — dim rad V, which occurs when
rad H = rad V
or both
H n rad V £ rad H
and
H n rad V £ r a d V ;
3. rank// B = ranky B when dim rad if = dim rad V — 1, which occurs when
rad H £ rad V.
If B is nondegenerate on V, then rad V = 0, which simplifies the possibilities to
1. rank// B = ranky B — 2 when dim rad if = 1;
2. rank// 5 = ranky B — I when rad H = 0.
If B is degenerate on V, then the nonzero rad V will intersect a generic hyperplane
properly. By generic, I mean that the the set of such hyperplanes is dense and open
11
in the space of all hyperplanes. In this case, the set
{H\ radV C H}
is a proper closed linear subvariety of P(V) V because rad V is nonzero. Therefore, a
generic hyperplane H does not contain radV^, so
H + rad V = V.
Then rad if must be orthogonal to all of V and not just to H, which implies
rad H C rad V.
If the above containment rad if C rad V is not strict, then radV is contained in H.
The hyperplane H generically does not contain radV, so the generic situtation is a
strict containment of radicals, which corresponds to the last case:
3. rank^ B — ranky B when rad H £1 rad V.
The point of Corollary 1.2.24 is that if one has enough hyperplanes (i.e., a complete set), then the generic situation where rank^ B = ranky B must hold for one
of them. If it held for none of them, then rad V would be contained in all of the
hyperplanes and hence contained in their intersection, which is zero. The remainder
of this section goes through this argument slowly and methodically and generalizes
to higher codimensions.
For a set of vectors {v\,...,
v^}, I will denote the span by (vi,...,
v^). In the
following examples, I restrict bilinear forms to the coordinate hyperplanes. There is
nothing special about the coordinate hyperplanes beyond forming a complete set of
hyperplanes. I use the coordinate hyperplanes for convenience of calculation.
Example 1.2.5 (Generic Symmetric Matrix of Rank k). Let V = Kn with the standard basis {ei,. .., en}. Let {X\,...,
Xn] be the dual basis of Vv.
Consider the
12
symmetric bilinear form
* = £*?
i=l
of rank k. As a matrix in the standard basis, B has Is on the diagonal in the first k
diagonal entries and Os everywhere else:
A o o ...
0 1 0
o\
...
0 0 1 ...
1
0
o)
Vo
The radical rad(V, B) is (c^+i,.. ., en).
Let Hi be the hyperplane spanned by all e?- with j ^ i so that Xj|//. = 0; more
simply, Hi — KerXj. Then the matrices for B\jj. in the standard bases for H{ are
the matrices obtained by deleting the ith row and column from the above matrix.
In particular, for i < k,
rank//. B = ranky B — 1
because the dimension of the space has decreased by 1 but the radical has not
changed. For i > k,
rank//. B = ranky B
because both the dimension of the space and of the radical have decreased by 1.
In the complete set of hyperplanes {Hi,...
,Hn},
there are n — k (so in par-
ticular at least one) with rank//. B = ranky B. Similarly, in the set of coordinate
subspaces of dimension fc, there is only one with rank k. This fact is generalized
to arbitrary symmetric bilinear forms and arbitrary complete sets of hyperplanes in
Corollary 1.2.24. In particular, the hyperplanes need not be the coordinate hyper-
13
planes in the basis for which B has the above form.
Example
1.2.6 (Two-dimensonal Case). Consider V = K2 with the standard basis
E = {ei,e2}. Let { X i , X 2 } be the dual basis of V v . Consider a generic bilinear
form
B = aX\ (g> Xi + bXi ® X2 + cX2 <8> X1 + d X 2 ® X 2 ,
with a, b,c,dE
K. In the basis E1, £? is represented by the matrix
\c
d) '
Let H\ = K e r X i = (e 2 ), and H2 = (ei). Then
5 ^
= d X 2 <8> X 2
(It would be more precise to write X2\JJ,
B\H2
an
= a X i (8) X i .
d ^IIF2
nere
' but I opt for a less
cumbersome notation.)
Suppose that B\JJ. is rank 0 for both hyperplanes Hj. Then a = o? = 0. The rank
of £? on V is then
1. r a n k y J B = 0 if b = c = 0,
2. ranky 5 = 1 if exactly one of 6 or c is 0, and
3. ranky B = 2 if both 6 and c are nonzero.
If B is symmetric, then b and c are equal and both zero or nonzero. In particular, if B
is symmetric and B\jj. is degenerate, then B cannot be rank 1 on V. Corollary 1.2.24
explains this fact in general.
The precise statement is that if ranky B > r a n k # B for all hyperplanes if in a
complete set and B is symmetric, then B is nondegenerate. The symmetry condition
can be relaxed to rad^ V — rad^> V. I also note two logically equivalent statements:
1. If B is symmetric and degenerate on V, then in every complete set of hyperplanes, there is at least one hyperplane H with ranky B = rank// B.
14
2. If r a n k y B > rank// B for all hyperplanes in a complete set and if B is degenerate, then B is not symmetric (in fact, rad^> V ^ rad^ V).
In particular, determining the rank of a degenerate symmetric square matrix can be
done by taking the maximum rank of one of the minors obtained by deleting the ith
row and column; there is no need to consider minors with different rows and columns
deleted. This must be well-known, but I do not know a reference.
However, to understand the way that I use this result, the reader should not think
of a degenerate symmetric matrix restricted to the coordinate hyperplanes. Later I
study degenerate symmetric matrices restricted to complete sets of hyperplanes, but
these hyperplanes will not be coordinate hyperplanes in general. In fact, there will
usually be more hyperplanes in the complete set than the dimension of the vector
space, so it will be impossible to choose them to be the coordinate hyperplanes.
Example 1.2.7 (Three-dimensional Case, Unequal Radicals). Consider V = K
with
be the dual basis of V v .
the standard basis E = {ei,e2,e3}. Let {X\,X2,X%}
Consider the bilinear form B = X\ <g> X2 + 2X2 <S> X2- That is,
B(ei,e2)
= l
B(e2,e2)=2
B(ei, ej) = 0 for all other
i,j.
As a matrix in the standard basis, B is represented by
/o 1 o\
0 2
y0 0
0 .
0/
This bilinear form has rad^ V = (2e\ — &2-,ei)
and corank 2.
an
d radjj V = (ei,e^).
It is rank 1
15
Consider the three coordinate hyperplanes
H\ = (e 2 ,e 3 )
Hi = (ei,e 3 )
# 3 = (ei,e 2 ).
In particular,
X\\H
X{\JJ.
is 0 for each i. The bilinear form B\u
is zero. Again, I should write
Du
•>
AT2|JJ
is 2A"2 ® AT2 because
t I prefer to keep the notation to a
minimum. As a matrix in the standard basis,
B
'
»
.
-
(
:
:
)
•
The left and right radicals on H\ agree:
r a d L # i = r a d j R / / 1 = (e 3 ).
Moreover, the radicals of H\ are subspaces of the radicals of V. The rank and corank
of B on H\ are 1. In particular, the rank on H\ agrees with the rank on V, and the
corank has decreased by 1.
The bilinear form B\JJ is zero because A"2|#
1S z e r o
- Now the rank has decreased
by 1 and the corank has remained the same. The radicals are both equal to 7/ 2 , which
is the same as the right radical of V. The left radical of V is neither a superspace
nor a subspace of if2.
The case of B\fj
= X\ <g> X 2 + 2AT2 <g> X 2 is similar to the case B\H except that
radj^ H3 y£ rad^ H3. In particular, the matrix of
B\JJ„
in the standard basis is
Therefore, radjj H3 — (e\) and rad^ H% = (2ei — e 2 ).
The main fact the reader should take away from this example is that the rank of
16
the degenerate form B matched the rank of B restricted to at least one hyperplane
in a complete set of hyperplanes. Corollary 1.2.24 establishes that this is always true
if radjj> V = rad^ V. This example shows that the condition r a d ^ V = rad^ V is not
necessary for that conclusion. The next example shows that the conclusion does not
hold in general.
A secondary point that will be used along the way is that rad^ H-t C rad^ V or
rad_R V C r a d ^ Hi for each hyperplane (and similarly with lefts and rights switched).
In this example with rad^ V = (2e\ — e2, e 3 ) and rad^> V = (e^, 63), one containment
in each row must hold:
r a d L Hi = (e 3 ) C (2ei - e 2 , e 3 )
radLH2
radLH3
(ei, e 3 ) £ rad f i Hi = (e 3 )
= ( e i , e 3 ) £ (2ei - e 2 , e 3 )
(ei,e 3 ) C mdRH2
= (ei,e 3 )
= (2ei - e 2 ) C (2ei - e 2 , e 3 )
(ei,e 3 ) £ radRH3
= (ei).
I demonstrate that at least one of the two containments holds for each hyperplane in
each of the following examples. Corollary 1.2.20 establishes that this holds in general
and is a key element of the main lemma.
Example
1.2.8 (Necessity of Equal Radicals). Again let V = K
set of coordinate hyperplanes {Hi, H2, H3}.
with the complete
Consider the bilinear form
B = Xi ® A"2 + X2 <8> A"2 + AT3 ® X\
on V. As matrices in the standard bases, B and its restrictions are
/o 1 o\
£ =
0 1 0
(1 0
5
#2
Si # 1
V
-f°
°)
~\1 0
SI # 3
In this example, B has rank 2, rad^ V = (ei — e2), and r a d ^ V = (e 3 ). The bilinear
17
forms B\H
and B\JJ„ are similar to the first example, although their radicals are
not all contained in the radicals of V in this example. Namely, they are rank 1 and
r a d L Hi n r a d L V = (e 3 ) D (ei - e 2 ) = 0
rad^> H\ = (e 3 ) = rad^ V
r a d L H 3 = (ei - e 2 ) = r a d L V
vadR H3 n r a d ^ V = (ei) n (e 3 ) = 0
For the other hyperplane, /f2> the bilinear form is rank 1 and
radx H2 = (ei) ^ r a d L V = (ei - e 2 )
vadR H2 = (e 3 ) = r a d ^ K
The key point of the example is that B is a degenerate bilinear form of rank
2 for which there is a set of complete hyperplanes on which the rank decreases.
Corollary 1.2.24 shows that this cannot happen if rad^ V = rad^> V.
Note the following containments of radicals, where (ei — e 2 ) = rad^ V and (e 3 ) =
rad^> V:
r a d L Hi = (e 3 ) £ (ei - e 2 )
(e 3 ) C r a d ^ Hi = (e 3 )
r a d L # 2 = <ei) £ (ei - e 2 )
(e 3 ) C r a d ^ H2 = (e 3 )
r a d L # 3 = (ei - e 2 ) C (ei - e 2 )
(e 3 ) £ radRHs
= (ei).
I give two more examples. The first gives a degenerate, nonsymmetric bilinear
form of rank 2 on A"3 for which rad^ V = rad^> V, and the conclusion of Corollary 1.2.24 is demonstrated. Then Example 1.2.10 demonstrates the contrapositive:
a bilinear form with rad^ V = ra.dR V whose rank decreases when restricted to each
hyperplane in a complete set. Corollary 1.2.23 implies that such a form must be
nondegenerate, which is clear in the example.
Example 1.2.9 (Demonstration of Corollary 1.2.24). Consider the bilinear form
B = X2 ® Xi - Xi <g> X2 + Xi <g> X3 - X3 <g> X1
on V = K 3 . As matrices in the standard bases, the bilinear form and its restrictions
are
B =
1
[-1
B\ H
2
0
0
0 0'
B
\HX =
0 Oj
0
-1
'o - 1
1
0
,0 0,
B\H3 =
1
0
On V, the bilinear form has rank two with radicals r a d ^ V = rad^ V = (e2 + 63).
When restricted to H2 or H3, the form still has rank two, and the radicals are zero.
When restricted to Hi, the bilinear form is 0.
Note the following containments of radicals, where (e2 + 63) = rad^ V = rad# V:
r a d L Hi = (e 2 , e 3 ) £ (e 2 + e 3 )
(e 2 + e 3 ) C rad f l Hi = (e 2 , e 3 )
rad^ F 2 = 0 C (e 2 + e 3 )
(e 2 + e 3 ) ^ r a d ^ # 2 = 0
r a d L H3 = 0 C (e 2 + e 3 )
(e 2 + e 3 ) £ r a d ^ # 3 = 0.
Example 1.2.10 (Demonstration of Corollary 1.2.23). Consider the bilinear form
B = Xi <g> X3 + X3 <g> Xi + X2 ® X 2
on V = /C . As matrices in the standard bases, the bilinear form and its restrictions
19
are
/o o i\
B =
O 1 0
I#1
0 0/
\l
B\ H
2
5
-~\1( ° 0A
5
#3
The bilinear form Z? is symmetric, so there is no need to consider the left and
right radicals separately; denote either radical on a subspace W by r a d W .
the hyperplane H2, B is nondegenerate, so r a n k ^ 2 B = 2 and r a d i / 2
=
On
0. On the
hyperplanes H\ and H3, B is degenerate of rank 1, r a d i f i = (63), and rad-H^ = {e\).
On V, the bilinear form is nondegenerate, so ranky 5 = 3. In this case, r a d V = 0
is contained in all r a d f / j .
In this example, notice that rankjy. B < ranky B for each hyperplane Hj. In fact
under the condition rad^ V = rad/j> V, the inequalities rank/j. B < ranky B are a
necessary and sufficient condition for B to be nondegenerate on V. The inequality
is necessary for nondegeneracy because if
ranky B = rank#\ B < dim Hi = dim V — 1,
then r a n k y B < dim V. I prove sufficiency in Corollary 1.2.23. Example 1.2.8 shows
that the inequality without the condition rad^> V = rad^ V does not imply this
nondegeneracy.
I now proceed to prove the lemmas before the reader starts to wonder whether I
will resort to proof by example.
L e m m a 1.2.11. Let B be a bilinear form on a vector space V, and let W be a
subspace of V.
Then
W n r a d # V C rad f i W,
and similarly for left radicals.
20
Proof. Consider an element w £ W^flrad^ V. As an element of rad/j V, B(v, w) = 0
for all v e V and, in particular, for all v G W C V. Because w is in W, this proves
that w G rad/j W.
Alternatively, note that W D rad^> V is the kernel of the diagonal map,
(r^)\\y,
in the diagram 1.1. That this kernel is a subspace of r a d ^ W is a consequence of the
commutativity of the diagram.
•
In particular, the diagram (1.1) can be extended:
0
W
nradjiV
n
•1.2)
rad# W
r[Blw
\wv
^w
n
0
rad^> V
In fact, there are three exact sequences that fit into the following commutative
diagram (note the middle sequence is only exact at the left and middle):
radftW
W n radR V
W
W
r{Dlw)
(n?)l w {V/{Wnrad Vj)
L
n
rad/j> V
V
• (W/ rad L W)w
n
r-B
(V/radLV)"
v
21
I also record the following useful corollary:
Corollary 1.2.12. / / rad^> V C W, then radjj- V C rad^VF, and similarly for left
radicals.
Proof. The assumption implies W n rad^> V = r a d ^ V , so Lemma 1.2.11 proves
r a d # V C r a d ^ W.
D
Lemma 1.2.11 is a reformulation of the well-known fact that the rank of a bilinear
form cannot increase when restricted to a subspace. In other words:
Corollary 1.2.13 (Rank Upper Bound on a Subspace). Let B be a bilinear form
on a vector space V, and let W be a subspace of V.
with equality if and only if radjj V — rad^> W,
rad^ V = Tadi
Then r&nky B > rank^- B
or equivalently with left radicals:
W.
Corollary 1.2.13 is well-known, but I still show how it can be deduced from
Lemma 1.2.11. It follows from another familiar lemma from linear algebra:
L e m m a 1.2.14 (Intersecting with a Codimension k Subspace). Suppose U\ and U2
are subspaces of a vector space V.
Then
dim U\ n U2 > dim U\ + dim U2 — dim V.
In terms of codimension,
this inequality is
codim U\ n U2 < codim U\ + codim U^.
Mixing codimension
and dimension,
the inequality is
dim U\ H U2 > dim U\ — codim U2,
which can be summarized
when intersecting
as the dimension
of a subspace may drop by at most k
with a codimension k subspace.
22
Proof of Lemma 1.2.14- F ° r every pair of subspaces U\ and U2 in V, there is a
diagonal map
A:UlnU2^Ui®U2
u I—> (u, u)
and a difference map
D:Ui@U2^>Ui
(ui,u2)
+
i-> u\ -
U2QV
u2,
which fit into the exact sequence of vector spaces
o -»• U\ n u2 ^ Ui © u2 -^ ?/i + c/2 ->• 0Taking dimensions in this exact sequence gives a lower bound on the dimension of
the intersection:
dim Uif]U2
= dim Ui®U2-
dim([/i + c/2)
> dim U\ + dim U2 — dim V.
D
Proof of Corollary 1.2.13. Taking the dimension in the Lemma 1.2.11 shows that
corankiy B is bounded below by dimVY D r a d ^ V .
Apply Lemma 1.2.14 to the
subspaces W and rad/j> V to get
coranki^ B > dim W (1 r a d # V > dim W — codim radjj> V
= dim W — ranky B
Gathering the W terms in this inequality gives rank^/ B < ranky B. The inequalities
are equalities if and only if radjj> W = W l~l rad^; V = rad^> V, or similarly with left
23
radicals.
•
I also note the well-known lower bound on rankiy B:
Corollary 1.2.15 (Rank Lower Bound on a Subspace). Suppose W is a subspace of
V of codimension
k.
Then
r a n k ^ B > v&nky B — 2k,
with equality when rad/^ V C W and dim r a d ^ V = dim rad/^ W — k. In other words
for the case of equality, ifvadji V = rad^ V, then B defines a nondegenerate
bilinear
form B on V/ r a d ^ V that has corank k when restricted to Wf rad^> V.
If ranky B — 2k < 0, then zero is obviously a better lower bound, and equality
cannot be achieved in the high codimension
cases k > ^ r a n k y B.
Proof. The image of rad^> W under {TB)\W
must go to zero in the map Vv —> W/V
by the commutativity of diagram (1.1), so
rB(r&dRW)
C (V/W)v
C
Vy.
Therefore, the following left exact sequence gives bounds on the ranks:
0 -)• W n x&dR V -> r a d ^ W
(rg),H/
(V/W)y.
)
Namely, an upper bound on the corank of B on W is
c o r a n k ^ B = dim r a d ^ W = dim W fl r a d ^ V + dim rR (rad/j W)
dim(V/W)v
< dim W n r a d # V +
< dim r a d ^ V +
d\m{V/W)v
= coranky B + k
with equality when r a d ^ V C W and r # ( r a d # W) = (V/W)v.
the dimensions in the exact sequence show that rR(radR
If rad^> V C W, then
W) = (V/W)v
if and only
24
if
k = dim(^/HO v
= dim r # (radR W)
— dim r a d ^ W — dim W n rad/j> V
— dim radjj W — dim r a d ^ V;
that is, rad/j> V" has codimension k in r a d # W. Translating the corank bound to rank
gives
n — k — rankjy B < n — ranky B + k
rankjy B > ranky B — 2k.
D
Definition 1.2.16 (Orthogonal Subspaces). Let S be a subset of V.
Define the
subspace right orthogonal to S,
±]
R(S)
= Pi
Kerr
s ( s ) = lv
G
V\B(s,v)
= 0 for all s e 5 } .
There is a similar definition for the subspace left orthogonal
to S, ±l(S).
The dependence on the bilinear form B is implicit in the notation. I note the
following properties to familiarize the reader with the definition.
P r o p o s i t i o n 1.2.17 (Properties of Orthogonal Subspaces). The following
are satisfied for right orthogonal subspaces.
orthogonal
There are similar properties for left
subspace.
1. rad^> V C _L^(5) for all subsets
S.
2. For a single element s E S, -L)j(s) = Ker r_g(s).
3- -l£(S) = rW-Lg(*)-
properties
25
I mdRv = ±l(v) = nveV±l(v).
5. B is reflexive if and only if ±^(v) = ±^(v),for
all v <EV.
The proof is straightforward and will not be given. If S is a subset of a subspace
W C V, then the subspace of W right orthogonal to S is
±%(S) = nseS
Ker r ( B |
)(s) = {w G VF | £ ( s , iu) = 0 for all s e S}.
In this case, i_jji (£*) is a subspace of l-^(S) simply because W is a subspace of V.
If the subset S is rad^ W, then the definition of the left radical shows
W =
±%(radLW).
W C
±^(radLW).
Therefore,
At this point, the formulas have begun to involve both right and left spaces. I did
not assume symmetry of the bilinear forms so that the role of right and left in these
formulas would be clear.
Lemma 1.2.18 (Subspace Orthogonal to a Subspace Radical). Let B be a bilinear
form on V, and let W be a subspace ofV.
W + r8idR V C ±£(rad L W),
and similarly with rights and lefts switched.
Proof. As noted above, W C T^(rad L W) and rad^ V C 1^{S)
Therefore, the sum is also contained in ±n(rad^ W).
for every subset S.
•
Corollary 1.2.19 (Criterion for Radical Containment). If W + radjj> V = V, then
rad^ W C rad^ V, and similarly with rights and lefts switched.
26
Proof. By the assumption and Lemma 1.2.18,
V = W + r a d # V C ± ^ ( r a d L W) C V,
s o V = -LjMradj^ W). By definition of right orthogonal, every element w £ rad^ W
and every element v £ V satisfy B(w,v)
= 0, which is the definition for w to be an
element of rad^r V.
•
Corollary 1.2.20 (Hyperplane Radicals Contain or are Contained). If H is a hyperplane in V, then r a d # V C r a d ^ H or rad^ H C racl^ V-, and similarly with rights
and lefts
switched.
Proof. By the hyperplane assumption, if rad/j>V ^ H, then radj^ V + H = V.
Therefore, either (a) r a d # V C H or (b) if £ r a d ^ V + if = V.
In case (a),
Corollary 1.2.12 implies that r a d # V C r a d ^ f f . In case (b), Corollary 1.2.19 implies
that rad£ H C rad^ V.
•
L e m m a 1.2.21 (Corank Criterion for Nondegeneracy). Suppose that B is a bilinear
form on V with rad^ V = rad^> V. If
coranky B < coranky B
for every hyperplane in a complete set, then coranky B = 0. In other words, if the
corank of such a bilinear form does not increase on any hyperplane in a complete
set, then the bilinear form is
Proof. Let {Hi,...,
nondegenerate.
Hi} be a complete set of hyperplanes for which
coranky B < coranky. B
for a l H € {1, ...,£}.
In other words, the dimension of the radicals satisfy
dim rad^ V < dim rad^ Hj,
27
so it is impossible for rad^ H-i £ rad^ V. Let / be the subset of { 1 , . . . , £} for which
rad^ Hi = rad^ V.
If i £ I, then rad^ Hi ^ r a d ^ V , so r a d # V C r a d ^ Ht by
Corollary 1.2.20. Therefore, using the assumption rad^ V = rad/j> V,
™dLV
C I f | r a d L i f i ) f| ( f l r a d f i i / i ] C f| ^
= 0.
A logically identical proof can be given switching L and R.
•
The interaction of the left and right radical in the previous proof is the reason that
I chose to include nonsymmetric forms. The condition rad/j> V = rad^ V imposes a
mild form of symmetry on the form and is clearly satisfied by symmetric forms. Note,
however, that this condition need not hold on a subspace of V. It may seem like
Lemma 1.2.21 should be true without this condition because it is simply a statement
about (co)ranks, which do not depend on left and right, but Example 1.2.8 shows
that the lemma may fail without the condition.
For comparison, Example 1.2.7
shows that it may hold without the condition.
It will be useful to have the contrapositive statement.
Corollary 1.2.22 (Corank Bound for Degenerate Forms). Suppose that B is a degenerate bilinear form on V with rad^ V = r a d ^ V . Then in every complete set of
hyperplanes,
there is at least one hyperplane H for which
coranky B > 1 + corank JJ B.
The reader may find the statement of Lemma 1.2.21 clearer in terms of rank. I
stated it in terms of corank because the proof works with the radicals, so the proof
is more direct in those terms. For the rank version, first note that Corollary 1.2.15
applied to a hyperplane H C V gives three possibilities for the rank of B on H:
ranky B — 2 < r a n k ^ B < ranky B.
The assumption of Lemma 1.2.21 rules out r a n k # B = ranky B.
28
Corollary 1.2.23 (Rank Criterion for Nondegeneracy). Suppose rad^ V = r a d ^ V.
If
ranky B — 1 >
for all H in a complete set of hyperplanes,
rankfjB
then B is nondegenerate
on V. That is,
if the rank of B drops on every hyperplane in a complete set of hyperplanes, then B
is nondegenerate
on V.
Note that the converse is straightforward by comparing dimensions: if B is nondegenerate on V (hence rank n), then its rank must drop on every hyperplane H
because n — 1 is an upper bound for r a n k # B.
Proof. The assumed inequality can be rewritten
dim V — coranky B — 1 > dim V — 1 — corank^ B
coranky B < corank^ B
for all if in a complete set of hyperplanes, which is the same as the condition for
Lemma 1.2.21. Hence, I make the same conclusion.
•
Again, the contrapositive stated in terms of rank will be useful.
Corollary 1.2.24 (Hyperplane Rank Condition, B Degenerate). Suppose rad^ V =
radjiV.
If B is degenerate on V, then every complete set of hyperplanes
has an
element H for which r a n k y B = rank// B.
Proof. The precise contrapositive is that r a n k y B < rank// B, but the inequality
cannot be strict because the rank cannot increase when the form is restricted to a
subspace (Corollary 1.2.13).
•
Corollary 1.2.24 says that under the assumption rad^ V = r a d ^ V, there is always
a hyperplane in a complete set on which the rank of a degenerate form does not drop
when restricted to it. In matrix terms, it says that a degenerate square matrix M
with equal left and right radicals has the same rank as one cofactor matrix M ? ;J,
i.e., the matrix obtained from M by deleting the ith. row and column. Because the
29
dimension of a hyperplane is one less than the dimension of the whole space, it is
possible for the degenerate bilinear form to become nondegenerate when restricted to
this hyperplane. Then the rank may drop when restricting to a codimension 2 space.
However, it no longer suffices to suppose rad_j, V = rad^> V. I also need rad^ H =
r&dftH
for the hyperplane H in the conclusion of Corollary 1.2.24.
For higher
codimension, I will need to repeatedly make this hypothesis on each hyperplane in
a flag. I simplify and assume B is reflexive.
Corollary 1.2.25 (Existence of Nondegenerate Subspaces in a Complete Set).
Suppose B is a reflexive bilinear form
dim V > k > 0. Let {Hi,.
on V.
Suppose B has rank k on V and
. . , Hg} be a complete set of hyperplanes in V. Then there
is a set J C {1, . . ., £} such that
1. \J\=n
— k,
2. dim if j = k, and
3. B\JJ
is
nondegenerate.
Proof. Proceed by induction on coranky B (k < dim V, so the minimum corank is
1). If coranky B is 1, then the rank k is d i m V — 1. By Corollary 1.2.24, in every
complete set of hyperplanes, there is an element H for which
rank// B = k = dim H,
so B is nondegenerate on H.
Now suppose corank^ B = d is greater than 1. In particular, k = dim V — d <
dim V — 1. By Corollary 1.2.24, there is a hyperplane Hi C V from the complete set
for which rank//. B = k. Therefore,
corank//. B = (dim V — 1) — k
= (dimV= d-
1.
l)-(dimV-d)
30
Because d > 1, A; is strictly smaller than dim Hi = dim V — 1. Thus, we can apply
the inductive hypothesis to Hi and the complete set of hyperplanes in Hi
{HjHHilHi^Hj}
to conclude that there is a dimension k subspace
Hj = H (Hj n Hi)
of Hi satisfying the conclusion of the corollary. The set J is contained in { 1 , . . . , £}—i.
Let J = J U {«}, and note that
H
J= r\(HjnHi)=
jeJ
(p[HA nHt=
f]H3=Hj.
\jsJ
)
J^J
The A;-plane Hj is also a subspace of V, which finishes the inductive step.
•
The contrapositive of Corollary 1.2.25 is
Corollary 1.2.26 (Nonexistence of Degenerate Subspaces in a Complete Set). Suppose that B is a reflexive, degenerate bilinear form on V and k is a positive integer
less than dim V. Let {H\,...,
Hg} be a complete set of hyperplanes in V. If B\JJ
is degenerate for all subsets J with \J\ = n — m and dim Hj = m, then
ranky B ^ m.
In particular, if the hypothesis is true for all m > k, then ranky B < k.
The conclusion that ranky B < k cannot be made by assuming the hypothesis
for k = m alone as the following example shows.
Example 1.2.27. Let V — K 3 with the standard basis E = {e\, e2, 63}. Consider the
31
dual basis {X\, X2, X3} whose kernels define a complete set of hyperplanes
Hi = Ke2 ® Ke3,
H2 = Ke\ © Ke:i,
H3 = Ke\ © Ke2.
Consider the symmetric bilinear form B = X\ <g> X2 + X2 <g> X\.
In the standard
basis, a matrix representative is
/o 1 o\
B =
1 0 0
\0
0 Oy
The rank of B on V is two. In agreement with Corollary 1.2.26, taking J = {3},
B\u
also has rank two. For all J C {1,2,3} with | J \ = 2 (i.e., the Hj are the
coordinate axes), the bilinear forms B\}j
are identically zero. In particular, when
I J I = 2 and dim if j = 1, all the forms B\JJ are degenerate, but the rank of B on
V is greater than one.
From Corollary 1.2.26, the proof of Lemma 1.2.4 follows.
Proof of Lemma 1.2.4- The assumptions of Lemma 1.2.4 imply the assumptions of
Corollary 1.2.26 for all m > k, so the last comment of Corollary 1.2.26 proves the
lemma. As an aside, note that there are more assumptions in Lemma 1.2.4 than
are strictly necessary to get the assumptions of Corollary 1.2.26. Namely, B\JJ is
assumed degenerate for | J \ < n — k, but Corollary 1.2.26 only needs to hold for
those J for which
1. I J I = n — m < n — k, and
2. dim if j = m.
•
CHAPTER 2
FAMILIES OF BILINEAR FORMS
The sections of this chapter recall two ways to construct families of bilinear forms.
In Section 2.1, the degeneracy loci of a map of vector bundles are introduced. In
Section 2.2, I describe quadratic linear series and their duals. This material may be
well-known to experts, who will prefer to skip this chapter. The following chapters
study specific examples of these families.
In particular, configuration and graph
hypersurfaces are degeneracy loci, and in special cases, they are duals to quadratic
linear series. I focus on the simplest properties and examples, which suffice for the
constructions in the following chapters.
I warn and apologize to the reader that in the remaining chapters, I abuse sheaf
and vector bundle notation. My motivation for abusing notation is that multiple
vector spaces are involved at all times, and keeping them straight can be confusing.
I choose the notation to make the vector spaces prominent. For example, for vector
spaces V and W, I will use
¥(V) x W
to denote both the trivial vector bundle over F(V) with fiber W and its sheaf of
sections.
Similarly, I use Of>/y\( — \) to denote both the tautological subbundle
of the trivial bundle F(V) x V and its sheaf of sections. I try to indicate which
interpretation I mean by the context.
2.1
Degeneracy Loci
Let V, W, and U be vector spaces. Suppose
a : V -> Hom(W, U)
32
33
is a linear map. The points of V can be partitioned by the rank of their image in
Hom(VF, U). For v EV
rank of a(kv).
and a nonzero scalar k, the rank of a{y) is the same as the
Therefore, it is natural to partition F(V) by this rank, as well. In
fact, a defines a map of vector bundles:
Onv)(-l)
® (F(V) x W)
a
F(V) x U
F{V)
The fiber of 0 ,F(vy •1) ® (F(V) x W) over a point L G 1P(V) (i.e., a line L in V) is
L ® VF. Fiberwise, the map of vector bundles takes v ® w G L <8> H-7 to cr(^)(iu) G (7.
There is a duality
H o m 0 p ( l / ) ( 0 P ( y ) ( - l ) ® W, /7) - Rom0nv)(W,
0 P ( y ) ( l ) ® C/)
where I have used W and U to denote the corresponding trivial bundles. Thus, the
map a may also be considered with values in the linear forms on V. In particular,
picking bases of U and W, one may think of this family as a dim U x dim W matrix
with entries that are linear forms on V.
This simple situation with U = W and a symmetry condition suffices for the
applications in Chapters 3 and 4. However, I will mention the general definition:
Definition 2.1.1 (Degeneracy Locus). Let X be a projective variety and let £ and
T be vector bundles over X.
Suppose a : £ —> T is a map of vector bundles. The
degeneracy locus (of rank k associated to a) is
Dk(X,
It is standard to endow D^{X)
a) = {x G X | r a n k e d < fc}.
with the scheme structure given locally by the ideal
of (k + 1) x (k + 1) minors of a, though I will not have much to say about the scheme
structure.
34
If X or a is clear from context, I may omit it from the notation.
Example 2.1.2. Let V = Hom(W/, U). Consider X = F(V), i.e., a generic projective
space of linear transformations. Let £ = X x W and J7 = (X x U) ® O j ( l ) . There is
a tautological vector bundle map whose degeneracy loci D^{X)
are the (projective)
set of all linear transformations from W to U with rank at most k. In particular,
each bundle Dji(X)
x W has a tautological map of vector bundles of rank at most k
to{Dk(X)xU)®0Dk(x){l).
Definition 2.1.3 (Determinant). Let a : V —> Hom(W, W) be a linear map of vector
spaces. Let m be the dimension of W. Then D m _ i (X) is a hypersurface defined by
the determinant of a matrix of linear forms on V representing a , a global section of
Op/y\(m).
The determinant
of a, deter, is the ideal defining this hypersurface.
I will also use det a to denote one of the generators of det a. The dependence on
a choice of basis leaves some ambiguity in this notation.
The above constructions also work for more specific types of linear transformations where rank makes sense. For a linear transformation a : V —> Sym W^, there
is an analogous map of vector bundles and degeneracy loci in F(V).
This example
is central in Chapters 3 and 4.
Example 2.1.4. Let X = P(Sym 2 Wy),
£ = X x (W ® W), and T = Ox(l)-
There
is a tautological bilinear map of vector bundles
X
and the degeneracy loci of this map, called the generic symmetric
degeneracy loci, are
defined as in Definition 2.1.1. These degeneracy loci have been thoroughly studied.
For comparison with the results of Section 3.3, I note the following theorem without
proof. Example 22.31 in [Har92] gently introduces this result and proves part of it;
full proofs can be found in [Ful98] or [JLP82].
35
T h e o r e m 2.1.5 ( [JLP82]). Suppose the dimension
of W is m.
The
m(m+l)
of the generic symmetric
degeneracy loci in P(Sym W ) = P
codim Dm_k{a)
If Qo £ -Cfc(cr) ~~ Dk-l(a)>
=
2
codimensions
1
are
.
then QQ is a cone with vertex P(radQo) over a smooth
quadric QQ in F(V/ r&dQo) (see Section 1.1). Then Dk(o~) is smooth at QQ, and the
tangent space to Dk(a)
at QQ is
TQoDk(a)
= {Qe
More generally, if QQ £ Dk_i(a)
Dk(a)
| r a d Q 0 C Q}.
— Dk_£_i(a),
then the tangent cone to Dk(a)
Qo is
TCQoDk(a)
and the multiplicity
of Dk(a)
= { Q e Dk(a)
| r a n k Q | r a d Q o < £},
at QQ is
m~k—l
rm — k+£+a\
a=0
In the case k — m—\
V a
)
and £ = j — 1, Equation (2.1) says that if
= D(m-l)-(j-l)(a)
= Dm-j((T)
~
-
£>( m _i)_(j_i)_i(ff)
D
m-j-l(°~),
then the multiplicity of QQ in the generic symmetric determinantal variety X
at
36
Dm-i(cr)
is the corank of QQ on W:
Multo n 3C = Mult Q n £) m _i((T)
m-(m-l)-l
TT
^
n
= n M^
a=0
/m-{m-l)+j-l+a\
m—(m—1) —a /
/2a+l\
V a /
^
a=0 V a 7
=i
This particular case of Theorem 2.1.5 will provide a useful comparison in Section 3.3,
so I point out that the generality of Theorem 2.1.5 is not needed to see this simple
result. Namely, if QQ has corank j , then there are coordinates in which
where l m _ j j m _ j is the (TO — j) x (TO — j ) identity matrix, and 0X)y is the x x y zero
matrix. In paticular, I may assume A\\
A\\
/
= 1. In these local coordinates, X is defined by the determinant of
1
A12
\
^ 0 and pick local coordinates such that
A12
...
1 + A22
...
Aim
A2m
^l,m—j
± T -^•m—j,m—j
^m—j,m—j+1
•••
^m—j,rn
^l.m—j+1
^-m,—j,va—j+l
^m—j+l,m—j+l
•••
-^m—j+l,m
^-m—j,m
^m—j+l,m
•••
^lm
\
Amm
J
Therefore, the leading term of the polynomial defining X at QQ is the determinant
37
of the generic j x j symmetric matrix
•™m—j+l,m—j+l
•••
^m—j+l,m
•••
\
•™-m—j+l,m \
Amm
J
which has degree j .
In Chapter 3, I will describe a projective linear subspace L of P(Sym W /V )
spanned by rank one quadratic forms. Corollary 3.3.10 will show that if QQ is in
LC\X with corank j , then
M u l t g 0 LHX
=j.
Because the multiplicity of QQ is the same in both X and Lf)X,
the tangent cone in
L n X is the linear section of the tangent cone in X:
TCQoLr\X
=
LnTCQQX.
In fact, L may be any subspace spanned by rank one quadratic forms as long as the
radicals of the quadratic forms intersect at zero. More generally, if the radicals have
a c/-dimensional kernel, then the multiplicity increases by d when intersecting with
L.
At this point, I note a few results on nondegeneracy and connectedness of degeneracy loci, which are well-known to experts but are not often cited in the literature
on graph polynomials.
T h e o r e m 2.1.6 (Fulton-Lazarsfeld [FL81]). Let £ v eg) T be an ample vector bundle
on an irreducible complex projective variety X of dimension n. Suppose £ is rank e
and T is rank f. For each section a of £ v <S> T',
1. if n > (e — k)(f — k), then D^(a)
is nonempy,
2. if n > (e — k)(f — k), then D^{o)
is connected.
and
38
The theorem applies to the examples mentioned above because the trivial bundles
tensored with 0^iy\(\)
axe ample. The theorem may be refined under symmetry
assumptions:
T h e o r e m 2.1.7. Suppose A = (Sym £v)
complex projective
variety X of dimension
(g> C is an ample vector bundle on a
n where £ is a rank e vector bundle on
X and £ is a line bundle on X. Let a be a section of A,
•
[IL99, Gra05] If n> e — k — 1, then D^{a) is
•
[Tu89, HT90] If n > ( e ~2 + 1 )>
then
D
k(°)
nonempty.
is connected.
Remark (1.8) in [FL81] notes that with further assumptions on Dfc(a), connectedness implies irreducibility using a theorem of Hartshorne [Har62]. In at least one
simple yet important case that I will introduce in Example 3.1.5, a connected D}t(a)
is not irreducible. This example is not nonsingular in codimension 1, a condition
assumed in the cited remark. It would be interesting to find when the irreducibility conclusion can be drawn in the case of configuration or graph hypersurfaces.
Lemma 3.1.10 provides a condition that implies reducibility for configuration hypersurfaces.
2.2
Quadratic Linear Series
The point of this section is to recall the basic construction of a quadratic linear
series and the corresponding map to projective space. In the case that the map to
projective space is an embedding, I describe the dual variety and a resolution of its
singularities. Most of the material could be developed easily without the quadratic
assumption, but I focus on the quadratic case because that is all that I use in the
following chapters. The idea to apply these constructions in Chapters 3 and 4 comes
from [BEK06, Section 4], which is a suitable reference for the material of this section.
A more detailed explanation of linear series can be found in most general algebraic
geometry references, for example [Har77, Chapter II.7] or [Laz04, Chapter 1.1.B].
39
Recall that the quadratic forms on a vector space W are naturally identified with
the space of global sections
r(P(wo,Op ( M O (2)).
I abbreviate the notation for this space by T(0(2)) throughout this section.
Definition 2.2.1 (Quadratic Linear Series). Let qo, . . . ,qn be quadratic forms on
W. A direct sum
L = ®?=0Kqi
is a quadratic linear series.
Every quadratic linear series comes with a linear map
f:L-+T(0(2)),
and the image of / is the span (qQ,..., qn) in the space of quadratic forms. If the
qi are independent, then the map / is injective and the quadratic linear series is a
subspace of Y{0{2)). Similarly, the dual map
/ v : T(C(2)) V - • L v
will be surjective if the qi are independent. Letting U be the kernel of this map,
there is the birational map, projection from F(U),
7Tfs/ : P ( r ( 0 ( 2 ) ) v ) - F(U) -+ F (L) v
whose image is the projective space of lines in the image of / v . Recall that for a
vector space V, every line in V
(i.e, point in P(V V )) is an equivalence class of linear
functionals differing by nonzero constant factors. These functionals all have the same
1. Most sources define a linear series to be a subspace of a space of global sections of a
line bundle. Note t h a t the definition here may differ if the sections are linearly dependent.
40
kernel, a hyperplane in V, and conversely, every hyperplane defines a nonzero linear
functional up to constant factor. These identifications allow the identification of the
lines in \ / v with the hyperplanes in V, and it is conventional to use the notation
P(Vy)
or P ( V ) V for either space.
Each point x of P ( W ) defines the subspace of sections vanishing at x in L or
r ( C ( 2 ) ) . If there are sections that do not vanish at x, then the subspace of such
sections is a hyperplane. Let B be the set of x for which qi{x) = 0 for all i, that is,
n
B = p | rad%.
i=0
In linear series terminology, F(B) is the base locus of the linear series. There is a
rational map
QL:F(W)-*F(LV),
regular on P ( W ) — F(B), which sends x to the hyperplane of sections vanishing at x.
It is well-known and easy to show that there is a quadratic form on W not vanishing
at x, so there is a regular map (in fact, an embedding)
¥(W)
^P(r(Q(2))v).
Then Qi factors as
P(W0-P(B)
p(r(c(2)) v )-p(f/)
Note that when F(B) is empty (i.e., B = 0), 7TJ-V is regular on i(F(W))
Ql is regular on P(H/"). In this case, the restriction of 7r^v to L(F(W))
because
is affine and
proper, hence finite.
These maps can be defined in coordinates, and the quadratic linear series has
41
a natural basis of coordinates for this definition. The qi define a basis of L, and I
denote the elements of the dual basis by Xqi. An equivalence class in P(X/V) may be
represented in homogeneous coordinates as [OQ : • • • : an], which corresponds to the
scalar multiples of the linear functional
n
In particular, the independence of the Xqi in L v implies that these linear functionals
are nonzero as long as some aj is nonzero. In these coordinates, Qi is the map
QL : P(W) - P(B) -> P(L) V
[w] (->• [q0(w) : ••• : qn(w)].
In a basis for W, each quadratic form % corresponds to a symmetric k x k matrix
Mi such that
qi(w) = w
MjW.
The following proposition is proved in [BEK06, Section 4]: it is the main result of
this section and is motivation for Section 4.4. I include its proof to prevent confusion
with my notation, which differs from that of the reference [BEK06].
Proposition 2.2.2 (Dual of Quadratic Linear Series). Assume Qi is an embedding
ofF(W)
in P(L V ), and let X denote the image. The dual variety X^ is defined
by det(^" = 0 v4jMj) = 0, where A{ are the coordinates on P(L). That is, it is the
2. The linear functional
E iX Aw
a
q
1=1
may be zero for nonzero an if the base locus is nonempty, i.e., when the qt are not linearly
independent in T(0(2)).
42
degeneracy locus Dn(o~) for the quadratic map of bundles a^ =
\L)
x
°L
(W®W)
J2j/^icli:
-onL)(i)
P(L)
Before proceeding to the proof, I explain some basic facts about dual varieties
and some consequences in the case that the map Qi is an embedding. For a more
thorough explanation of dual varieties and generalizations, see [GKZ94,Har92,Zak93,
Kat73j.
For a variety X
C F(V),
the dual variety Xv
C P(V /V ) is defined as
the closure of the set of hyperplanes that are tangent to X at a smooth point.
More precisely, let Xsm
denote the smooth points of X.
Over Xsm.
define YQ in
F(V) x P ( V V ) as
{(x,H)\xeX8m,HDTxX}.
The incidence variety Yx is the closure,!^), and
X=W2(Yx)=Pt2(Yo)i
where pr2 is the second projection map in the following diagram:
YQQYXQ
F(V) x P(\/ v )
-£/•.
^v
<?
^
Xsm^XC
F(V)
Pr2(r0)cx
v
cp(^
Note that YQ is a projective bundle over Xsm
whose fiber over a point x is the
projective space of hyperplanes containing TXX.
That is, for x <E Xsm,
p^1(x)nY0
=
xxT(V/TxX)v,
43
which has dimension c o d i m X — 1.
The fundamental theorem of dual varieties is
T h e o r e m 2.2.3 (Biduality Theorem). For every closed, 'irreducible subvariety X in
P(V), the incidence varieties are the same:
YX=Yxv,
and therefore
( X v ) v = X.
Moreover, if x is a smooth point of X and H is a smooth point of X^,
tangent to X at x if and only if x is tangent to X^
then H is
at H.
For the proof of the theorem, see [GKZ94] or [Har92]. If X is a hypersurface, then
the map pr^ |y
from Yx to X is birational because the generic fiber is a projective
space of dimension 0, i.e., a point. By the identification Yx = ^ x v ' P r 2 \YY *S a ^ s o
birational if Xv
is a hypersurface. In the case that X is smooth,
^0 =
YX,
and Yx is a projective bundle over X, hence smooth. If X is smooth and X v is a
hypersurface, then Yx = ^ v
is smooth and birational to X^,
i.e., a resolution of
singularies. That is the key point:
T h e o r e m 2.2.4 (Resolution of Dual Hypersurface). If a hypersurface is the dual Xv
of a smooth variety X, then the incidence variety Yx is a resolution of
singularities
v
forX .
Assume Proposition 2.2.2 for the moment. Then Theorem 2.2.4 implies
Corollary 2.2.5 (Proposition 4.2 of [BEK06]). Suppose that Qi is an embedding.
Let X denote the image of QLlarities for X
v
= V(det(£
The incidence variety Yx is a resolution of singu-
A.M,)).
44
In case the forms qi are all rank 1 and the base locus is empty, the polynomial ^(A) = det(^yljMj) is known as a configuration polynomial, which will be
explained in Section 3.1. If Qi is an embedding, the theory of dual varieties constructs a resolution of singularities for the configuration hypersurfaces defined by
the configuration polynomial. In particular, this resolution of singularities given by
the incidence variety might be a useful tool for studying graph hypersurfaces. In
Section 4.4, I show, however, that Qi is not an embedding for most graphs.
I now proceed to the proof of Proposition 2.2.2. The proof follows from an
alternate description of Yx in the case when X is smooth. For a bundle £ on X,
let ¥(£) denote the projective bundle of hyperplane sections of 8. Let J\fx/F(V) ^ e
the normal bundle of X in P(V), and thus P(A/V/p/m) is its projective bundle of
hyperplane sections.
Lemma 2.2.6 (Proposition 4.1 [BEK06]). For a smooth subvariety X of¥(V),
the
incidence variety Yx is
Yx =
The fiber ofYx
¥{Mx/nv))-
over a point H in P(l / V ), thought of as a hyperplane in V, is
nMX/F(V))\H
=
Smg(Xr\H).
A general treatment without the quadratic case in mind can be found in [Kat73,
Section 3].
3. Some authors prefer to include the hyperplane sections in the notation. For example,
the notation in [Zak93] is
PCA/"x/p(v)(-l)),
and the notation in [Kat73] is
P(TVV).
T h e notation I present is also common. The key is that the fibers of the bundle are
hyperplanes in the fiber of the normal bundle or, equivalent ly, linear functionals up to
scalar on the fibers of the normal bundle.
Proof. For a smooth X, there is the normal bundle sequence over
0
-• Tx -* ?p(y)lx -> NX/v{V) ~^ °-
The dual sequence provides the inclusion
Recall the Euler sequence for P(V):
o -> onv) -+ (P(v) x v) ® onv)(i)
-+ rnv) ->
The first map defines a global section
s
e v ® v v = r(p(\o, (P(V) x v) ® o P(K) (i))
whose vanishing defines the projective tangent bundle:
V(s) = P ( T P ( y ) ) c p ( y ) x P ( y v ) .
In a basis t>j of V and its dual basis X^ of F v , s is the form
The projection
pr 2 : F(V) x P(V V ) -»• P(\/ v )
has three restrictions:
n^x/nv)) ^ nTnv)\x)c-
PCV))
46
Suppose the hyperplane H has coordinates [ag : • • • : an] in ¥(V ), i.e., H corresponds to the scalar multiples of the linear form
The vanishing of this form on V defines the fiber of try over H by the Euler sequence (2.4):
nTnv))\H
= V ( J > ^ ) x {H} C F(V) x P(F V ).
The fiber of TIX restricts the fiber of ny to X:
nTv(y)\x)\H = ( x n v E ^ i , ) ) x{H}cXx
p(yv).
Let i x ff be an element of this fiber. The normal bundle sequence (2.3) is exact,
so x x H is in P(A/V/p(m) if and only if ^ 04X; is zero on T ^ , i.e.,
TX,xCKer(52aiXi) = H,
which means that x x H is in Yx by the other definition of the incidence variety.
Thus,
Yx = n^x/nv))A hyperplane H contains the tangent space to X at x if and only if H fl X is singular
at x\ otherwise, the intersection H fl X is transverse and thus smooth.
•
Proof of Proposition 2.2.2. Take X = ¥(W) embedded by QL into ¥(V) = P(L V )
in Lemma 2.2.6. In particular X
Xj of V
is a subvariety of P(L) VV = P(L). The basis
in the proof of Lemma 2.2.6 becomes the basis qi of L in this context.
Evaluating qi on an element a = 5Z a ^<ji °f Lv gives (ft (a) = aj. By definition of
Ql, if a =
QL(W),
then qi(a) = qi(w), which can be written as the matrices M{ in a
basis for W.
Over a point £ in P(£), represented by q — Yl aiq% in L, the fiber of the incidence
47
variety Yx is
Yx,e = P (-A6f/P(LV)) if = ( * n V ( ] T a^)sing) x {£}
= v(YJWi\w)
.
V*—'
x {£} C P(WO x {£}.
/ sing
I use qi\w to emphasize the step at which I consider the fact that q{ is a quadratic
form on W, at which point it makes sense to replace qi\w by the matrix Mj. Note
that the singular locus of a quadric V{Ylai^i)
m
PfW) is the vertex of the quadric
P(rad^ajAfj) (see Section 1.1). In particular, q = Y2cHqi i s
an
element of the dual
X v if and only if q is degenerate on W, so
V(det Q r ^ M , ) )
=IVCP(L).
For a generic q in X v , the radical radg is one-dimensional, so the fiber of Yx over
X v is generically a point.
•
To apply the previous results, it is necessary to know whether Qi is an embedding. When the base locus is empty and
2 ( d i m W - 1) > n,
the following theorem of Fulton and Hansen shows that it suffices to know whether
Ql is an immersion. Corollary 4.4.2 applies this theorem. The proof may be found
in [Laz04, Theorem 3.4.1] or in the original paper [FH79].
T h e o r e m 2.2.7 ( [FH79]). Let X be a complete irreducible variety of dimension k,
and let f : X —> Fn be an immersion. If 2k > n, then f is a closed embedding.
CHAPTER 3
CONFIGURATIONS
A based vector space is a pair of a vector space V and a preferred basis E, often
abusively called the basis. The basis defines a family of symmetric bilinear forms on
this vector space, and that family restricts to every subspace of the vector space. A
configuration
is a subspace of a based vector space; it is not based in general, i.e.,
it has no preferred basis. One example of a configuration is the first homology of a
graph, which is a subspace of the based vector space over the edges of the graph. This
example will be treated in detail in Chapter 4. It determines the graph polynomial
and graph hypersurface
I begin in Section 3.1 with the description of the family of bilinear forms and
the configuration polynomial, whose vanishing determines the locus of degenerate
forms in this family. Two natural operations on a configuration are restriction and
projection, which I describe in Section 3.2. These operations are reflected by the
configuration polynomial in simple ways, and there are corresponding operations on
graphs that reflect these operations, which I will discuss in the following chapter.
The operation of restriction is especially useful for understanding the singular locus
of the configuration hypersurface, the hypersurface defined by the configuration polynomial. I use this restriction operation in Section 3.3 to prove the main theorem of
this chapter, Theorem 3.3.6, which relates the singularities and the rank of the points
on the configuration hypersurface in a precise way. In Section 3.4, I discuss some
simple consequences of Theorem 3.3.6 by counting dimensions of subconfigurations.
48
49
3.1
Configuration Hypersurfaces
Let K be a field, and let V be a finite dimensional vector space over K.
that E = {e\,...,
Suppose
en} is a basis for V. There is a based vector space isomorphism
from V, with preferred basis E, to K
particular, I use V, (V, E), and K
, the vector space generated by the set E. In
to denote the same based vector space depending
on what I would like the notation to emphasize. A general element v of V can be
written as v = ^Ji=\viei
f ° r unique V{ 6 K, i = 1 , . . . n; Vj is the coordinate of v
along ej. I also write this as v = J2eeEvee
w
h e n I would like to avoid unnecessary
indexing.
Each element e of the preferred basis E defines a linear functional Xe on K
by
assigning to v its coordinate along e:
Xe : V ->• K
Xe(v)
= Ve.
These elements also define symmetric rank one bilinear forms X£ on V:
Xl : V x V -> K
Xe (v, w) = Xe(v)Xe(w)
= vewe.
There is a natural map from V to the span of these forms inside the space of bilinear
forms on V. I write this map as
BE = Yl AeXe
eeE
:V
"^ Sym2 VV
a —
i > V ] oeXe .
eeE
1. A morphism of based vector spaces takes basis elements to basis elements or zero.
50
In the context of Section 2.1, BE is a map of vector bundles on ¥(V)
F(V)x(V®V)^0¥{v)(l).
There is also the corresponding map QE from V to the space of quadratic forms
Y{Onv){2))
on V by letting QE(a)(v)
The bilinear forms BE{a)
=
BE(a)(v,v).
are represented by the diagonal matrices
M,a,E
[ a\
0
0
a2
•••
0
0
in the preferred basis E = {e\,...,
o\
0
an J
e n } . Therefore, det BE — (^41^2 • •' A i ) 5 and the
corresponding hypersurface in V or F(V) is the union of the coordinate hyperplanes
(up to a choice of coordinates).
Definition 3.1.1 (Configurations). A configuration
W is a subspace of a based
vector space. It is not assumed to be a based subspace. A generalized configuration is
a linear map (p from a vector space W to a based vector space. In particular, for every
generalized configuration <p, the image f(W)
is a (nongeneralized) configuration in
the based vector space.
I study how the vector spaces of bilinear forms BE or quadratic forms QE behave
when restricted to configurations.
That is, each configuration W C V defines an
inclusion of trivial vector bundles over P ( l / ) :
P(V) x (W ® W) C P(V) x (V (8) V).
I denote the composition of BE with this inclusion by BE\\y,
QE\W
r
f ° composing the inclusion with
and similarly I denote
QE.
Bear in mind that there may be no basis elements e G E in the configuration.
However, some of the dual elements Xe will define nonzero linear functionals on
51
nonzero configurations, and therefore restricting Bj?(a) or Q_e(a) to a nonzero configuration defines a bilinear or quadratic form on the configuration that is nonzero
for some a. If W is a configuration, then I will write X e | ^ , QE(a)\wi
an
d
Bj?(a)\\y
when I want to emphasize that I have restricted to W.
I can also write BE(CL)\W
as a
w(a)
matrix ^E
m
some
basis of W. The choice
of basis for W is suppressed from the notation. I reiterate that W is not a based
vector space, so there is not a preferred basis in which to represent B^(a)\^/.
matrix M^w^a)
then ME w(a)
The
7
is still symmetric but is not diagonal in general. If dimlT = £,
is £ x £. As matrices with variable entries, the space of M g jy have
entries linear in A\, . . . , An. I do not define or use the notation ME\W-I
suggest using the diagonal n x n matrix ME = Mgy
which might
restricting to row and column
vectors from W and not depending on a basis for W.
Example
3.1.2. Let E = {^i,^2iez)i
W7 be the configuration spanned by
and ^
£\ = e\ + t2 and £2 = 2e% — e2- Then
*i(h)
= 1
xlih) = 1
Xl(£x) = 0
X*(£2)=0
Xl(£2) = 1
Xl(£2) = A
Xf(£1,£2)=0
X22(£1,£2)
=
-l
Xl(£1,£2)=0
Therefore, B^\y
in this basis is
A
+ A2[
V° 0/
-l\
(l
0\
ME,w = A1[
1
V"
/o ()\
+ A3
!/
(Ai + A2
-A2
\
=
0 4
•
~M
A2 + 4A3J
52
An element ME w(a)
detMEwi0)
°f the family ME W ^S degenerate if and only if
=
( a l + G 2)( a 2 + 4a3) — a 2 = a\a2 + 4a\a^ + 4a 2 a3 = 0.
Notice what happens if I write ME W
m
a
different basis, say i\ — 2>t\ and
£2 = £2. Then
/9
()\
\0 Oy
/ 9
-3\
V-3
W
/0
()\
V° V
/ 9 A i + 9^2
\
- 3 ^2
-3^2
^2 + 4A3/
As explained for arbitrary bilinear forms in Section 1.1, the determinant of MEW
differs from that of MWE
by the square of the determinant of the change of basis
matrix:
det ME,w(a)
=
( 9 a l + 9a 2 )(a 2 + 403) — 9a 2
= 9 a i a 2 + 36ai<23 + 36a 2 «3
=
9detMEjW.
D e f i n i t i o n 3.1.3 (Configuration Hypersurface/Polynomial). Let W be a nonzero
configuration in K
. The configuration
ideal of W is the homogeneous ideal
de\,BE\wQK[Al,...,An}.
The configuration
hypersurface
Xw
of W is V(detE>E\W)
tion -polynomial *i>w(A) is a generator of det BE\Wdet ME W
m
some
element of K[Ai,...,
— P(Vr) A configura-
I n particular, the determinant
basis for W is a configuration polynomial, and \I/jy(v4) is an
AildimVF' the homogeneous polynomials of degree dim J47.
The condition that W be nonzero is included to avoid ambiguity about the determinant in this case. In particular, the formula of Proposition 3.1.13 does not make
sense for W = 0. T h a t formula is essential to many of the other proofs. I note,
however, that defining ^\Y(A)
to be constant if W = 0 is consistent with all the
53
following theorems about configurations polynomials by using some natural conventions. However, I have no use for the additional generality of allowing W to be zero,
so I prefer to assume W is nonzero and avoid discussing the exceptional conventions
for this degenerate case.
Every configuration W C K
is a configuration in multiple based vector spaces,
for example by changing the preferred basis or including K
in another based vec-
tor space. The map of vector bundles that defines the configuration hypersurface
depends on the preferred basis, so the configuration hypersurfaces do, as well. Therefore, it is important to understand that all statements about configurations are made
with respect to a specific based vector space, which is usually implied and not explicitly mentioned or notated. When I need to keep track of the based vector space,
I write the configuration as the pair (W, K ) .
Remark
3.1.4. As described above, a preferred basis of V defines a map from V to
v
Sym V . For every subspace W C V, the natural surjective restriction map from
Sym 2 Vv to Sym 2 Wv makes P(Sym 2 Vv) a cone over P(Sym 2 Ww) with vertex F{Z)
where Z is the kernel of Sym 2 Vy —> Sym 2 Wv.
rational map from F(V) to P(Sym
2
Therefore, the basis of V induces a
v
W ):
n : P(V) - P(Z) n F(V) -* P(Sym 2 Wy).
The image is a linear subspace of P(Sym 2
(3.1)
Wv).
The generic symmetric degeneracy loci in P(Sym 2 W /V ) are
•£Sym,fc = Dk{a)
= {xE
P(Sym 2 Wv)
| rank a < k}.
Here a is the tautological map
P ( S y m 2 i y v ) x (W®W)
• Op(Sym2
P(Sym 2
Wy)
wV)(l)
54
are
Suppose dimW^ = m, so X s y m , m - 1
the points that are not full rank. In terms
m(m+l)
of matrices, P ( S y n r Ww) is the projectivization P
2
,
of the space of m x m
symmetric matrices, and 3£sym,fc *s the projective set of matrices of rank at most k.
The map ix in (3.1) restricts to rational maps of configuration hypersurfaces
m,m— 1'
and there are similar statements about the other degeneracy loci. In particular, a
point x of F(V) - F(Z) n F(V) is in Xw
if and only if TT(X) is in X S y m i m _ x .
If P ( Z ) and F(V) do not meet, then n embeds F(V) as a linear subspace of
P(Sym 2 Wv).
In this case,
XSym^nF(V)
=
Dk(BE\w).
Note that if the subspace F(V) is tangent to X g y m ^ at a smooth point of X g y m £,
then that point will be a singular point in D^ (B^\^).
Theorem 3.3.6 shows that
this cannot happen for k ~ m — 1.
Example
3.1.5 (Trivial Configuration). For the configuration W — K
ready shown that tyw(A) = YieeE
A
e-
, I have al-
In particular, a configuration hypersurface
need not be irreducible.
Example 3.1.6 (Nontrivial Configuration). I computed two configuration polynomials
in Example 3.1.2:
$w(A)
= A\A2
+ ^A\A?> + 4 A 2 ^ 3
$w(A)
= 9 A i ^ 2 + 36AiA3
+
36A2A^.
I reiterate that I suppress the dependence of these polynomial on the choice of basis
from the notation, and writing $f\y(A)
does not specify a unique polynomial.
Remark 3.1.7 (Generalized Configuration Polynomials). Suppose that <p : W —> K
is a generalized configuration. The rank one bilinear forms X\ pullback to W by
<p*X*(w) = X*(<p(w)).
55
The image of ip is a configuration in KE,
2
and the image of f* is a linear subspace
V
of Sym VK spanned by rank one quadratic forms. All of the following definitions
and results about configurations and configurations polynomials apply to the configuration
f(W)
C K
. In particular, a configuration polynomial for a generalized
configuration could be defined as ty^ryy^A). The configuration hypersurface X<n(w)
in P \KE\
would correspond to forms of rank diraf(W)
rank dim W — 1 on W. The pullbacks f*X\
— 1 on p(W),
not forms of
will define degeneracy loci according to
the rank on W, but K e r ^ is contained in all the radicals, so the maximum rank is
dim ip(W). T h a t is, consider the subspace L of P(Sym 2 Wy) = P
by f*Xg.
Every element of L vanishes on Keif. In particular,
r(r+l)
2
spanned
L C P(Sym2(H//Ker^)V) C P(Sym2W/v).
In terms of matrices, pick a basis {w\,..,
,wn}
of W whose first k elements are a
basis for Ker f. In this basis, Sym (W/ Ker f)^
corresponds to matrices of the form
(° °)
\0 MJ
where M is a symmetric (n — k) x (n — k) matrix. Then L must be contained in the
projectivization of this subspace.
In what follows, all results are stated for configurations, configuration polynomials, and configuration hypersurface unless specifically stated for generalized configurations. The generalized situation does not apply to the applications in Chapter 4.
The reader interested in the generalized situation can translate the configuration
results to generalized configurations using the preceding comments.
The following proposition shows that every subspace of Sym W^ spanned by
rank one forms comes from a generalized configuration if : W —* K
, and it is a
configuration under a simple condition.
P r o p o s i t i o n 3.1.8 (Constructing Configurations from Rank One Forms). Suppose
{qi,. . ., qn} are a set of rank one quadratic forms on W. Let E be a set { e i , . . ., en},
56
and let {X\,.
.. ,Xn}
be the basis dual to the basis E of K
KE
<p : W - •
making a generalized configuration
tion is a configuration
. There is a linear map
such that q{ = <p*X^. The generalized
configura-
when
n
f)mdql =0.
i=l
Proof. There exist linear forms i{ such that qi = if. Define a linear map
n
'•
W
Ke
-" 0
i =
RE
n
W
i=.
i=l
Then the pulled back quadratic forms are
(<p*X?)(w) = X?(<p(w))
(
n
= il(wf
= Qi(w).
The generalized configuration is a configuration when ip is injective. The map if
is injective if and only if
P|Ker^ = 0.
i=\
From the factorization q^ = if
rad^j C Ker£j
57
because
wEradqi
=>
q~i(w,-) = 0
=^> Qi{w) = 0
==^>
£j{w)
=0.
Both radgj and Ker^j are hyperplanes, so containment implies equality.
Remark
•
3.1.9. The essential fact about rank one forms is that qi = if because the
linear forms ^ are needed to define the linear map </?, which makes W a generalized
configuration in K
.
L e m m a 3.1.10 (Direct Sum of Configurations). If W\ is a nonzero
in K
1 and W2 is a nonzero configuration
configuration
in K
1
2 with configuration
VW(A)
= *Wiew2(A)
in K
2
configuration
, then W\ © W2 is a nonzero
polynomial
=
VWl(A)VW2{A).
As an application, I could have inductively computed Example 3.1.5, the trivial
configuration polynomial corresponding to W = K
.
Proof. Select a basis for W = W\ © W2 consisting of a basis for W\ and a basis for
W2. For e E Ei, Xe\\y2
= 0 and similarly for e E E2 and Wi, so in this basis,
and therefore, the determinant of Bg\\y-.^Wo
the BE\Wi.
IS
* n e P r °duct of the determinants of
D
A subspace W C V of dimension f is an element of the Grassmannian of £dimensional subspaces of V, Gr(£, V). When V has a preferred basis, the configuration polynomial for W is defined up to a constant, and you can compute a formula
for the configuration polynomial tyyy(A) using the coordinates of the Pliicker embedding of GT(£, V) into P ( A ^ ) • The Pliicker embedding is the map that takes an
^-dimensional subspace W of V to the line det W = f\ W in f\ V. From an ordered
58
basis of V, denoted E = {e\,...,
e
I
=
e
e
en}, there is an induced basis on f\ V defined by
r
i\ A ' ' ' A i / f ° every set / = {?4,.. ., ip} such that zi < • • • < ig. Let Vj
denote the subspace spanned by the e^ for i G / . By picking the basis J51, there is a
projection
717 : V
Vi
eel
In particular, det Vj is the line spanned by e j .
The image of an £-plane W under the Pliicker embedding has a coordinate along
each ej, and this set of coordinates is the set of Pliicker coordinates of W.
For
each I, write Pluckerj(W) for the coordinate of d e t W along I. That is, for an £dimensional subspace W of V, its coordinate Pliicker/(W) is the element of K such
that det W —» det Vj is multiplication by Pliicker/(W):
These coordinates are well-defined up to a constant representing a change of basis
for W, and Pliicker/(W) is 0 if and only if W —*• Vj is not an isomorphism. A change
of basis for V changes the basis vectors ej of / \ V, so the coordinates of W change.
In the context of configurations, the based vector space K
has a fixed preferred
basis, so a general change of basis is not allowed. The preferred basis is not assumed
to be ordered, and a change of basis of K
by reordering may change the Pliicker
coordinate for the index set I by ± 1 . The dependence of the coordinates on these
choices of bases are suppressed from the notation. The preceding discussion simply
explains that Pliicker coordinates are projective coordinates:
L e m m a 3.1.11 (Pliicker Coordinates are Projective Coordinates). If p and 8 are
59
bases for W and Pliicker/ ^ ( W ) and Pliicker^ z(W)
dinates, then there is a constant C, the determinant
are the respective Plilcker coorof the change of basis matrix
from P to j3, such that
Plucker / ) / 3 (W) = C P l i i c k e r ^ W O
for all It-subsets I of E, a basis for V. In particular, these coordinates are well-defined
projective coordinates for the £-plane
W.
In terms of matrices, once I have chosen the basis for V, I can write a basis for W
as £ row vectors in the coordinates of the basis for V. Arranging these row vectors
into a £ x n matrix, the Pliicker coordinates of W are the £ x £ minors of this matrix.
Example
3.1.12. Consider the configuration W of Example 3.1.2. In the ordered
basis E = {ei, e2, 63} of K
and the basis {£\, £2} of W, the matrix representing W
is
In the induced basis of f\
/l
K
'l
1
0N
.0
-1
2,
, det W is
l \
det W = det
(l
0\
\ e\ A e^ + det
\0
-l)
/ 1
0\
\-l
2)
\ e\ A eq + det
\0
2)
e9 A ex
= —e\ A e2 + 2e\ A e% + 2e2 A e$.
Therefore, its Pliicker coordinates are [ — 1 : 2 : 2 ] . Notice the similarity between the
last line and the configuration polynomial
$w(A)
= AlA2
+ 4 ^ i A3 +
4A2A3.
The next proposition explains this similarity.
P r o p o s i t i o n 3.1.13 (Configuration Polynomial in Pliicker Coordinates). Let W be
60
a nonzero configuration
in K-E . A configuration polynomial for W is
VW(A)
Yl
Plucker F (W / ) 2 ]J
FCE
f£F
F\=dimW
=
Af.
Remark 3.1.14. The configuration polynomial is only denned up to a constant, so the
ambiguity of which basis for W is used to compute Plucker^(W^)
is irrelevant—it will still generate the principal ideal det BE\W.
on the right side
The basis for K
is
fixed up to order, which can only change the Pliicker coordinates by ± 1 . Therefore,
the values Pliicker^?(W)
are well-defined in the preferred basis up to a nonzero
constant corresponding to a change of basis of W. The preferred basis removes any
ambiguity about which Pliicker coordinates are used in the formula.
Also, note that there is some nonzero Pliicker coordinate for every nonzero configuration W (i.e., the Pliicker embedding is a well-defined regular map, so not all
projective coordinates may vanish), so the configuration polynomial is never identically zero.
Proof. Pick a basis (3 = {wi,...
,wp} for W, and work with the matrix Mgj,y,
which is the bilinear form B^\w
written in this basis. As noted above, the entries
in M^^w
are
l i n e a r in t n e ^ e ,
so
the determinant is homogeneous of degree dim W.
The bilinear form X\ has rank at most 1 on W. If it is rank 1, then there is a basis
for W in which XJ is represented by a matrix with 0s everywhere except a 1 in the
(1,1) entry, so
(l
0
0
0 0
ME.W
E
— At
A x
fr
feE-e
\0
0
...
0/
The variable Ae only appears in one entry of Mg^y,
at most 1 in det Afg y/. If X | is rank 0, then
M E,W
AP0
feE~e
and so it appears with degree
61
and it is still true that Ae appears with degree at most 1 in det Mp\y.
This suffices
to show that the monomials of det Mp \y are products of £ distinct variables, and
hence there are constants cp such that
detME^w=
C
F\\A!-
J2
FcE
feF
\F\=dimW
Evaluate both sides at Ae = 1 for e G F and Ae = 0 for e ^ F to determine
the constants. Evaluating the right side gives cp. Evaluating detMpiy
&ves the
determinant of a bilinear form:
det [ Y.X%V
\ •
JeF
That is, cp = detY^f(=F^-'f\wbilinear form Xf\\y
Y
feF
X
j\w =
With respect to the basis (3 = {w\,...
is the matrix with i, j t h element XJ{WJ)XAWJ).
Y
X
f(wi)Xf(wj)
= (Xfk^)
\Xfk^W^) =
,wi}, the
Thus
n
F\w^F\w-
/
V/GF
The matrix (Xf (WJ)) is the £ x £ matrix representing the map
TIF\W
:W ^ K
F
projecting W onto its F coordinates (so k is the row index, and j is the column
index). Similarly, {XJAWJ))
is the matrix representing the transpose map
*F\W- (KFY ^WV
(so here k is the column index, and i is the row index). Taking determinants yields
det
Y
feF
X
f\w
=
(detnF\w)2
= PhickerF(^)2.
62
•
The formula of Proposition 3.1.13 will simplify the analysis of the singularities
of the configuration hypersurface.
First, I need to explain how it behaves under
restriction.
3.2
Restrictions and Projections of Configurations
Every based vector space K
has natural based subspaces K
for F C E. Moreover,
the basis of a based vector space determines a canonical isomorphism with the dual
space: e —
i > Xe
and extend linearly.
Therefore, these based subspaces are also
naturally based quotient spaces. For every subset F of E, there is a short exact
sequence of based vector spaces
0 -»• KF -> KE
'nE-F)
KE~F
-»• 0.
There is a natural splitting of this vector space from the isomorphism of these based
vector spaces with their duals, and K
is isomorphic to
Kp.
For a configuration W, there are two induced configurations in K
tion W
= K
fl W and the projection
Wp = np(W).
: the restric-
There is an induced exact
sequence:
0 -> WF - • W ->• WE_F
-> 0.
There is not a natural splitting of this sequence. Note that Wp^p
a subspace of W and W
Both W
is not generally
is not generally isomorphic to Wp.
and M ^ are subspaces of K
, so they can also be considered configu-
rations in K
. Every configuration is a pair of a based vector space and a subspace,
so ( W
J and ( W
(\Vp,K
,K
J and (Wp,K
notation, so I write W
,K
J are two different configurations, and similarly for
) . I prefer to suppress the based vector space from the
and Wp when I mean that the based vector space is K
When I need to use the configurations (W
, KE)
the based vector space explicitly in the notation.
and (Wp,Kj,
.
I will include
63
One reason for suppressing the based vector space is that the configuration polynomials are the same. For example, if W C KF
C KE for some subset F C E, then
the formulas for them given by Proposition 3.1.13 are
V W ^
)
=
S
Plucker G (M/) 2 J ] Ag,
GcF
geG
\G\=dimW
and
VW^)
=
£
Pl^kerG(VF)2 J ] Ag.
GCE
\G\=dimW
geG
There are two apparent differences between these two formulas:
1. There are more subsets G of E to sum over than subsets G of F.
2. ty,yr „E\{A)
is a polynomial in more variables, namely the Ae for e G E — F.
These apparent differences do not actually affect the polynomial:
L e m m a 3 . 2 . 1 . Suppose W is a nonzero configuration
every GCE
in K
and F C E.
such that \ G | = dim W and G D (E — F) ^ 0, the Plucker
For
coordinate
vanishes:
Plucker G (W/) = 0.
In
particular,
Proof. By assumption, Xe\w
n
G\w
'• W ~^ K
= 0 for all e € .E — F . Let e 6 Gn(E
— F). Therefore,
cannot hit the eth coordinate, so it is not a surjection and the
Plucker coordinate is 0.
•
Remark 3.2.2. Note that this result corresponds to the fact that if W C K
then det W is contained in / \ K
, a linear subspace of / \ K
.
C K
,
64
Example 3.2.3. I will continue with the configuration W of Example 3.1.2, whose
configuration polynomial I computed to be
VW(A)
= AlM
+ 4 ^ i ^ 3 + 4A2A3.
Let F = {e2,e%} C E. The basis {e\ + e2,2e3 — e2} of W maps to the basis
{e2, 2e3 — e2} of Wp under the projection to K . In this basis for Wp, the bilinear
form is the matrix
M
F,WF = A2^e2\wF
+
A
3^e3\\VF
( 1
-l\
/o 0N
\-l
1/
VO 4,
A2
-42
Therefore, a configuration polynomial for the projection configuration Wp is
1V F (i4) = A 2 (A 2 + 4A3) - A\ = 4A2A3.
Notice that this is ^ ^ ( O ,
A2,A3).
Now consider the restriction configuration W . In the basis 2e 3 — e2 for W
the bilinear form A2Xk + A 3 Xi restricted from K
is
M „; W F = M ( l ) + A 3 (4) = [A2 + 4^l3
The corresponding configuration polynomial is ^wF(A)
basis for WF, the bilinear form AiXf
M
£,(^,i^)
=
^
+ A2^2
(°) + ^
= A2 + 4^43. In the same
+ ^3-^f restricted from i ^ is
( l ) + ^3 (4) = [A2 + 4A3
65
Thus, ^^vF_KE^Ai,A2,A3)
= ^wF{A2,A3).
In fact,
d
— Vw(Ai,A2,A3)
=
*WF(A2,A3).
For a subset F C E, let j4_p denote the set of Ae for e G F, so Ajr = 0 means
that Ae = 0 for all e G F . Similarly, let <9^F, or simply <9f, be the derivative with
respect to each Ae with e E F. The one well-known fact about projections that I
will note is
Lemma 3.2.4 (Projection of a Configuration). Let W be a configuration in K . If
W = WQ, or equivalents WE'G
= 0, then
VW(A)\AE_G=O
=
*WG(A).
Proof. The exact sequence
0 -> WE~G
->W->WG^>0
establishes the mentioned equivalence. Now set Api_G = 0 in the formula for
^iy(A)
given by Proposition 3.1.13. Setting those variables to zero removes all terms that
contain factors Ae for e G E — G. So
^W{A)\AF_G=Q
IS o r u
F
F contained in G. For every F C G, the projection to K
«
I
TVF\\Y
W
• W
"G\W
n W
r
> uWQ
^ G
F
> KK
y
a s u m over tne
subsets
factors as
.
In particular,
Plucker^(H/) = det (nG\w)
Pliicker F (WG).
Therefore, the formula from Proposition 3.1.13 gives
^W{A)\AE_G=Q
= det {TTG\W)2
^WG(A).
This suffices to prove the proposition because the configuration polynomial is only
well-defined up to a constant.
Note that if W = WG,
•
then W = WG
because WG C W. However, even when
W = WQ, the configurations are not equal in general, and WQ is not a subspace of
W in general.
I now provide a more detailed analysis of the restrictions, which play a prominent
role in analyzing the singularities of configuration hypersurfaces. I start with the
case of W
for simplicity. By the general formula for configuration polynomials
(Proposition 3.1.13), the configuration polynomial for a nonzero restriction W
is
^wE-e{A)
=
Yl
FcE-e
\F\=dimWE~e
For comparison, the partial of ^w(A)
de^w{A)=
Plucker F (wE~e)2
J ] Af.
f£F
with respect to Ae is
Y,
PliickerHWO2 I I
F<zE
| F | =dim W
eeF
A
f-
feF-e
Using the bijection F —
i > F — e between subsets of E of size dim W containing e and
subsets F of E — e of size dim W — 1
de*w(A)=
>T
VmckeryUe(W)2l[Af
FcE-e
\F\=dimW~l
This last polynomial differs from ^wE-e
(3.2)
feF
f° r two possible reasons:
1.
Pliicker^ Ue (WO ± Pliicker^
(wE~e
in general, and
2. if W C KE~e,
then VK = W ^ - 6 , and the sets F indexing the sum in
have size dim W — 1, but the sets F indexing the sum in ^W^(A)
de^w(A)
have size
67
dim W.
The following lemmas clarify these differences.
Lemma 3.2.5 (Trivial Restriction, Simple Case). Suppose that W is a nonzero
configuration in K
1.
. The following conditions are equivalent.
WCKE~e,
2. W =
3. VW(A)
WE~e,
=
*wE-e(A),
4. PliickerF(WO = Plucker^ (wE~A
for all F C E with | F | = dim W,
5. Pliicker^py) = 0 t / e 6 f , | F \ = dim W,
6. Ae does not appear in the polynomial
7.
^\y{A),
de^W{A)=0.
Proof. The following implications are straightforward:
(1)^(2)^(3)^(4)^(5)^(6)^(7).
Actually, assuming (2) W = WE~e
really implies (3') ^\y — ^(u/E-e. KE\-> ^ u t ^y
Lemma 3.2.1, ^ / w E - e j^E\ = ^\yE-eimplies (1). Note that P (/\eKE~e^j
To complete the equivalence, I show that (5)
is the linear subspace of P (/\eKE^j
defined
in the Pliicker coordinates by Pliicker^ = 0 for those F containing e. In particular,
if Plucker F (H/) = 0 for all F containing e, then det W E P (f\e KE~e\.
W C KE~e.
Therefore,
U
The previous lemma generalizes by induction to subsets of E with more than one
element.
Lemma 3.2.6 (Trivial Restriction, Intermediate Case). Let W be a nonzero configuration in K
. The following are equivalent:
1. there is a subset H C E such that W C K
,
2. W = WH
3. *w(A)
=
*wH(A),
4. Plucker F (W) = Pliicker^ (\VH]
for all F C E with \ F | = dim W,
5. PluckerF(WO = 0 tf F n ( F - H) ^ 0, | F | - dim W,
6. Ae does not appear m
7. deVw(A)
^\Y(A)
= 0 for all
for all e £ E — H,
eeE-H.
Proof. This is a straightforward induction on the size oiE—H from Lemma 3.2.5.
•
A further generalization that I use is:
Lemma 3.2.7 (Trivial Restriction). Suppose W is a nonzero configuration in K .
The following are equivalent:
1. there are subsets H C G C E such that WG C KH,
2. there are subsets H C G C E such that WG = WH,
3. ^wG(A)
=
^wH(A),
4. Plucker F (\VG\
= Pliicker F (wH\
5. Pliicker F (\YG\
= 0 if F n (G - H) ^ 0, | F | = dim WG,
6. Ae does not appear in
7. deWwG(A)
^WQ{A)
= 0 for alleeG-
for all F C E with \F\=
dim WG,
for all e G G — H,
H.
Proof. Apply Lemma 3.2.6 to the configuration W
in KG.
a
69
The case when W is not contained in K
is described by formally differenti-
$w(A)-
ating
L e m m a 3.2.8 (Nontrivial Restriction, Simple Case). Suppose W is configuration
in K , and suppose that diiaW > 2 so that W
cannot be zero and ^urE-e
ts
defined. The following conditions are equivalent:
W£K E-e
1.
2. W
e
is a hyperplane in W, and there is a nonzero constant C such that for
allF C E - e with \F\ = dimW E ~ e
Pliicker^ (wE~e\
3. de*W{A)
=
= CPlucker FUe (WO,
*WE-e(A).
Proof. (1) =>• (2): The dimension of a subspace drops by 1 when intersecting with a
hyperplane that does not contain the subspace (Lemma 1.2.14).
The projections
•KF\wE-e:WE-e^KF
have factorizations:
KF\wE~e
•
W
Extend a basis {w\, . . . , u> m -l} of W
W
K
K
•
to a basis for W by adding an element
wm. The matrix representing the map from W to K
Ue
in this basis for W and the
canonical basis of K
is
<Xfl(wi)
Xfl(w2)
...
Xf2(wi)
Xf2(w2)
...
Xe(w2)
...
\Xe(wi)
(Xh(wi)
Xf2(wi)
V
Here I let F — {f\,...,
which has Xe(wm)
Xfl{wm)\
Xf2(wm)
Xe(wm))
Xh(w2)
...
Xf2(w2)
...
o
Xh(wm)\
Xf2(wm)
Xe{wm)
o
j
/ m _ i } . Expanding the above determinant in the last row
as its only nonzero element gives
Pliicker^WO = (-l)2mXe{wm)
In particular, C = Xe(wm)
Pliicker F (wE~e\
= Xe(wm) Plucker F
(wE~e
satisfies the conclusion of (2). The choice of wm can
change the overall constant, but it is independent of the subsets F.
(2) => (3): As noted in Equation (3.2),
de*w(A) =
™&erpue(W)2
£
FcE-e
F\=dimW~l
H Af.
feF
By the assumption (2), the coefficients can be simplified to
de^w{A)
= C2
Plucker^ (\VE-e)
Yl
JJ
Af.
feF
FcE-e
F \=dim.W — l
That is,
de*w(A)
=
cHwE„e(A).
There is an ambiguity about an overall constant on both sides because they are
71
configuration polynomials, so I can omit the nonzero C .
(3) => (1): By assumption, de<S>w(A) = $wE_e(A).
By Lemma 1.2.14, WE~e is
nonzero because its dimension is at least one less than W. Thus, one of its Pliicker
coordinates is nonzero, so de^\y(A)
^ 0 by the general formula for the configuration
polynomial (Proposition 3.1.13). If W were contained in KE~e,
then de^/\y(A) = 0
by Lemma 3.2.5, which would be a contradiction. Thus, W <£. K
.
•
Corollary 3.2.9 (Isolating a Variable). IfW is a nonzero configuration in K
that
and We = 0 (i.e., e £ W), then
is not contained in K
VW{A)
= AeVwE_e(A)
+
*WE_e(A).
Proof. The two terms on the right side correspond to partitioning the subsets of
E based on whether they contain e. The first term is determined by the preceding lemma using the assumption W <jt. K
. The second term is determined by
e
Lemma 3.2.4 requiring the assumption W = 0.
•
Again, I generalize to larger subsets of E and higher derivatives.
Lemma 3.2.10 (Nontrivial Restriction). Let W be a configuration in KE.
H
following conditions are equivalent for subsets H £ G C E assuming W
The
is nonzero:
1. for all sets H' with H £ H' C G and all e <E H' - H,
WH'
2. W
£
KH'~e,
is a codimension \G — H\ subspace of W , and there is a nonzero con-
stant C such that for all F C H with \F\ = dim WH,
Plucker F (wH\
3- d{G_H)^wG(A)
=
^wH(A).
= CPliicker^^.jy)
(wG
72
Proof. Consider the configuration W
in K
and induct on the size of \G — H \.
The case | G — H | = 1 is the content of Lemma 3.2.8.
Now consider H C\ G with | G - H \ > 1.
(1) => (2): Assuming the condition in (1) holds for H, then it also holds for
H U e for every e E G — H because every H' that properly contains H U e also
properly contains H , and every element of H' — H U e is also an element of H' — H.
Therefore, the induction hypothesis can be applied to HUe to conclude that W
is codimension I G — H — e | in j y
e
and there is a nonzero constant C\ such that
Pliicker^ ( V f f U e ) = d P l u c k e r / , u ( G F _ e ) (FK G
for all F <Z HUe
with I F I = d i m V t ^ 0 6 .
Condition (1) for H implies that WHUe <£ KH
to Lemma 3.2.8 applied to the configuration W
by taking H' = HUe.
e
in K
Ue
, W
According
is a hyperplane
in VK-"Ue, and there is a nonzero constant C2 such that
Plucker F (\VH)
= C2 Pliicker F U e
(w1
for all F C. H with | F | — dim W
. Such F satisfy the conditions for F above.
Therefore, the codimension of W
in WG
is 1 + \G — H — e | = | G — # |, and
letting C = C1C2 gives
Pliicker F ( l y ^ ) = C i C 7 2 P m c k e r F U e U ( G _ F _ e )
(\VG
= CPluckerfu(G_F)(^G
(2) =>• (3): In general, using the bijection between subsets of G containing G — H
73
of size dim W
and subsets of H of size dimWG
d(G-H)*WGW=
Pliicker
E
-\G-H\,
F{WG)2
II
FCG
\F\=dimWG
G-HCF
A
f
fep-{G-H)
P l u c k e r ^ u ( G _ ^ (wGf
^
FCH
=dim\VG-\G-H\
F
E
Pmcker
— \G — H \ become
F {WHf I I Af
FCH
p\=dimWH
=
Af.
feF
By assumption (2), the Pliicker coordinates and dim W
d(G-H)*WG(A) = °2
J]
feF
C2*wH(A).
Again, I invoke that configuration polynomials are only well-defined up to a constant
to ignore the C .
(3) => (1): By assumption, W"
is nonzero, so one of its Pliicker coordinates
is nonzero, and thus the general formula for t h e configuration polynomial, Proposition 3.1.13, shows that ^WH{A)
is not identically zero.
I prove by contradiction, so I will assume (1) fails and show that this implies that
(3) fails. In particular, I will show that the failure of condition (1) implies
0(G-ff)*WG(^)=O,
which contradicts condition (3) that
74
By assuming condition (1) fails, there are sets H' such that
H ^H'
<ZG
and elements e G H' — H such that
WH'
£• ^ f l ' / - e i
Let if be such a subset with the largest size, and let e G if — H be an element for
which
Either H = G or condition (1) is satisfied by H, i.e., for all if' with
H £H'
and all e G H' -
QG
H,
WH' £ i^i/'"e.
In case H = G, Lemma 3.2.5 implies that
d^wG(A)=0,
so
9(G-H)^WG(A)
= d(G~H-e)de^WG(A)
= 0.
In case H c; G, the fact that i7 properly contains H implies that the induction
hypothesis applies to H, and H satisfies condition (1), so by induction, condition (3)
holds for H:
Then apply Lemma 3.2.5 to W
and e to find
75
Therefore,
d{G-H)^WG(A)
=
d(&_-_H)d-ed{G_&)^wG{A)
d
(H-e-H)d^WH
(A)
0.
In either case, diQ_Fl\i$>wQ(A) = 0, but condition (3) assumes
which is nonzero for a nonzero configuration W H
D
The following corollary summarizes the preceding results:
Corollary 3.2.11 (Restrictions of Configuration Polynomials). Let W be a nonzero
configuration in K
dFmw(A)
=
. For all subsets F C E,
VWE-F{A)
ifWh~*
tj t
<^K-E-F'-e
- " e for all F' £ F and all e £ F',
otherwise.
Conversely,
if WE~F'
dF*w(A)
*WE-F{A)
= <
F' £F
*WE-F'(A)
= dE_F,Vw(A)
£ KE~F'-e
for all
and all e e F',
for some F' £ F.
In particular, for every integer k, the following ideals in K[A] are the same
dF*w(A)
F\ <k
*WE-F{A)
F\<k
76
3.3
Singularities of Configuration Hypersurfaces
This section describes the singularities of the configuration hypersurfaces in terms of
their rank. I assume that the characteristic of the field K is zero. If the dimension of
a configuration W is m, then the configuration hypersurface X\y is the degeneracy
locus -Dm-1 (-£>£; IW-0 ^ f(K).
The scheme structure of a degeneracy locus is defined
by the vanishing of minors, so in this case, the configuration polynomial defines the
scheme structure. There is a chain of degeneracy loci
D0(BE\W)C
C-.-C
D^BEIW)
C Dm-2(BE\w)
C Dm-iiBslw)
C Dm{BE\w)
=
F(KE).
Definition 3.3.1 (Order of Singularity Ideal). Let X be a projective variety defined
by a homogeneous ideal / in K[Ai,.
S(I),
. ., An].
The first order singularity
ideal of /,
is the ideal generated by the set
{ # — -}•
The kth order singularity
= 5(5(fc_1)(/)).
ideal of / is S^(I)
The phrase "fcth order singularity ideal" seems an unwieldy choice; the phrase
"Jacobian ideal" is already taken [Rim72] and may have the wrong connotation as I
discuss below.
L e m m a 3.3.2 (First Order Singularity Ideal defines Singularities of Projective Hypersurface). Let X be a projective hypersurface.
SingX
=
Then
V{S(I)).
Proof. In [Har77, Chapter 1.5], the singular points of an affine Y in A n of codimension
r defined by fi, • • • , ft are shown to be defined by the typical rank condition on a
77
Jacobian matrix:
rank ( ——- ) < r.
dA3
If Y is an affine hypersurface, then this condition is that the rank of the Jacobian
matrix is 0 at singular points y. T h a t is, the rank condition can be replaced by the
condition that the matrix is identically 0. Therefore, the singular points of Y are
defined by
8/
<v) = o
BAj
for all / G / (i.e., y is in the affine variety defined by
S(I)).
For a projective hypersurface A , take Y to be the affine cone over A and note
that y G (SingF) — 0 if and only if [y] G Sing AT. Thus, S(I)
also defines the
singularities of the projective variety.
•
According to the lemma, the singularities of projective hypersurfaces are
V(S(I)).
However, Sing X will not be a hypersurface, so generally,
V(S(2)(/))C_Sing(SmgA).
In terms of the Jacobian matrix for Sing A , Sing(Sing A ) is defined by a rank condition, but V(S^ >(I)) corresponds to the Jacobian matrix being identically 0.
Instead, the order of singularity ideals define the order of vanishing of the polynomials defining X in the following sense:
Definition 3.3.3 (Order k Singularities). Let X be a projective variety defined by
a homogeneous ideal I. The locus of order at least k singularities
is the scheme defined by S^
'(I).
As noted above,
Sing>/e X ^ Sing(Sing A ._ 1 A").
In the case where A is a hypersurface, Lemma 3.3.2 states
S i n g > 1 A = V(S(I))
= SmgA.
of X, S i n g ^ A ,
78
Let X be a projective hypersurface defined by a homogeneous polynomial / . If
a = [a\ : • • • : an] is a point in X and a?; j^ 0, then the multivariate Taylor formula
in the local coordinates with az- = 1 expresses / near a as
A
Al
deg/
1gJ/
m=0 | J^J = m
J\ dAJ
= Ev
A
a
j
Therefore, when a 6 Sing>^._^ X, all the terms with m < k are zero and
A
Al
A
^, ,^
J
m=fc J \=m
A _ a \ A
J! <9A
(3.3)
^i
i
Then Sing>£_i X is the locus of points where / has multiplicity at least k, and the
set of points of multiplicity fc, Mult/. X, is
Multfc X = Sing >A ,_ 1 X - Sing>fc
(3.4)
Picking out the kth term of Formula (3.3) defines the affine tangent cone
TCaV(f)
1 dJf
=V
A
A
^ k
J l d A J
-
a
a
\ A-
(3.5)
I use V to denote either affine or projective varieties, letting the context determine
which one is meant; Equation (3.5) is affine. One may homogenize the leading term
at a
A^
Ja.k
Al
v
1 d 7f
'
A
-
a
A
a.J
A •
n-'
\J\=k
1
A
i ojf
E
^ UftkJ'dAJ
J
A_a(alA-aAl)
~^i
a
i
,
79
so the projective tangent cone to V(f) at a is
T,
A = a_
A~ a{
|J|=fc
The notation (a^A — aAj)
{a,A - aAiY
(3.6)
for a tuple J = (JQ, ... ,jn) is shorthand for
{atA - aAi)J = ]J {aiAe -
aeAi)je.
£=0
In particular, if ji > 0, the product is zero.
Proposition 3.3.4 (Configuration Tangent Cone). LetW be a nonzero configuration
in K . If a = [a\ : • • • : an] is a point of multiplicity k with a^ ^ 0, then the tangent
cone to X]Y at a is
\
TCaX\y
E
=V
a\
*WE-J
( A
A
a
<hj \ i
JCE,\ J\=k
\dim W7
—dim W—fc
a
J
i
The projective tangent cone to Xyy at a is
(
TCaXw
£
=V
^WE-J
( —1
(aiA~aA,y
JCE, | J\=k
Vdim WE~ J = d i m W-k
J
Proof. Proposition 3.3.4 follows from Equations (3.5) and (3.6) with /
* W-
Namely, the fcth order term of ty\y at a is
V W.a,k
A
A
£
J\=k
1 dJ$>
w
J! dAJ
A
A a
- at
A~i
a
u
n\A~~M
i
The configuration polynomial has degree at most one in each variable (Proposition 3.1.13), so only tuples J that are sequences of 0s and Is need to be included
80
in the sum. Such tuples correspond to subsets of E.
Corollary 3.2.11 simplifies
the derivatives to the configuration polynomials ^WE-J
• Note that if | J | = k but
E
dimW ~^
dj^w
7^ dimVF — k, then Corollary 3.2.11 shows that the partial derivative
is identically zero.
•
Euler's formula for homogeneous polynomials states
E A ~ = deg(/)/.
(3.7)
Therefore, each homogeneous / in / is also in S(I) using the assumption that the
characteristic of K is zero so that deg(/) is invertible. The ideal / is homogeneous,
so every g E I can be decomposed into its homogeneous pieces, and g is also in S(I)
by applying the same reasoning to its homogenenous pieces. Therefore, the order of
singularity ideals form a chain
I CS(I) c sW(I) c ...,
and so do the order k singular loci
X D Sing>! X D Sing>2 X D ....
In particular, note that the fcth order singularity ideal can be defined by
grrif
5W(/)
dAl1 •••dAJnnn
n
f G /, m < k, and N , 3i =
i=\
\
m
I
Lemma 3.3.5. If I is generated by a set of homogeneous polynomials {f\,..
., fj,},
then S(I) is generated by
Proof. The set L is contained in S(I) by definition. First note that each fj is also in
the ideal generated by L by Euler's formula 3.7. Consider a homogeneous element
81
f £ I. Then there are homogeneous polynomials h\,...,
h^ such that
k
3=1
Differentiating gives
df__A(dh1
dfj_
Both terms on the right are in the ideal generated by L.
•
The main result of this section is
T h e o r e m 3.3.6 (Singular Loci are Degeneracy Loci). Let W be a nonzero configuration of dimension
m, in K
and X\y
its configuration
hypersurface
in ¥(K
).
Then
Sing>kxW
=
D
m~k,-l(BE\w),
for 1 < k < m — 1. In other words, the locus of points where tyyy has multiplicity
least j is Dm_j(B^\^).
The case k = \ states
SingXvj/ =
Remark
at
Dm_2(BE\w).
3.3.7. Consider the particular case k — 1, that is, the singularities of the
degeneracy locus X\y coincide with the locus of next lower rank. This result is true
for the generic determinantal varieties X C P(VKV <g) Wy)
and X g v m
c
IP(Sym2
Wv),
but the behavior is not true for all degeneracy loci. For example, intersecting the
generic determinantal variety with a hyperplane H will cause
Dm__2(Xn/f)
CSingXntf.
If the hyperplane H is tangent to X at a smooth point, then the inclusion is strict:
there will be singular points of X D II that are rank m — 1. The set of hyperplanes
that are tangent to X at a smooth point are contained in the dual variety of X, which
82
has positive codimension in the dual projective space. In particular, the set of such
hyperplanes is not open nor dense in the set of all hyperplanes, so getting a strict
inclusion is not generic. As discussed in Remark 3.1.4, the image of the X\y
under
a particular rational map
7T
:F(V)
--+P(Sym2W/V)
is the intersection of a linear subspace with Xsym- Theorem 3.3.6 implies that this
linear subspace cannot be tangent to Xsym at a smooth point. The proof does not
take this geometric perspective, and it would be nice to have a more direct geometric
understanding of why this particular linear subspace is not tangent to Xsym- A key
feature seems to be that it is spanned by rank one forms.
The method of proof that I present for Theorem 3.3.6 works for the generic
determinantal variety.
In particular, if D is the generic determinant defining X,
then dj\.D is the minor given by removing the unique row and column containing
A{. These all vanish at x if and only if x has rank at most m — 2. This argument
generalizes to higher derivatives and other minors.
In the case of the configuration hypersurface, the argument is more subtle. In
particular, not all minors can be found as d^.^\y
shows that dj\.^w
for some A{. Corollary 3.2.11
are determinants of matrices, and Corollary 1.2.24 shows that if
enough of these determinants vanish, then ty\y vanishes, as well. That is the basic
argument, which I complete in detail below.
The next lemma follows from the analysis of restricting configurations in Section 3.2.
L e m m a 3.3.8 (Singularities Determined by Restricted Configurations). Let W be
a nonzero configuration
in K
.
Sing>fc Xw = V ((VWE-F
\F\<kJj
= P|
FCE
\F\<k
X{WE-F^KE)
83
Proof. The lemma restates Corollary 3.2.11, which established the equality of the
defining ideals.
•
Proof of Theorem 3.3.6. First note that the condition on k, k < m — 1, implies that
dim WE~F
= dim W D
KE~F
> m —k
> m — (m — 1)
= 1
for | F | < k by the usual bound on the dimension of the intersection (Lemma 1.2.14).
In particular, W
is nonzero, and its configuration polynomial is defined.
By Lemma 3.3.8, it suffices to show that a point a is in V ((^TTT-E-F | I F | < k))
if and only if Bj^(a) has rank at most m — k — 1 on W.
If BE (a) has rank at most m — k — 1 on W, then it has rank at most rn — k — 1
on a subspace WE'F
(Corollary 1.2.13). If | F \ < /c, then ^\mWE~F
(Lemma 1.2.14), so BE(CI) is not full rank when restricted to W
>
m-k
. Therefore, the
determinant
det
BE{O)\WE-F
= %WE-F{a) = 0.
If a E V {{'fyyyE-F | I F | < k)), then Bjr(a) is degenerate when restricted to all
subspaces W
for | F \ < k. Note that the subspaces W
E e
of the complete set of hyperplanes < W ~
are intersections
E e
| e E E, W ^ W ~
conditions of Lemma 1.2.4 are satisfied, so the rank of Bfi(a)
>. Therefore, the
is at most m — k + 1.
•
Note that
dim F(V)
= n - l > m - k - l
for all nonnegative integers k because m < n. In particular, the hypotheses of the
nonemptiness part of Theorem 2.1.7 are always satisfied, so the D^{W) are nonempty
for all k between 1 and m—1.
By Theorem 3.3.6, Sing>^ X^
is therefore nonempty
for such k, as well. The hypotheses of the connectedness part of Theorem 2.1.7 are
84
more subtle, so using it to decide whether Sing>^ X\y is connected will depend on
the specific values of n, m, and k.
Example
3.3.9 (Trivial Configuration). The configuration W = K^
introduced in
Example 3.1.5 has the union of the coordinate hyperplanes as its configuration hypersurface. For each k between 1 and m — 1, Sing>^ X\y is the union of the coordinate linear subspaces of codimension k. In particular, Sing>^. has codimension 1 in
Sing>^._i. For example in K , there are three distinct coordinate hyperplanes whose
union is X\y.
These planes meet in three distinct lines whose union is SingXyj/.
Corollary 3.3.10 (Multiplicity of Configuration Hypersurfaces). Suppose W is a
nonzero configuration
in K
M\AtkXw
. The set of points of multiplicity
= Dm_k{BE\w)
-
k m X\y is
Dm_k_i(BE\w).
Proof. The corollary simply restates Equation (3.4) using Theorem 3.3.6:
Mult*. Xw = Sing^fc.! Xw
= Dm-k(BE\w)
- Sing> fc
~
Xw
D
m-k-l(BE\w)-
D
Corollary 3.3.11 (Intersection of Tangent Cones). Let W be a nonzero
in K
. Let Z be the kernel of the restriction
configuration
map
Sym2(iv^)V^Sym2^v.
Let L = F(n(BE(A)))
be the image of the family BE{A)
X be the generic symmetric
= KE in P(Sym 2 Wy).
2
v
degeneracy locus in P ( S y m V y ) . Note that Xw
cone over L n X with vertex V = F(KE)
Multa; Xw
fl F(Z). For every x G X\\r — V,
= M u l t ^ ) X,
Let
is a
85
and
therefore,
TC7r{x)(Lnx) = LnJCn{x)x,
and similarly for the affine tangent
cones.
When V = 0, n em beds F{KE)
into
P ( S y m 2 V ^ v ) ; and
TCXXW
Proof. When TCntx\X
= LH
TCXX.
is cut by the linear section L, the leading term at x may
vanish leading to a higher multiplicity. By Corollary 3.3.10 and Theorem 2.1.5, both
Mult x X\y
and Mult^^-j X are the corank of
bilinear form on W, so the
multiplicity does not increase on X-yy- Therefore, the leading term defining X\y
at
x is found by linear substitution of L into the leading term defining X at x, which
means that the tangent cone of L n X is just the linear section.
Remark
•
3.3.12 (Generalized Configuration Hypersurfaces). If
ip : W -»
KE
is a generalized configuration and dim Ker tp = d, then the results of this section still
hold for the configuration (f(W) C K
. The locus of corank k in (p(W) is the locus
of corank k + d in W, so putting the formulas in terms of W instead of >p(W) is
possible by modifying k appropriately.
3.4
Subconfigurations
Let U and W be configurations in K
defined by the basis for K
with U C W.
The bilinear forms By (a)
restrict to both U and W. I write D^{U) and
D^{W)
for the corresponding degeneracy loci of forms of rank at most k. I use the convention
that Dk(U)
= 0 if k < 0 and Dk(U)
= r(KE)
if k > dimU, and similarly for W.
This section deduces some simple relationships between the singularities of Xjj and
X\Y
by simply counting dimensions.
86
Suppose U is codimension £ in W. By Corollaries 1.2.13 and 1.2.15, a rank k
form on W restricts to a form with rank between k — 2£ and k, so
Dk(W) - Dk^(W)
c Dk(U) - £>fc_2£-i(J7).
In particular,
and
Dk_u_x{XJ)
C Djfe.iCW).
(3.8)
Suppose dim W = m, so the configuration hypersurfaces are
XW = D m _i(lV)
Jfy = A„_,_i(E0-
Note that XJJ and Xjy are not necessarily contained in one another. However, the
containment
Dm-i-i(W)
C
Dm_e^(U)
can be restated as
Sing>£ Xw
C JQy
using Theorem 3.3.6. More generally,
e-k-i(W)
c
Dm_i_k_i(U)
implies
In particular, when [/ is a hyperplane in W (a case that will have applications for
graph hypersurfaces),
SingXpy C XJJ,
and
S^g>k+1Xw
C Sing> fc X[/.
87
I have proved the following lemma:
Lemma 3.4.1. Let U and W be configurations in K
with U C W. Suppose that
U has codimension £ in W. Then
Sing> f c + ^X W C Sing>fc Xv.
In particular, if U is a hyperplane in W', then
SingXpfA C XJJ.
The lemma follows directly from Dk{W) C Dk(U). The other containment (3.8)
Dk-2l-l(U)
C
Dk_x(W)
proves a similar lemma:
Lemma 3.4.2. Let U be a codimension £ subspace of W and W a configuration in
KE.
For all k,
Sing>fc+£X(7 C Sing>fcXVy.
In particular, if U is a hyperplane in W, then
SingXu
C
Xw.
C
Dk.^W)
Proof. Replace k in (3.8)
Dk-2e-l(U)
by m + I — k to get
Dm_t_k_x(!J)
C
Dm+i_k^(W).
88
By Theorem 3.3.6,
Dm_e_k_i(U)
= Sing> fc XJJ and
D m + ^_ f c _l(Vl/) = Sing>fc_^
Xw
Therefore,
Smg>kXu
C Sing>fc_^y.
In particular, when U is a hyperplane in W (i.e., £ = 1), taking k — 1 gives
SingX[/ C X ^ / .
D
Putting together the previous two lemmas proves the next theorem.
T h e o r e m 3.4.3 (Relationship Between Subconfiguration Hypersurfaces). Let U be
a codimension I subspace of W and W a configuration
( S i n g > f c + ^ X ^ ) U (Smg>k+iXu)
In particular,
if U is a hyperplane in W,
(Sing Xw)
in KE.
C (Sing> fc A" w ) n
For all k,
(Smg>kXu).
then
U (Sing Xu)
CXwnXu.
Example 3.4.4. Consider the configuration W of Example 3.1.2 whose configuration
polynomial I computed to be
VW(A)
= AlA2
+ 4^iA3 +
4A2A3.
The singular locus of its configuration hypersurface is
Sing Xw
= V(A2 + 4 ^ 3 , Ax + 4A3, AAi + 4A2) = 0.
89
This is a subconfiguration of the trivial configuration K . In accordance with Theorem 3.4.3,
SmgXKE
= {[1 : 0 : 0], [0 : 1 : 0], [0 : 0 : 1]} C
Xw.
CHAPTER 4
G R A P H HYPERSURFACES
The study of configuration hypersurfaces in Chapter 3 was motivated by the example
of graph hypersurfaces, which I present in this chapter. Section 4.1 introduces graphs
from both topological and combinatorial perspectives. The material is necessary for
the following sections but is elementary and well-known. In Section 4.2, I discuss
how the homology of graphs defines configurations. In particular, I derive the usual
combinatorial formulas for the graph polynomials as configuration polynomials. The
results about configuration hypersurfaces in Sections 3.3 and 3.4 apply to the case of
graph hypersurfaces, and I restate them here. In Section 4.3, I recall how some basic
graph operations are reflected in the graph polynomial and graph hypersurface. I
conclude in Section 4.4 with the proof that the quadratic linear series defined by the
graph is not an embedding for most graphs.
4.1
Graph Preliminaries
The material in this section is well-known graph theory and algebraic topology; for
more details, see [Mas91, Section VIII.3] for the algebraic topology and [Lov93] for
the graph theory. I use the following definition of graph.
Definition 4.1.1 (Graph). A graph G is a finite cell complex consisting of 0 and
1-cells. The 0-cells of G are its vertices, denoted V or V(G).
edges, denoted E or E(G).
The 1-cells of G are its
To avoid trivialities, I assume V ^ 0.
A typical example of a graph is shown in Figure 4.1. There are many definitions
of graph in the literature, and Definition 4.1.1 includes many of these definitions as
special cases. I emphasize that the generality of Definition 4.1.1 does not include
90
91
Figure 4.1: A typical graph.
graphs with an infinite set of edges or vertices. Otherwise, Definition 4.1.1 is quite
general and includes graphs underlying Feynman diagrams (although I do not distinguish between internal and external edges, but this feature could easily be included
by partitioning the set of edges). A more traditional, graph-theoretic definition is
that a graph is a set of vertices and a set of edges where each edge is a set of two
vertices. This definition fits into Definition 4.1.1, but Definition 4.1.1 is more general. In particular, multiple edges may connect the same pair of vertices, and an
edge may connect a vertex to itself. One may redefine graphs in Definition 4.1.1 as a
set of vertices and a multiset of edges where each edge is a 2-multiset of vertices; this
definition corresponds to a multigraph or pseudograph in the graph theory literature.
The homology of a graph G is its homology in the usual topological sense. As a
cell complex, its homology may be computed using cellular homology. In particular,
the homology of G is the kernel and cokernel of a map
d : If
-)• Z
92
defined as follows. For each edge e G E(G), the gluing map from e to the vertices of
G is defined by mapping the boundary of e to two elements of V(G) (not necessarily
distinct). Denote these elements by h(e) and t(e) (there is a choice to be made here
if h(e) 7^ t(e), but the homology groups do not depend on these choices). Let ZE
and Z
be the free abelian groups on the sets E = E(G) and V — V(G).
Define the
group homomorphism
d : ZE -> ZV
by d(e) = h(e) —t(e) and extend linearly. The homology of a graph G is the homology
of d : ZE
-> Z y . That is, the first homology of G is Hi{G,Z)
zeroth homology of G is HQ(G,Z)
= Kerd, and the
= Coker<9. The reader seeking more details of
this construction may consult [Mas91].
L e m m a 4.1.2. If G is a connected graph, then the image of
d : ZE -^
ZV
is
zv$
={xezv
Y, xv = °
veV
where xv denotes the coefficient of x on the basis element v.
Proof. The image of each edge e is h(e) — t(e), which is in Z • . By the connectedness
assumption, HQ(G,Z)
> Z. The kernel of the degree map is Z ' , which agrees
with the image by exactness.
•
The homology of a graph is a direct sum of the homology of each connected
component of the graph, so the lemma generalizes to disconnected graphs:
Corollary 4.1.3. The image of d : ZE -> Zv
zy'° = I x e z V
y ^
veV(Ga)
is
xv = 0 for all connected components Ga of G
93
The first homology is a subgroup of a free abelian group, thus free abelian itself.
The zeroth homology is a direct sum of copies of Z, one for each connected component
of G. In particular, the homology has no torsion.
To get a configuration, tensor with a field K:
0 - • Hi(G, Z) ® K -> KE
^ ^
Kz
-> H0(G, Z) <g> AT -^ 0.
By direct calculation or the universal coefficient theorem, Hi(G, K) = Hj(G, Z) <g)K.
I write Hi(G) for i/j(G, K ) , and h{{G) for dim^- Hi(G).
In some of the lemmas, I
apply the Euler characteristic of this chain complex, which in this case gives
X(G)
= h0(G)-h1(G)
= \V\-\E\.
(4.1)
The next step is to develop a dictionary to translate between the topological definitions and combinatorial definitions. First note the following notational convention:
N o t a t i o n 4.1.4 (Removing the Interior of An Edge). If e is an edge in a graph G,
then by definition, both vertices h(e) and t(e) are in e. In particular, e is a subgraph
of G. The notation G — e will be used as a shorthand for removing the interior of
e from G; the vertices remain. In particular, V(G — e) = V(G) because no vertices
are removed, and thus the vertices of every edge is still in G — e, so G — e is still a
subgraph. If G — e denoted removing all of e, it would fail to be a graph if h(e) or
t(e) had multiple incident edges.
The following terms are basic in graph theory, but I prefer the simplicity of
the topological definitions.
I prove below that the graph-theoretic definitions are
equivalent.
Definition 4.1.5 (Graph Terminology). A forest is a graph G with H\(G) = 0. A
tree is a connected forest. A spanning forest of a graph G is a maximal subgraph F
with respect to inclusion among all subgraphs that are forests. A spanning tree of
a graph G is a connected spanning forest. The complement of a spanning forest is
94
called a chord set] many arguments could be given in terms of chord sets, but I have
favored spanning forests instead.
A cycle is an element of Hi(G, Z), or more generally H\{G). A cyclic graph is a
graph with H\(G) ^ 0; a graph is acyclic if H\{G) — 0, i.e., it is a forest. A circuit
is a connected cyclic graph with no cyclic subgraphs.
In particular, for every edge
e in a circuit C, h\(C — e) = 0.
A two-forest is a spanning forest minus the interior of an edge. If each connected
component of G has a single vertex, then G has no two-trees because the spanning
forests have no edges. If the spanning forest is a spanning tree, then a two-forest is
called a two-tree. The complement of a two-forest is a minimal cut set. A minimal
cut set C will not contain the vertices of the complementary two-forest F, and thus
there can be edges whose interior is in C but whose vertices are in F; in particular,
C may not be a graph.
Every subgraph Y C G induces an inclusion Hi(Y) C Hi(G).
In particular,
subgraphs of forests are forests. The subgraphs that are circuits and forests play an
important role in identifying specific elements of the homology groups. They will also
provide a combinatorial description of the first graph polynomial (Proposition 4.2.4).
The two-forests and minimal cut sets are useful for describing the second graph
polynomial (Proposition 4.2.12). For the combinatorial perspective, I present the
following lemmas.
Lemma 4.1.6 (Homology of Attaching 1-Cells). Suppose Y is a subgraph ofG, and
e £ E{G) — E(Y) with both vertices of e in Y. Then either
1. hY(Y U e) = hi(T) + 1 and hQ(Y Ue) = h0{Y), or
2. hi{Y U e) = hi{Y) and hQ(Y U e) = h0{Y) - 1.
In particular, if Y is connected, the second case cannot occur.
1. This definition differs from some but not all graph theory literature. The graph theory
literature does not seem uniform about the definition of a cycle, circuit, etc., whereas the
topology literature is a little more consistent in its definition of cycle. I have used the
topological definitions as motivation here.
95
Proof. The Euler characteristic satisfies
x(rue) = x ( r ) - i
because there is one new edge and no new vertices. Regrouping the terms in this
equation gives
(Mr) - M r u e)) + (Mr u e) - MH) = 1.
(4.2)
Because e is connected to T,
Mrue)<MH.
Because F is a subgraph of T U e,
M r ) < /ii(rue).
Therefore, both terms on the left side of Equation (4.2) are nonnegative, and so
exactly one term is equal to 1 and the other to 0.
•
L e m m a 4.1.7 (Forests Extend to Spanning Forests). Every forest F in a graph G
is contained in a spanning forest of G.
Proof. Consider the set T of forests that are contained in G. Every vertex of G is
in J7, so T is nonempty (recall that I have assumed all graphs have V ^ 0). The set
T is a subset of the finite set of all subgraphs of G partially ordered by inclusion.
Therefore, T has maximal elements, which are spanning forests by definition.
Corollary 4.1.8 (Existence of Spanning Forests). Every graph has a spanning
•
forest.
L e m m a 4.1.9 (Spanning Forests Span Vertices). A spanning forest F of a graph G
hasV(F)
= V{G).
Proof. The spanning forest is a subgraph so V(F) C V(G).
Adding a 0-cell to a
graph cannot change the first homology. Therefore, if v G V(G) — V(F), then the
graph F U v is a forest properly containing F, which is a contradiction.
•
96
L e m m a 4.1.10. Suppose F is a spanning forest ofG.
For every edge e G E(G) —
E(F),
hi{F\Je)
= \ and h0(F U e) =
h0(F).
Remark 4.1.11. By the maximality of the spanning forest F, the larger graph F U e
is not a forest, so
hi(FUe)
> 1.
The Euler characteristic shows that it must be an equality, but intuitively it is
impossible to increase the first homology by more than one when a single 1-cell is
added.
Proof. By definition of spanning tree,
ftl(FUe)
> h1{F) = 0.
The vertices of e are in F (Lemma 4.1.9), so applying Lemma 4.1.6 gives the conclusion.
•
L e m m a 4.1.12. A spanning forest of G is a spanning tree if and only if G is connected.
Proof. Suppose T is a spanning tree. It is a spanning forest and contains all vertices
of G by Lemma 4.1.9. Therefore, it connects all vertices of G, and every edge of G
is connected to a vertex, so G is connected.
Suppose G is connected and T is a spanning forest. By Lemma 4.1.9, V(T)
V(G).
=
If T has multiple components, then some pair of them, T\ and T2, can be
connected by an edge e in E(G) — E{T) because the vertices of all edges are in T.
In particular, HQ(T U e) < ho(T).
h\(T U e) = h\(T)
By Lemma 4.1.6, IIQ(T U e) = IIQ{T) — 1 and
= 0. Therefore, T U e is a forest, contradicting the maximality
property of the spanning forest T.
•
L e m m a 4 . 1 . 1 3 . A tree contained in a connected graph G is a spanning tree if and
only ifV(T)
= V{G).
97
Remark 4.1.14. Note that this is not true for a disconnected graph G and a forest F
in G. For example, Lemma 4.1.16 shows that two-forests have V(F) — V(G), too.
Proof. Lemma 4.1.9 shows that a spanning tree must have V{T)
versely, suppose T is a subtree with V(T)
and h\(T)
= V(G).
= V(G).
Con-
Because it is a tree, IIQ(T) = 1
= 0. To show it is a spanning tree, I must show that h\(T U e) > 0 for
every edge e € E(G) — E(T).
The tree T contains the vertices of e by assumption
and is connected, so h\(T U e) = 1 by Lemma 4.1.6.
•
L e m m a 4.1.15 (Spanning Forests are Componentwise Spanning Trees). A subgraph
F of G is a spanning forest if and only if F n G ? is a spanning tree of Gj for each
connected component Gj of G. In particular,
ho(F) = /irj(G).
Proof. If F is a spanning forest of G, then F D T is a spanning forest for every
subgraph T because a larger forest contained in F would also be contained in G. In
particular, for each connected component G\ of G, FflG,; is a spanning forest of Gj.
By Lemma 4.1.12, F C\ Gj must be a spanning tree because G-t is connected.
If F fl Gj is a spanning tree of Gj for all components, then F, which is the union
of each of these spanning trees, is a forest. For every edge e in E(G) — E(F),
e is in
one of the components Gj, so
1 = hi{(F n Gj) U e) < hi(F U e)
because F fl Gj is a maximal forest in Gj. Thus F satisfies the maximality condition
of a spanning forest.
•
L e m m a 4.1.16 (Alternative Characterization of Two-Forests). A subgraph F of G
is a two-forest if and only if
1. h0(F)
= ho(G) + 1,
2. hi(F)
=0,
3. V(F) =
and
V{G).
98
Proof. Suppose F is a two-forest. Let F be a spanning forest such that F — e = F.
By definition, F — e is shorthand for removing the interior of e, and by Lemma 4.1.9,
V(G) = V{F) = V{F - e) = V(F).
As a subgraph of a forest, F is a forest, i.e., hi(F) < h\(F) = 0. All vertices are in
F, so Lemma 4.1.6 implies
h0(F) = ho(F) + 1
= /i 0 (G) + l.
Now suppose F satisfies
1. h0(F) = h0(G) + 1,
2. hi{F) = 0, and
3. V(F) = V(G).
In particular, F is forest containing all vertices of G. For each component G\ of G,
F n G j is a forest containing all vertices of Gi, and in particular, ho(F n Gz) > 0.
The decomposition
H0(F) =
Q)Ho(FnGl)
i
shows that in fact all h^{F D Gi) = 1 with one exception HQ(F n Gj) = 2. Then
for z ^ j , F n Gj is a spanning tree of Gj by Lemma 4.1.13. The component Gj
is connected, but F n Gj has two components and all vertices of Gj.
Therefore,
there is some edge e that connects these two components, and (F n Gj) U e must
be a spanning tree. By Lemma 4.1.15, F U e is a spanning forest, and thus F is a
two-forest.
D
Lemma 4.1.17 (Alternative Characterization of Minimal Cut Sets). A subset C of
G is a minimal cut set if and only if it is a minimal subset of the set of interiors of
edges of G with the properties
99
1. hi(G-C)
= 0, and
2. h0{G-C)
= h0(G) + l.
In particular,
C contains no vertices and is never a graph.
Proof. This is just a restatement of Lemma 4.1.16 for the complement. Perhaps two
points are worth noting. If C is a minimal cut set corresponding to a two-forest
F = G — C, then C is a set of edge interiors because V(F)
= V(G).
The set
C is minimal among such sets because a two-tree cannot properly contain another
two-tree.
•
Recall that C — e means removing the interior of e so that when C is a graph,
C — e is guaranteed to be a graph.
L e m m a 4.1.18 (Alternative Characterization of Circuits). A graph C is a circuit
if and only if
1. /ii(C) = l,
2. h\{C — e) = 0 for all edges e G E(C),
3. h0(C - e) = 1 for all edges e E
and
E(C).
Proof. The definition of circuit is that h\(C)
^ 0, every proper subgraph is acyclic
(hi(T) = 0 for T C\ C), and C is connected, so ho(C) = 1. If C is a circuit, then the
proper subgraph C — e satisfies h\(C — e) = 0, so the only condition that must be
verified is IIQ(C — e) = 1 (i.e., C — e is connected). Both vertices of e are in C — e,
so Lemma 4.1.6 implies IIQ(C — e) = ho(C) — 1.
Conversely, both vertices of e are in C — e, and thus
h0(C)
using h\(C)
h\(C)
= h0(C - e) = 1
= 1, h\(C — e) = 0, and Lemma 4.1.6. In particular, C is connected and
7^ 0. Every proper subgraph T of C is missing some edge e of C, so T C C — e,
and thus hi(T) = 0. Therefore, C is a circuit.
•
100
Figure 4.2: A circuit with 6 edges and vertices.
L e m m a 4.1.19 (Combinatorial Description of Circuits). A graph C is a circuit if
and only if there is an labeling of the vertices of C':
vi,..-,
vn
such that E(C) is the set of exactly n edges {e\,...,
en} where ez- connects vi to t'j + i
for i < n and en connects vn to v\.
Proof. Induct on the number of edges in C. A graph with no edges is automatically
acyclic.
There are two connected graphs with one edge e depending on whether the vertices are distinct. If h(e) ^ t(e), then the graph is contractible, and thus acyclic. If
h(e) = t(e), then the graph is cyclic, the set of vertices is trivially ordered, and there
is one edge connecting v\ to itself.
Suppose graphs with fewer than n edges are circuits if and only if the vertices
and edges satisfy the description in the statement of the lemma. Let C be a graph
with n + 1 edges. If C satisfies the condition on vertices and edges, then C satisfies
the conditions of Lemma 4.1.18.
2. In the graph theory, such a graph is often called an n-cycle.
terminology differs because a topological n-cycle is different.
As I have noted, my
101
Conversely, suppose C is circuit, and let e be an edge of C, and in particular a
subgraph. If e has a single vertex, then H\(e) 7^ 0, and thus e = C by the definition
of circuit; this contradicts the fact that n + 1 > 1. Therefore, e has two distinct
vertices. Therefore, C is nomotopic to C/e, and thus
Hi(C)^Hi(C/e).
Similarly, C — f is homotopic to (C — / ) / e = (C/e) — / for every edge / ^ e, so for
all i,
Hi(C-f)<*Hi((C/e)-f).
The topological space C / e is still a graph; it has n edges and one vertex fewer
(the vertices of e get identitified). The isomorphisms of homology imply that the
conditions of Lemma 4.1.18 are satisfied for C/e, and thus C/e is a circuit. By the
induction hypothesis, C/e satisfies the conclusion, so in particular the vertices can
be ordered so that v\ is the vertex to which e was contracted. In C, one vertex of e
is adjacent to V2 and the other to vn; label these v\ and vn+i,
Remark
respectively.
•
4.1.20. Lemma 4.1.19 makes a key connection between the combinatorial
and the topological perspective; it is unnecessary for the topological arguments that
I make. However, the combinatorial perspective is essential. For example, Theorem 4.4.8 is a famous theorem from graph theory for which I use this equivalent
combinatorial definition of circuit.
L e m m a 4.1.21 (Cyclic Graphs Contain Circuits). A graph G has h\(G) ^ 0 if and
only if G contains a circuit.
no
Conversely,
a graph is a forest if and only if it contains
circuits.
Proof. If G contains a circuit C, then the inclusion H\{C)
C H\(G) proves h\(G) is
nonzero.
Suppose hi(G)
^ 0. If Gi,...,G
n
are the components of G, then the isomor-
phism
#l(G9^#i(Gi)
©•••©#!(£„)
102
Figure 4.3: A circuit with three edges.
a
a
a
^
e2
Figure 4.4: Spanning trees of the circuit with three edges.
shows that there is at least one connected component of G with h\(Gi)
^ 0. Consider
the set C of all connected subgraphs of G with h\(G) =^ 0, which is nonempty because
it at least contains G{. The set C is finite because the set of subgraphs is finite, so
every minimal element of C is a circuit by definition.
Example
•
4.1.22 (Circuit with Three Edges). Consider the graph G of Figure 4.3
where I have used arrows to indicate an orientation on the edges that I will use
for computing homology.
The homology H\(G)
is generated by the circuit £ =
e\ + 62 + e%. The spanning trees of G are shown in Figure 4.4, its two-trees in
Figure 4.5, and its minimal cut sets in Figure 4.6. In this case, the minimal cut sets
of G are the spanning trees without the vertices, but this is not true in general.
Definition 4.1.23 (Circuits as Integral Homology Classes). Every circuit C has
Hi(C,Z)
= Z.
I will ambiguously use C to denote the circuit C or either of the generators of
Hi(C,Z).
The context should determine whether I mean the graph or an integral
103
e2
/
Figure 4.5: Two-trees of the circuit with three edges.
Figure 4.6: Minimal cut sets of the circuit with three edges.
homology class. When a statement about the integral homology class depends on
the choice of generator, I will clarify how the statement depends on the choice.
Recall that Xe : K
—> K are the linear forms on K
They restrict to the subspace
defined by the dual basis.
H\(G).
D e f i n i t i o n / L e m m a 4.1.24 (Circuit Basis). Let G be a graph.
1. For every circuit £\ contained in G, there are circuits £2, • • •, £\u (G)
that < £1,...,
%n
G
su
°h
Ifr IQ\ > is an integral basis for H\(G, Z). / call such a basis is a
circuit basis.
2. Every circuit £\ can be completed to a circuit basis \£\, • • • Ah-ilG) f
suc 1
^
^ia^
there are edges ex, . . . , e^ /Q\ with e^ E £j if and only if i = j . In this case, the
set of edges
£-{ e l>-"> e fti(G)}
forms a spanning forest F, and I call the basis < £\,. . . , £L ,Q\ > a circuit basis
with respect to the spanning forest F.
3. With circuits and edges as in the previous part, the quadratic
X? : Hi(G)
-+ K
forms
104
for i = 1 , . . . , hi(G) are given by
xf{b1ii + '-- + bhl{G)ehl{G)) = bf.
This does not make any claims about Xf for i > h\(G).
Proof. First note that if h\{G) > 1, then G must have a circuit l\ by Lemma 4.1.21.
Removing the interior of an edge e\ from the circuit £\ leaves a tree i\ — e\. The
remaining tree can be completed to a spanning forest F (Lemma 4.1.7). The edge
e\ is not in the spanning forest because
1 = M ^ l ) = M ( 4 " ei) U e i ) < hx(F U ei).
The Euler characteristic determines that there must be h\{G) edges not in the
spanning forest, so label those that are not e\ as e 2 , . . . , %-.IQ\- For each edge
ej, h\{e{ UF) = 1 (Lemma 4.1.10), so there is a circuit £{ that generates H\{ei UF).
Note that e^ £ £j because
/ i l ( ^ - e i ) < / i i ( F ) = 0.
Moreover, e^ £ £j if and only if i = j because e^ ^ F U ej for z =£ j .
In particular, these circuits satisfy
Xi(tj) = ±^j>
which proves the last part of the lemma.
If E j = i G ) fy^' = 0, then for each i = 1,.. ., hi(G)
±b; = 0,
so these circuits are linearly independent. They must span because there are h\(G)
of them.
•
105
Figure 4.7: A simple graph to demonstrate the first graph polynomial.
4.2
Configurations from Graphs
The first homology group of a graph G is a configuration in K
. Therefore, the
following definition is possible:
Definition 4.2.1 (First Graph Polynomial). Suppose G is a graph with h\(G) > 0.
The first graph hypersurface
uration Hi{G) C K
mial for H\(G)
C K
is the configuration hypersurface defined by the config-
. The first graph polynomial
^Q(A)
is a configuration polyno-
computed in a basis for i 7 i ( G , Z ) . This polynomial is also
called the first Kirchhoff polynomial
or the first Kirchhoff-Symanzik
polynomial
in
the literature after Kirchhoff who used it to study electrical networks and Symanzik
who applied it to quantum field theory.
Example 4.2.2. Consider the graph in Figure 4.7 where I have used arrows to indicate
an orientation of the edges that I will use to compute H\{G).
e
e
The circuits l\ = e\
and £2 = 2 ^ 3 form a basis for H\(G, Z). Let L\ and L2 be the dual basis elements
106
of H\(G)y.
The linear forms from the edges can be written in this basis as
eil#i(G) : = Li
e2l#i(G) : = L2
e^HiiG)
:=
-L2
Therefore, the restricted quadratic forms are
X
"= Ll
X
=
e1\Hi(G)
e2\H1(G)
X
e3\H1(G)
L
2
=L2-
Thus, the first graph polynomial is
3
*G(A) = detY,AiXl.\Hl{G)
i=l
= det (AiLi
+ A2L2
det l ' *
°
= AiA2
0
A2 + A3y
+
A1A3.
+
A3L2
The term ^ 1 ^ 2 corresponds to the spanning tree e3, and the term A\A3
corresponds
to the spanning tree e2. I verify that this correspondence is general in Lemma 4.2.3.
By Lemma 4.1.24, the circuit bases provide a nice combinatorial object to use
for the basis of H\(G) when computing the first graph polynomial. However. I must
first prove that the first graph polynomial is well-defined.
L e m m a 4.2.3 (First Graph Polynomial is Well-Defined). The first graph polynomial is well-defined,
polynomial
^JJ SQ\ .
i.e., all integral bases of Hi(G,Z)
yield the same
configuration
Proof. The coefficient of YiceC ^o in ^G(A)
is
Plucker,^./^ (G)) 2 computed in an
integral basis for H\(G, Z), where | C | = h\(G). The projection
H1(G)C
• KE
KC
is induced by tensoring the projection of free abelian groups
Hi(G,Z)
c
»ZE
if
with X. Consider the commutative diagram of free abelian groups with short exact
108
sequences in the vertical and horizontal directions:
0
jE-C
0
- Hi(G,Z)
• ZE
d
ZVfi
»0
if
0.
The map ft is surjective if and only if a is injective. The map a is injective if and
only if the subgraph with edges E — C has h\(E — C) = 0 because a is induced by
d. In particular, ft is surjective if and only if E — C is a forest.
The map ft is injective if and only if a is surjective, which happens when the
edges E — C are connected and spanning in each component of G.
Therefore, if H\(G)
—> Kc
is an isomorphism, then ft : H\{G,Z)
—> lF
is an
isomorphism of Z-modules and E—C is a spanning forest. In particular, det(/3) = ± 1 ,
so P l i i c k e r a ( i f i ( G ) ) 2 = 1.
•
The proof shows more:
P r o p o s i t i o n 4.2.4 (First Graph Polynomial, Combinatorial Formula). Let G be a
109
graph with h\{G) > 0. The first graph polynomial
is
chord sets e£C
CcE
=
n-4.
£
spanning forests
FcE
e^F
The argument for Lemma 4.2.3 is taken from [BEK06]. I have included it here
because it provides the foundation for a similar treatment of the second graph polynomial. I note the well-known combinatorial corollary:
Corollary 4.2.5. For a graph with h\(G) > 0,
^G(^)UP=1
=
^ e number of spanning forests
ofG.
The first graph polynomial appears in the denominator of the following parametric Feynman integrals in quantum field theory:
^G(PA)
(43)
'U-h^AT
The chain of integration a is all positive real values of the A variables. The integral
is a period of the complement of the first graph hypersurface relative to where that
complement meets the boundary of the chain of integration, namely the coordinate
hyperplanes.
Note that the coordinate hyperplanes are the trivial configuration
hypersurface corresponding to the configuration K
C K
. The quantum field
theory behind this integral can be found in Section 6-2-3 of [IZ80]; the condensed
presentation of [BEK06] is sufficient and focused on the applications I have in mind.
The expression $Q(P, A) is the second graph polynomial. This second polynomial
receives less attention in the literature because the integral is not a period of the
complement of the second graph hypersurface. Therefore, the literature often refers
to the graph hypersurface or graph polynomial without reference to the first or
no
second; in such cases, the first one is meant.
I now show that the second graph polynomial is also a configuration polynomial.
Actually, I will define it to be a configuration polynomial, and then Proposition 4.2.12
will show that the configuration polynomial has the same formula as the second
graph polynomial defined elsewhere in the literature (for example [BK08], [Nak71],
or [IZ80]).
E
d(K )
In fact, it is a family of configuration polynomials parametrized by
v
C K.
d{KE)
By Corollary 4.1.3,
=ZV>°®K
= {PeKV
|
J2
Vv = 0 for all components C of G};
veV(C)
I denote this space by K ' .
Remark
4.2.6. The set K ' u is often called the space of momenta or external mo-
menta for the graph G. This terminology is motivated from physics where external
edges of a graph carry nonzero momenta to a vertex in the graph subject to conservation of momentum. More specifically, the idea is that an element of K
labels
the momenta along the edges of the graph (if q = YleeE Qee' then there is momenta
qe along edge e). Then the coordinate pv of dq = p at a vertex v indicates the net
flow of momentum into the vertex v (or out of v depending on sign and convention).
The condition YlveV(C) Pv = ® for connected components of G corresponds to the
conservation of momentum for each component of the system. The vertices v for
which pv is nonzero indicate where momentum is flowing into or out of the graph.
Fix a nonzero p € Kv,{i,
H\{G) + Kq.
and suppose dq = p.
Let H\(G,p)
= d~ (Kp)
=
This is the relative homology group for Kp considered as a chain
complex concentrated in degree zero included in the chain complex for the graph G.
Ill
More specifically, the kernel and homology of that inclusion of chain complexes give
Hi(p) = 0
0
• Kp
• H0(p) = Kp
n
d
K1
Hi(G)
•*
K
V
H0(G)
n
K
Hi(G,p)
E
»KV/Kp
~H0{G,p).
There is an induced long exact sequence from the snake lemma:
0 -+ HX{G) -+ Hx{G,p)
-* H0(p) ^
H0(G) ^
H0(G,p)
-+ 0.
Because p G K V,0
H0(G) = Kv/Kv>°
and the map H\(G,p)
= KV/(KV>°
+ Kp) =
HQ(G,p),
—> HQ(P) = Kp is surjective.
Definition 4.2.7 (Second Graph Polynomial). Let p e K^^>^
second graph hypersurface
tion H\{G,p)
C K
YQP is the configuration hypersurface for the configura-
. The second graph polynomial
<&Q(P,A)
polynomial ^JJ. <Q p\ (A) with respect to a basis of H\(G,p)
basis for Hi(G,X)
and an element q E K
is the configuration
consisting of an integral
such that dq = p. In the literature,
this polynomial is also known as the second Kirchhoff
Kirchhoff-Symanzik
be nonzero. The
polynomial
or the second
polynomial.
The second graph polynomial may be defined when h\(G) — 0 as long as there
are nonzero p in Kv,°.
There are nonzero p G K - as long as G has a component
with more than one vertex. Only a graph composed of disconnected vertices and no
edges fails to define either graph polynomial.
112
Note that every circuit basis for H\ (G) extends to a basis for H\ (G, p) by selecting
any q G H\{G,p) such that dq = p. I will derive the standard combinatorial formula
for the second graph polynomial below. First, I present an example.
Example 4.2.8 (Circuit with Three Edges). Consider the graph of Example 4.1.22.
Let p = YlveV PvV ^ e
an
element of K ' . Suppose that q 6 K
is a lift of p:
p = dq
3
= d^2qiet
i=l
= («3 - Ql)a + (qi ~ q2)b + {q2 ~ 93) c
= paa + pbb + pcc.
Then £ = ei + e2 + e% and q are a basis for Hi(G,p) as in Definition 4.2.7. Let L
and Q denote the dual basis of Hi(G,p)^.
all i, the linear forms Xei\jj
Because Xei(£) = 1 and Xe-(q) = qi for
IQ ^ are
X
ei\Hl(G,P)
= L + Q'i.Q
Therefore, the bilinear forms X^.\^j IQ ,p\ are
Xl\Hl(G,P)
= L2 + 2qlLQ + qJQ'
113
Calculating the second graph polynomial in this basis gives
*G,p(A) = det J2 ML2 + 2qeLQ + q2Q2)
eeE
(
=
Ai+A2+A3
\qiAi
91^1+92^2+93^
+ q2A2 + 93^3 q\Ai + q\A2 + 93^3/
= (Ax + A2 + A2,){qlAi + q2A2 + 93^3) - (91^1 + 92^2 + 93^s) 2
= E «?A? + E QiAiA3 ~ E 9?A? " E mjAAj
i^j
i
i^j
i
J2qzAiAJ~^WAtAJ
=
J2^+g]~2q^A^AJ
=
i<j
= E ^
"
Qj)2AiA3
i<j
=
PIAXAS
+ p2hAlA2
+ P2CA2A3.
In particular, the AjAj correspond to the minimal cut sets of the graph, and their coefficients (qi — qj) correspond to the square of the value of p on one of the components
of the corresponding two-tree. I derive this general formula in Proposition 4.2.12.
Lemma 4.2.9 (Second Graph Polynomial is Well-Defined). The second graph polynomial is well-defined. That is,
Pliicker
2
c(Hi{G,p))
is independent of the choice of an integral basis for H\(G, If) and a preimage q of p
for all subsets C of E with \C\ = h\(G,p) = h\(G) + 1.
114
Proof. I use a diagram similar to the one in the proof of Lemma 4.2.3:
0
0
KE
• Hi{G,p)
• Kv-Q/Kp
• 0
KC
0.
Again, a is injective if and only if (3 is surjective, and a is surjective if and only if 3
is injective. The maps factor for every edge e G C:
a :KE~C
Pe -H^G)
~> AT(S-C)ue ^
^
H!(G,p)
^
RVfi
KC -
_^
KVfi/Rp
KC~e
As in the proof of Lemma 4.2.3, pe is an isomorphism if and only if (E — C)Ue
is a
spanning forest.
If Plucker^(i/x (G, p))
is zero, then p is not an isomorphism, and there is nothing
to check. If PliickerQ(Hi (G,p))
from H\{G)
isomorphism.
to K
is nonzero, then 8 is an isomorphism, and the map
is an inclusion. Then there must be some e such that 6e is an
For this e, (E — C) U e is a spanning tree, and thus (E — C) is a
two-forest. In particular, it suffices to compute Plucker ( ^(i : fi (G,p))
for minimal cut
sets C, though there may be minimal cut sets for which Pliicker(j(Hi(G,p))2
is zero
115
depending on p.
If C is a minimal cut set, then there is an edge e such that (E — C) U e is a
spanning forest, and hence j3e is an isomorphism.
Whether 8 is an isomorphism
depends on the particular value of p and the minimal cut set C. If G is connected,
then G — C is a pair of trees, but if G is disconnected, then G — C has more than
two components. In either case, there is a unique connected component T of G for
which r n (G — C) is has two components T\ and T<i- Let F]_ U F'2 be a partition of
the two-forest G — C such that Tj C Fj. For these forests, define
veF,
With the exception of Tj, the components of Ft cover the vertices of a component of
G, and summing pv over those vertices is zero by definition of K • . Therefore, it
suffices to sum over the vertices in Tf
m
Ft(p)
=^PvveTt
Note that
mFl (p) = ~m,F2 (p)
because
J2veTPv
0 for the connected component T that is disconnected by the
minimal cut set C Define the momentum
sc(p)
of p on the cut set C to be
=
mF.(p)2
for both forests JF\ and F2. The next two lemmas complete the proof by showing
that
PlnCkeTC(H1(G,p))2
=
sc(p),
which is basis independent. In particular, 3 is an isomorphism if and only if
is nonzero.
sc(p)
•
116
First, I need to put S(j(p) in a form that relates it to a lift q G K^E with dq = p
L e m m a 4.2.10 (Vertex Momenta in terms of Edge Momenta). Suppose C is a
minimal cut set of a graph G partitioning
element of Kv>®, and let q G K
G — C into F\ and F2. Let p be a nonzero
map to p: dq = p.
rn
Qe
Fl (P) = Y
qe
~ Y
eeC
h(e)eFt
for both i = 1,2. In
Then
eeC
t(e)<=Fi
particular,
(
£ *- £
sc(p)
. eeC
\h(e)eFl
eeC
t(e)£Fi
Qe
,
/
Proof. The proof is the same for F\ and F2; I prove it for F\.
v G Kv
has a dual element Xv G (Kv)y.
The basis element
Denote the restriction of Xv to Kv>° by
Xv, as well. By definition of q and nip (p),
m
Pv
F1 (p) = Y
veV(Fx)
= Y XM
VGV(FI)
= Y
X
viPq)
veV(Fx)
= Y
Xv
[ Y
veV{Fx)
=
Y
qeh
^ ~qet^
\eeE{G)
Y
veV(F!)eeE(G)
QeXv(h(e) - t(e))
117
The edges in F2 have vertices in ^ ( i ^ ) , which is disjoint from V(Fi), so
m
F1(p) =
E
E
1eXv(h(e)-t(e)).
(4.4)
veV(Fi)eeE(G-F2)
Note that
1
if v = h(e),
Xv(h(e) - t(e)) = { -l
ifv = t(e),
0
otherwise.
If both h(e) and t(e) are in V(Fi), then qe appears in the sum (4.4) with both a plus
and a minus, so it cancels out. For example, every edge e 6 F j has both h(e) and
t(e) in V(Fi), so
m
F1(p) =
E
E
qeXv(h(e) - t(e))
veV{F1)eeE(G-F1-F2)
=
E
^2 QeXv(h(e) - t(e))
veV(Fi)eeC
=E
E &~ E ^
eeC \/i(e)ey(Fi)
i(e)e7(Fi)
which completes the proof.
•
Lemma 4.2.11 (Pliicker Coordinates are Momenta). Suppose C is a minimal cut
set of a graph G partitioning G — C into F\ and F2. Let p be a nonzero element of
KV,Q.
Let (3 be the projection from H\{G,p)
to K . Then in a basis of H\(G,p)
consisting of q in the preimage of p and an integral basis of H~i(G,Z),
det(/?) = ±mFl(p)
In particular, Pluckevc(Hi(G,p))
= S(j(p).
= TrriF2(p).
118
Proof. Let m = hi(G). Write (3 in a basis as in the statement of the lemma:
/ X^q)
Xx(£2)
...
*l(Wl)
\
X2(q)
X2(£2)
...
X2(£m+1)
\xm+i(q)
xm+i(i2)
...
xm+i(£m+i)J
The columns correspond to a g such that dq = p followed by the integral basis
^2) • • • > ^m+1 °f H\{G, Z). The rows correspond to the elements of the cut set
C = {ei,...,em+i}.
Write qi for Xi(q).
Expand the determinant of (3 along the first column of this
matrix:
det/5 = qi Plucker c -_ e i (//i(G))-g 2 Plucker c „ e 2 (Fi(G'))+
.••±9mPluckfirc_em+1(^i(G)).
The Plucker coordinate Pliicker(7_e.(ifi(G')) is computed in an integral basis, hence
as in Lemma 4.2.3, it is 0 if (E — C) U e^ is not a spanning forest and is ±1 if
(E — C) U ei is a spanning forest.
Comparing the formula for det/3 with Lemma 4.2.10, I need to check that
1. the signs on qi and qj in det f3 agree if the heads of both or tails of both are in
Fy,
2. and the signs on qi and qj are opposite if the head of one and the tail of the
other is in Fy
3. but the coefficient ( - l ) i + 1 Plucker c _ e .(#i(G)) of q{ in det 3 is 0 if
(E-C)Uet
is not a spanning forest, which happens exactly when both the head and the
tail of ej are in F\ or both are in F2 (in agreement with Lemma 4.2.10 where
qe appears twice in the sum but with opposite signs).
119
To complete the proof, I compare the signs of q\ and 92 assuming ei and e^ fit into
one of the first two cases above. In particular, (E — C) U ei is a spanning forest. At
this point, the value of det (3 only depends on the integral basis of Hi(G,Z)
for an
overall sign, so I am free to choose the basis {£2, • • •, £m+l} to be a circuit basis for
Hi(G,Z)
with respect to the spanning forest (E-C)Uei
as in Lemma 4.1.24. That
is,
&i G £{ for i = 2,. . . , m -f- 1
ej £ t,- for j ^ i.
Note that i 7^ 1 in these formulas as e\ may well be in all £j.
Because I have assumed that e\ and e<i do not fit into the third case above, both
(E — C) U ei and (E — C) U e2 are spanning forests. Therefore, the circuit generator
£2 of ifi ( ( F — C) U e\ U e2) must contain both ei and e2 because
M^2-ei)<M(£-C)Ue2) = 0
^ l ( ^ 2 - e 2 ) < / i i ( ( ^ - C , ) U e i ) = 0.
Therefore,
-^1(^2) — -^2(^2) — i l if
-^1(^2)
=
on
ly
o n e OI e
l
or e
2
nas
its head in F\, and
± ^ 2 ( ^ 2 ) = ± 1 if both ei and e2 have their heads in the same Fj.
By changing the orientation of every edge in £2 if necessary, I may assume ^2(^2) = 1?
so that the cases are Xi (^2) = 1 or X i ^ ) = - 1 .
Note that the determinant begins
det 8 = q\ det 71 — q2 det 72 + • • •
120
where
(
X2(£2)
x2(£3)
x3{£3)
x3(h)
71
\Xm+\{£2)
x2{£m+i)
x3(£m+\
Xm+i(ellm+i)
Xm+i(£3)
(x2{£2)
0
\
X2(h)
X3{£3)
X2(£m+i)
••
..
(h)
•••
J
\
X^m+l)
(ellm+i)J
Xm+i
and
(
Xx{£2)
XY{iz)
x3{£2)
x3(e3)
X\{tm+l)
72
\
X3{£m+\
\Xm+i(£2)
Xm+i(£3)
(xx{h)
xx{£3)
.
0
X3(£3)
.
0
Xm+1(£3)
\
...
Xm+i(ellm+i)
Xl^nUrl)
J
\
X3(£m+i)
Xm+i(ellm+i)
J
With the exception of the first row, 71 and 72 agree, so by the multilinearity of the
determinant,
det(7i) ± det(72)
({x2±xl){£2)
det
0
0
(x2±x1)(e3)
X3(£3)
Xm+i{h)
(X2±X1)(£m+1)\
XMm+l)
Xm+i(ellm+i)
J
121
Now I consider the two cases noted above:
( X 2 - X 1 ) ( £ 2 ) = 0ifX1(£2) = l
(X 2 + X i ) ( ^ 2 ) = 0 i f Xi(£ 2 ) = - 1 In both cases, the first column of the preceding matrix is zero. Therefore,
det 71
det72
— det 72
if only one of ei or e 2 has its head in F i ,
if both ej have their heads in the same Fj,
and
det/3
(det 7i)(q i i — <?2) + • • •
if only one of e\ or e 2 has its head in F i ,
(det7i)(gi + 92) + • • •
if both ej have their heads in the same F,.
This completes the comparsion of the signs on q\ and g2 and hence the proof.
D
P r o p o s i t i o n 4.2.12 (Second Graph Polynomial, Combinatorial Formula). The second graph polynomial corresponding to a graph G and a nonzero momentum
p £ K '
is
$G(p,A) =
min.
=
S
C(P) I I Af
J2
cut sets
CCS
X]
2-forests
TCE
f 6C
S G _ T CP) I I
A
f
f<£T
Proposition 4.2.12 is false if p is allowed to be zero: d - 1 ( 0 ) is F i ( G ) , and the
configuration polynomial of this configuration would be the first graph polynomial,
but then the formula of this proposition would not hold for all p because SQ(0) = 0
for all C whereas the first graph polynomial is nonzero. Similarly, the sum in the
first graph polynomial is over spanning forests, not two-forests.
Corollary 4 . 2 . 1 3 . Let p = v\ — V2 for any two vertices v\ and i>2 in a connected
122
graph G. For each minimal cut set C, sG(p) is 0 if both vertices are in the same
component of G — C and is 1 if the vertices are in different components of G — C.
As a result,
^G
P(A)\AF=I
— the number of two-trees separating v\
andv2-
The first and second graph hypersurfaces are configuration hyper surfaces in
F(KE)
defined by a quadratic form on the trivial vector bundles P(K
E
and P(K )
x Hi(G,p)
) x H\(G)
respectively with values in the line bundle C L / ^ g J l ) . I
write the degeneracy loci of rank at most k forms as Dk(H\{G))
and
Dk(Hi(G,p)).
Theorem 3.3.6 restated for these configuration hypersurfaces says
Corollary 4.2.14 (Singularities of Graph Hypersurfaces). Let G be a graph, and
let p be a nonzero element of K'°.
Suppose h\(G) is nonzero and denote it by m.
Then
Sing> fc X G = £> m _fc_i(#i(G9)
and
Smg>kYG^p =
Dm_k(Hi(G,p)).
In particular,
SmgX G = Dm_2{Hi{G))
For all nonzero p E KV',
and
S i n g l y = Dm_i(#i(G,p)).
H\(G) is a hyperplane subconfiguration of
Hi(G,p),
so Theorem 3.4.3 in this context says
Corollary 4.2.15 (Intersection of First and Second Graph Hypersurface). Let G be
a graph, and p a nonzero element of K^ ,0 . For all k,
(Sing> fc+1 XG) U (Sing> fc+1 YG^p) C (Sing>fe XG) n (Sing>fc YGp).
In particular,
(Sing x G ) u (Sin g r G)P ) c xG n r GiP .
123
4.3
Polynomial Formulas for Some Graph Operations
The definition of the graph polynomials in terms of homology makes it possible to
analyze how some topological operations on a graph are reflected by the polynomials. Propositions 4.2.4 and 4.2.12 express the graph polynomials combinatorially,
so some more combinatorial operations can be analyzed too.
One fun aspect of
these polynomials is that many approaches (e.g., topological, combinatorial, algebraic) may prove the same result. I present multiple proofs for some of the results
in this section to show the range of techniques. One theme that emerges is that the
combinatorics of two-trees can get quite messy, and often the topological approach
produces simple proofs using linear algebra. Most of these results are well-known,
but Proposition 4.3.19 is particularly interesting and still needs an analysis of the
graph motive as discussed below.
L e m m a 4.3.1 (Restricted Graph Configurations). For every subgraph Y of a graph
G,
H1(G)nKE^
In other words, Hi(T) is the restriction
KE(G),
ForpeKV(T)fi)
H1(G,p)nKE^
=
to K
=
Hi(T).
' ' of the configuration
H\(G)
C
H1(T,p).
Proof. By definition, <9p is the restriction of 8Q to the subspace spanned by the edges
of T, so the following diagram commutes:
0
• H^T)
n
• KE^
H
- t
Kv™
0
• H^G)
KEW
- ^
#T(G),0
Because every element of H\(G) fl KE^>
r
goes to 0 in KV(G>'°,
0
. 0.
it must already be
124
0 in the subspace KV^',Q.
The same reasoning applies to the other equation using
the diagram
ffl(r!p)_^)i#(r),o/i{p
0
-
HI(G,P)
— KE^
-M
0
Kv^°/Kp
D
L e m m a 4.3.2 (Conditions for an Edge to Define the Zero Functional). Let e be an
edge in graph G. The corresponding linear
functional
XP:KE^K
restricts to both H\(G)
1. X
\H-\(G)
2. Xe\j],(Gp)
and Hi(G,p).
Then
= 0 i/ and only if e does not belong to a circuit, and
= 0 tf
an
d
on
ty tf
e
does not belong to a circuit and there is a lift
q of p such that
Xe(q)
=qe=0-
Contrapositively,
1. Xe\fl
<Q\ 7^ 0 if and only if e belongs to a circuit, and
2. Xe\p[ tQp\ 7^ 0 if and only if e belongs to a circuit or for every lift q of p
Xe(q) = Qe ^ 0.
Proof. If e belongs to a circuit c, then Xe(c) ^ 0, and c <G H\{G) C H\{G,p).
versely, if e does not belong to a circuit, then H\{G — e) = H\(G),
0.
and Xe\jj
Con-
(G-e) =
125
If Xe\H
/Qp\ = 0, then Xe\H
/G\ = 0, so e does not belong to a circuit. By
assumption, Xe(q) = 0 for all lifts of p, and in particular there is one. Conversely,
suppose e is in no circuit and there is such a q. Because e is in no circuit, Xe\jj
(Q\ =
0, so the value of Xe(q) is independent of the lift q. In particular, if there is one q
for which Xe(q) = 0, then Xe\H
I Q „ \ = 0.
•
For an edge e G G, I use G — e to denote removing the interior of e: its vertices
remain.
L e m m a 4.3.3 (Removing a Trivial Edge Disconnects). An edge e in a graph G is
not contained in a circuit if and only if HQ(G — e) > ho(G).
Conversely,
an edge e
in a graph G is contained in a circuit if and only if HQ(G — e) = ho(G).
Proof. If e is not contained in a circuit, then Xe\jj
Hi(G)
^KE~e,
/Q\ = 0 (Lemma 4.3.2), so
and thus
H1(G) =
H1(G)nKE-e,
which is H\(G — e) by Lemma 4.3.1. Apply Lemma 4.1.6 using G — e as the subgraph
T. Of the two possible conclusions of Lemma 4.1.6, the first one is ruled out because
h\(G) = h\(G — e). The second conclusion includes KQ{G — e) = IIQ(G) + 1.
If e is contained in a circuit, then Xe\H
KE~e,
rG\ ^ 0 (Lemma 4.3.2), so H\(G)
^
and
ffl(G-e)£#i(G)
by Lemma 4.3.1. Apply Lemma 4.1.6 using G — e as the subgraph I \ Of the two
possible conclusions of Lemma 4.1.6, the second one is ruled out because h\(G) >
h\{G — e). The first conclusion includes HQ(G — e) = HQ{G).
3. In the physics literature, a graph G whose edges are all contained in circuits are
called one particle irreducible, or 1PI. Therefore, if e is not contained in a circuit, then G
is not 1PI.
•
126
Proposition 4.3.4 (Removing a Nontrivial Edge). Suppose e is an edge in G contained in a circuit. If hi (G — e) > 0, then
dAe*G(A)
=
yG^e(A)-
If p is a nonzero element of K '- '' ,
Proof. Note that h\(G — e) = h\(G) (Lemma 4.3.3) and all vertices of G — e are in
G, so
KV(G),0
=
KV{G-e),0
and there is no restriction on p in Lemma 4.3.1.
The edge e is in a circuit, so Xe\JJ,(G)
K
¥" 0 (Lemma 4.3.2). Therefore, H\(G) <j£
, and the formulas follow from Lemma 3.2.8 for restricting configurations be-
cause
Hi(G-e)
= Hi{G)r\KE-e
and
HX(G - e,p) = Hi(G,p) n KE~e
by Lemma 4.3.1.
Alternatively, one may argue that e is not in a spanning forest T of G if and only
if T is a spanning forest of G — e, which establishes the formula for the first graph
polynomial. For the second graph polynomial, it suffices to consider the connected
component of G containing e, and thus I assume G is connected. The edge e is not
in a two-tree F of G if and only if one of the following conditions is satisfied:
1. There is an edge e / e such that F U e is a spanning tree of G, or
2. F U e is a spanning tree of G but F U e is not a spanning tree of G for all e ^ e.
In the first case, F is a two-tree i n G - e coming from the spanning tree F U e .
I claim that the second case cannot occur because e is in a circuit c. The second
case says that there is no edge e for which F U e is connected. For every edge e not
127
in the spanning tree F U e, the vertices of e must be in the same component of the
two-tree F because F U e is not connected. By Lemma 4.1.6,
/i 0 (FUe) = h0(F) = 2
and
hi(FUe)
= l.
Therefore, there is a circuit in F U e containing e and entirely contained in one
component of F. There is one such circuit for each of the h\(G) edges not in the
spanning tree F U e, these circuits are independent, and thus they span H\(G){ci.,
the proof of Lemma 4.1.24). The edge e is in none of these circuits, so Xe\jj• ?Q\ is
identically zero contrary to e being contained in some circuit (Lemma 4.3.2).
•
Suppose T is a subgraph of G. Let G//F denote the topological space that is
the quotient by each connected component of F. In other words, each component of
T is contracted to a vertex, and the topological space G//T is still a graph. If T is
connected, G//F is just G/T. Denote the components of F by F?-, and let V{ be the
vertex to which Fj is contracted in G//T.
T,
V
^
:K
Define the surjective linear map
-•
KV{-G/I^
by
(v
xvtv(r)
n{v) = <
[vi
and extend linearly. For p e
live
Ti
Kv^>,
2_^
pv = Tl(p)l
For each component C of G, there is a component of G//T that I denote C / / F . For
128
e^(G),
pv=
Y
veV(C)
Y
veV{C)-V{T)
=
veV(rnC)
Pv+
Y
pv
J2
PV+
Y ^k
v£V{c//F)
r?:cc
n v
^-
Y
veV(C//T)
Therefore, TT restricts to a surjective linear map, also denoted 7r,
n
The kernel of n is Kv^fi
y G
. Kv(G),o ^ Kv(G//r),o^
for both Kv^
and KV(G^°
because p e Kv^
goes to
r
0 in i r ( / / ) when both
1. Vv = 0 for all v £ V{Y) (i.e., p e KV{T)),
2-
TT(P)^
= YlveT-Pv
=
0 IOT
eacri
and
component T?; (i.e., p G
In particular, the following sequence is exact
0 _> KV^'°
-> KV^fi
^
KV(G//T),0
so for each nonzero p e A'' / (^ r )' 0 i exactly one of ^r,p
or
_> Q)
^Gl IT n(p)
1S
defined.
Lemma 4.3.5 (Projected Graph Configurations). Suppose T is a subgraph of G.
The projection of the configuration H\(G) to K ^ '
v
p is an element of K ^>-®.
Hi(G//T).
Ifp i Kv{?)$,
Proof. Note that E(G//T)
v
If p G K ^>^,
^ > is H\(G//T).
Suppose
then the projection of H\(G,p)
then the projection of #i(G,p) is H^Gf/T,
is
n(p)).
= E{G) - E(F), and let vt be the vertex to which the
129
connected component rz- is contracted. I claim that the following diagram commutes:
d
KE(G)
G
KV(G)
t
TX
^E(G-T)
d,
G//T
U
KE(G//T)
KV(G//T)t
The vertices of each edge in F are in a single component Tj of F. Therefore, for a
general edge e of G,
n(dG(e)) = ir(h(e) - t(e))
h{e)~t{e)
tih(e),t(e)<£V{T),
Vi-t(e)
Xh(e)eV(ri),t(e)tV(r),
<
h(e) -
Vi
if h(e) £ V{T), t{e) G V(Ti), and
0
ifee£(F).
Following the other direction around the diagram,
d
G//r(^E(G^r)(e))
fdG//T(e)
iie£E(r)
\dG//T(0)
iieeE(T)
h{e)-t{e)
=
iih{e),t{e)<£V(r),
Vi-t(e)
Hh(e) =
vieV(ri),t(e)iV(T),
h(e) - v,
if h(e) i V(T),t(e)
<
0
ifeeE(r),
e V(Ti), and
130
so the diagram commutes. Therefore, the following diagram commutes:
0
•
* KE^G)
ffx(G)
— ^ _
d H\
UGt
#i(G//r) — K E(G//T)
K ^ °
•0
KV(G//T),Q
I must show that the map
MG
ffi(G)
"r)>ffi(G//r)
is surjective; denote the image by VK. Combinatorially, this amounts to looking at
the edges of a circuit in G//T,
looking at them in G, and then filling in with a tree
in T to get a circuit in G, but making this precise can be messy. Instead, look at the
long exact sequence from the snake lemma:
0 -» # i ( r ) -+ KE(V
^
KV(V>°
-> Hi{G//T)/W
-+ 0.
The last map is zero because the first three nonzero terms are the short exact sequence for the subgraph T. By exactness,
jfi(G//r)/w = o,
that is, Hi(G)
surjects onto W =
Hi(G//T).
The same argument works for the second graph polynomial using one of the
following commuting diagrams. If p £ K*V->>®, then the diagram is
+ Hi(G,p)
Hi(G//T)
*
dG
KE{&
KE(G//T)
d,
G//^
U
* KV^°/Kp
KV(G//T),0
0,
131
and the exact sequence of kernels is
0 -+ H!(r,p)
- KE^
Kv™/Kp
%
-+ 0.
If p ^ K ( >'®, then the diagram is
0
0
• Hx{G,p)
. H^GI/TMp))
• KE™
—^
• KVM>°/Kp
•0
— KEW'T)
^
KVWM>°/Kn(p)
. 0,
and the exact sequence of kernels is
0->Hl(r)^KEM^Kv™^0.
n
Proposition 4.3.6 (Contracting a Contractible Subgraph). Suppose a forest T is a
subgraph of G. If'h\(G) > 0, then
*G(A)\AE(rro
Letp be an element of Kv^'°
=
*G//r(A)-
and p £ KV(T^°.
^G,p(A)\AE{r)=0
=
Then
*G//T,n(p)(A)-
Proof The projections of HY(G) and # i ( G , p ) onto i ^ ( G / / r )
are
Hi(G//T)
and
H\{G/ /T, 7r(p)), respectively (Lemma 4.3.5). The kernels of both projections are
Hi(T) = 0, so the formulas follow from Lemma 3.2.4 for projecting a configuration.
•
Remark 4.3.7. Consider a forest T that is a subgraph of G as in Proposition 4.3.6, but
132
suppose p e
KV(T),0^
A Spiitting a
. Hi(G//T)
->• Hi{G,p)
of the exact sequence
o - # i ( i » -> HX{G,P) - #i(G//r) -, o
will provide a direct sum decomposition
H1(G,p)^H1(T,p)®a(H1(G//T)).
Without additional assumptions, it is impossible to guarantee a splitting which does
not meet T, i.e., in general, there is not a splitting
a : Hi(G//T)
-> Hx{G,p)
D
E
K
E { T
^-
\
Therefore, the hypotheses of Lemma 3.1.10 are not satisfied for this direct sum
decomposition, and the second graph polynomial will not be a product in general.
In the case that the contractible subgraph V is a single edge, Proposition 4.3.6
states
Corollary 4.3.8 (Contract an Edge). Suppose e is an edge of a graph G that is not
a loop. If hi(G) > 0, then
*G(A)\Ae=0
Let p be a nonzero element
ofKv(G^°.
=
VG/e(A).
Ifp^Kv^°,
then
^GAA)\Ae=0 = ^G/eMpMy
Corollary 4.3.9 (Isolate Nontrivial Edge). Suppose G is a graph and e an edge that
is in a circuit but not a loop. If h\{G — e) > 0, then
^G(A)
= AeVG_e(A)
+
VG/e(A).
133
Ifp is a nonzero element of KV(G^°
^G,p(A)
and p £ Kv<"e>'°, then
= Ae^G^p(A)
+
qG/eAp)(A).
Proof. The variable Ae has degree at most 1 in $Q, SO the graph polynomials may
be written as
mG(A)
^GM)
= AeF + G
= AeH + I.
Then F and H are determined to be ^G~e
anc
' ^G-e,p by differentiating with respect
to Ae using Proposition 4.3.4. Similarly, G and / are determined to be ^ G/e and
^G/e n(p) ^ setting Ae = 0 using Proposition 4.3.6.
For a combinatorial argument, partition the set of spanning forests into those
containing e and those not containing e. A spanning forest T does not contain e
if and only if T is a spanning forest for G — e, and a spanning forest T contains e
if and only if T/e is a spanning tree for G/e.
Similarly for the second polynomial,
a two-forest T does not contain e if and only if T is a two-forest for G — e (the
possibility that T is not a two-forest for G — e as in Proposition 4.3.4 is excluded by
the assumption that e is in a circuit). A two-forest T contains e if and only if T/e
is a two-forest for G/e. In addition, the value of
s
G/e~T/e(7r(p))
m
e a c n case
SG~T(P)
is equal to SQ-T-e(p)
> which is straightforward from the definitions.
or
•
P r o p o s i t i o n 4.3.10 (Disjoint Union). Suppose G is the disjoint union of two subgraphs G\ and G<i, and suppose h\(G2) > 0. If h\{G\)
*G(A) =
Ifp G K
\ >- is a nonzero element of K
*GM^G2(A).
'• l)' , then
^GAA)-^G1AA)^G2(A).
> 0, then
134
In particular,
the graph hypersurfaces of G are joins of graph hypersurfaces
in the disjoint linear subspaces P
in P K
contained
).
Proof. The homology of a disjoint union is the direct sum of the homologies of
the components. Thus, the configuration H\(G)
configurations Hi{Gi)
C K
E
^ \
Hi(Gi,p)
because p has a preimage in K
C KE
Similarly, Hi(G,p)
© Hi(G2)
C KE{-G^
is the direct sum of two
is the direct sum
©
KE{-G^
^ l>. According to Lemma 3.1.10, the configuration
polynomial of the direct sum is the product of the configuration polynomials for each
factor.
Combinatorially, the bijection of the spanning forests of G with the pairs of
spanning forests of G\ and G2 proves the formula for the first graph polynomial. For
the second graph polynomial, the two-forests of G correspond to a pair of a spanning
forest in one G{ and a two-forest in the other. The assumption that p (E
guarantees that SQ2_p(p)
K*(Gi)fi
= 0 for all two-forests F in G2) so only pairs of two-forests
in G\ and spanning forests in G2 contribute.
•
There is also a formula for the second graph polynomial of a disjoint union for
all nonzero p G
Kv^>^:
P r o p o s i t i o n 4.3.11 (Disjoint Union, General Momenta). Suppose the graph G is
the disjoint union of G\ and G2, h\(Gj)
element of K
is nonzero for both i, and p is a nonzero
~' ' ' . Let pi be the image of p under the projection
from
KV(G),0
its subspace K
\ i'^.
*G,P(A)
If both pi are nonzero,
= *Gl,Pl(A)yG2(A)
to
then
+ *Gl(A)*G2,P2(A).
If either Pi is zero, a formula for ^ ^ ^ ( ^ 1 ) is given in Proposition
(4.5)
4-3.10.
135
Proof. Combinatorially, the two-forests in G correspond to a pair of a spanning forest
in one G{ and a two-forest in the other. If F is a two-forest in G\, then
=
3GI-F(P)
SGi-F(Pl)
because each component C of G2 has
veV{C)
by definition of KV^G'^.
The same holds for two-forests of G2, which completes
the combinatorial proof by the general combinatorial formula for the second graph
polynomial (Proposition 4.2.12).
For an algebraic argument, use the direct sum decomposition
KV(G),0
= KV{Gx)fi
under which p = p\ + P2- There are qi e K
0
KV(G2),0
^i> for which dG(qi) — pi for both i.
By the multilinearity of the determinant,
*G)P(^) = d e t £ £ | # l ( G - p )
= ^BE\H1(G,PI)
+AQXBE\HX{G,P2)-
Each Pi satisfies the condition in Proposition 4.3.10 for one of the G{, so
detBE\Hl{Gpi)
+detBE\Hl(G)P2)
= *GliPl(A)VG2(A)+
*Gl(A)VG2>P2(A).
•
P r o p o s i t i o n 4.3.12 (Vertex Union). Suppose G is the union of two subgraphs G\
and G2 that intersect at only one vertex and no edges. Suppose that h\{G2) > 0 so
136
that its first graph polynomial is defined. If h\{Gi) > 0, then
yG(A)
=
If p is a nonzero element of KV(G^°,
*Gl(A)*G2(A).
then
Proof. The proof for Proposition 4.3.10 works here, as well. The intersection Gi nG2
is a vertex, so h\(G\ n G2) = 0 and j^(GinG 2 ),0
=
Q. A Mayer-Vietoris argument
provides the direct sum decompositions
#l(G0 = # i ( G i ) © # i ( G 2 )
and
Hi(G,p) = H^G^p)
© #i(G2).
D
Corollary 4.3.13 (Loops). Suppose an edge e is a loop in a graph G, suppose
h\(G — e) > 0, and suppose p is a nonzero element of K '>> Then
VG{A) = Ae*G_e(A)
arid
VG)P(A) =
Ae*G_ejP(A).
Proof. Apply Proposition 4.3.12 with G\ = G — e, G<i = e, and use
^e(A)
= Ae
for a loop e.
•
Corollary 4.3.14 (Vertex Union, More General Momenta). Suppose G is the union
of two subgraphs G\ and G 2 that intersect at only one vertex and no edges. Let px
be a nonzero element of Kv^-i>$
fori = 1,2, and consider p = p\ + p2 £
Then
*GAA)
= 9GlJ}1(A)9G2(A)
+
^Gl(A)^G^2(A).
KV^),Q.
137
Proof. The same proof as for the disjoint union (Proposition 4.3.11) applies here.
Note the difference between the two results is that part of the hypothesis of this
corollary is that p decomposes into p\ and P2- In the disjoint union case, p automatically decomposes into p\ and P2- It i s necessary in either case to assume both
Pl are nonzero for the second graph polynomials to be defined.
•
P r o p o s i t i o n 4.3.15 (Trivial Edge). Let G be a graph and e an edge not contained
in a circuit. If h\(G) > 0.
*G(A)
= *G~e(A)
= *G/e(A).
If p is a nonzero element of
*G,p(A)
= *G-e,p{A)
=
*G/eMp)(A).
Remark 4.3.16. When e is not contained in a circuit, G — e is a disjoint union of two
graphs and G/e is a vertex union as in Proposition 4.3.12, so the graph polynomials
decompose further.
Even though the sets V(G — e) and V(G) are the same, the
v G e
is not the same as p G if^( G )>° because G — e has more
assumption p <E K ^ ~ >^
components than G.
Proof. The condition that e is not in a circuit implies that the connected component
containing it becomes two components m G — e by Lemma 4.3.3. Contracting e in
G joins those two components at the vertex to which e is contracted.
The hypothesis that e is not in a circuit implies that Xe\jj
£
(Lemma 4.3.2), so H^G)
E
to K ^-
e
is Hi(G-e)
G
e
/Q\ is identically zero
C K ( ) ^ . The restriction of the configuration
(Lemma 4.3.1), so HX{G) = Hi(G-e),
polynomials are the same:
*G(A) =
*G-e(A).
In particular, these polynomials do not depend on Ae, so
*G(A)
= *G(A)\Ae=0
=
VG/e(A)
Hi(G)
and the first graph
138
by Proposition 4.3.6.
If p e Kv(G~e^°}
then it has a lift q e KE^G^e,
zero (Lemma 4.3.2). Therefore, H\{G,p)
so Xe\H^G^
is also identically
is contained in KE^~e,
so its restriction
to this subspace is itself, but it is also H\{G - e,p) by Lemma 4.3.1. Therefore, as
long as p is nonzero, the second graph polynomials are defined and
^G^A)
=
^G-e,piA)-
In particular, these polynomials do not depend on Ae, so
®G,p(A) = ^G,p(A)\Ae=0
*G/e,n(p)(A)
=
by Proposition 4.3.6.
•
P r o p o s i t i o n 4.3.17 (Valence Two Vertex). Suppose edges e\ and e2 meet at a
valence 2 vertex and are contained in at least one circuit.
VG(A)
= (Al + A2)*G-ei(A)
+
Then
*G/eie2(A).
Proof. Every spanning forest must contain one or both of e\ and e2 in order to
include the vertex where they meet. Partitioning the sum in the graph polynomial
by those spanning forests containing just e\, just e2, or both e\ and e2, the graph
polynomial is
VG(A)
= A2VG_e2(A)
+ A^G_ei(A)
+
^>G/eie2{A).
In fact, every spanning forest T that does not contain ei gives a unique spanning
forest (T — e2) U ei that does not contain e2. Therefore,
^G~eM)
and the formula is verified.
= ^>G-eM)i
•
139
Proposition 4.3.18 (Parallel Edges). Suppose edges e\ and e2 have the same pair
of distinct vertices, and suppose h\{G — e\) > 0. Then
*G(A)
= AlA2^G^ei_e2{A)
+ A^{G_ei)/e2{A)
+
A2*G/ei_e2(A).
Proof. The pair of edges form a circuit, so they are not both contained in a spanning
forest. Thus, every spanning forest contains neither edge or only one edge. If F
contains neither edge, then F is a spanning forest of G — e\ ~ e2, which corresponds
to the first term on the right side of the equality. If F contains ej but not ej, then
F/ei is a spanning forest of (G — ej)/e^, which gives the last two terms.
•
Proposition 4.3.19 (Edge Union). Suppose G is the union of two subgraphs G\
and G2 that meet at exactly one edge e, riot a loop. Suppose that h\(G{ — e) > 0 for
both i = 1,2. Then
*G(A)
= Ae (*Gl-e(A)*G2/e{A)
+ v& G l / e (A)^ G 2 „ e (A)) +
*Gl/e(A)*G2/e(A).
In particular,
*G-e(A)
= *Gl-e(A)yG2/e(A)
+ 9Gl/e(A)*G2_e(A).
(4.6)
Proof. It suffices to consider G to be the connected component containing e because
the other components can be factored out of the equation using Proposition 4.3.10.
The decomposition of Corollary 4.3.9 applies to give
VG(A) = Ae^G_e(A)
+
tfG/e(A).
The graph G/e is the vertex union of G i / e and G2/e in the sense of Proposition 4.3.12, so
*G/e(^) =
S'd/eC^Ga/eW
The spanning trees of G — e correspond to the spanning trees of G not containing
e. Let T be a spanning tree of G not containing e, and let T{ = Gj D T for i = 1, 2.
140
As subgraphs of a tree, the X-L are forests. Also, T\ flT2 is a pair of disjoint vertices,
the vertices of e. The Euler characteristic of a forest is the number of components,
so the Mayer-Vietoris relationship among the Euler characteristics is
h0(Tl) + h0(T2)=X(T1)
+ x(T2)
= x(T) + 2
= h0(T) + 2
= 3.
Therefore, exactly one of the Tj has two components, and the other has one. Suppose
T\ has one component and T2 two components. Note that both G\ — e and G2 — e
are connected because e is contained in a circuit of G\ and of G2 (Lemma 4.3.3).
The spanning tree T contains all the vertices of G (Lemma 4.1.9), so the forests Tj
contain all the vertices of Gj. Therefore, T\ is a tree with all vertices of G\ — e and,
thus, a spanning tree of G\ — e (Lemma 4.1.13). The forest T2 has two components
and all vertices of G2 — e and, thus, a two-tree of G2 — e (Lemma 4.1.16).
To conclude, I need to show that T2 U e is a spanning tree of G2, and thus
T2 U e/e is a spanning tree of G2/e.
By adding the edge e to T, h\{T U e) = 1
(Lemma 4.1.6), and for the same reason, h\(T\ U e) = 1. If h\(T2 Ue) = 1, then
T U e will contain two circuits, one in G\ and one in G2, contrary to h\{T U e) = 1.
Therefore, h\(T2\Je) — 0, thus ho(T2Ue) = 1 (Lemma 4.1.6), and T2Ue is a spanning
tree of G2 as claimed.
•
Example 4.3.20 (A Banana, Degenerate Case). Consider the graph G in Figure 4.8.
Each edge is a spanning tree, so
*G{A) =AX+A2
and
For both i = 1, 2, j ^ i,
1. dA^G(A)
= 1,
2. dA^G^A)
=
*G_ehP{A)=p\Aj,
$>GiP{A) = pJAiA2.
141
v
Figure 4.8: A circuit with two edges; a banana.
ei
Ae
ei
Figure 4.9: Two banana graph.
The graphs G — ej have /&i(G — ej) = 0, so the graph polynomial is not defined. For
consistency among the formulas, it would seem that ^G~e- (^)
=
1 would be a good
convention, and this would follow by defining the configuration polynomial of the 0
configuration to be a constant (1 for graphs because of the integral basis). There
is no problem defining ^Q^e.p(A)
because H\(G,p)
p, but there is a problem defining ^Q/e.
P{A).
is always nonzero for nonzero
A graph with a single vertex has
no nonzero momentum and no two-forests, which suggests defining the second graph
polynomial to be identically 0 if p = 0. This convention fits the combinatorial formula
(Proposition 4.2.12), but it violates the configuration definition as the configuration
polynomial of H\(G, 0) = H\{G),
which is nonzero for h\{G) ^ 0.
Example 4.3.21 (Gluing Bananas, Degenerate Case). Let G be the graph in Figure 4.8
142
as in the previous example. Suppose G is identical to the graph G in Figure 4.8, but
all labels and variables are denoted with the accent. Identifying edge e2 with e2 in
these two graphs yields the graph Y in Figure 4.9. Letting the identified edge be e
with corresponding variable Ae, the graph polynomials are
i£ r = A1A1 + AiAe
+ A\Ae
and
*r^ =
p\A\A\Ae.
In particular, if I follow the suggested convention above that the first graph polynomial be 1 for acyclic graphs,
^
r
= Ae(Ai
+Ai)
+
= M*G/e2*G-e2
AiAi
+ *G-e2*G/e-2)
+
^G/e^G/e2
In other words, this convention would agree with the result of Proposition 4.3.19.
Moreover, F — e is also a graph isomorphic to G and G but with different labels, and
* r - c = *G/e2*G_e2 + *G-e2*G/e2In general, gluing a graph G\ to the banana graph G along an edge to produce a
new graph Y and then deleting that edge as in Proposition 4.3.19 produces T — e, a
graph isomorphic to G\.
Example
4.3.22 (Adding a Bubble). Let G be the graph in Figure 4.10. Gluing G
to another graph Y along an edge e and then deleting e produces a graph that is
F with a bubble on edge e. For example, let Y be the graph in Figure 4.9. This
operation produces the graph in Figure 4.11. To demonstrate some propositions of
this section, I compute the polynomial of "adding a bubble" following two methods.
First, hit it with the hammer of Proposition 4.3.19 assuming h\(Y — e) > 0 so the
first graph polynomials are well-defined. Note that
^G-e
= Ai + A2
and
^G/e
= AXA2
+ AtA3
+
A2A3.
143
Figure 4.10: Two bananas, one edge split.
Then the bubbled graph H has graph polynomial
*H = ^G/e^T-e
+
^G-e^T/e
(AXA2 + AiA3 + A2A3)VT_e
+ (Ai +
A2)VT/e
As a second method, put the bubble on V by a sequence of two operations:
subdivide e into e2 and e%, then attach an edge e\ parallel to e2. Initially,
q?r = ApV
e
^ r „ e + ^^Tr / e -
Let F be the graph F with e subdivided into e2 and e3. The formula for a valence
two vertex (Proposition 4.3.17) shows that the graph polynomial of F is
VF = (A2 + A3)Vr_e
+ VT/e.
Note that e% is trivial in F — e2 in the sense of Proposition 4.3.15, so ^JF_e^ = $Y-
144
Figure 4.11: The graphs of Figures 4.9 and 4.10 glued along e, then with e removed
to create a bubble on edge e% of the two banana graph.
Then the formula for parallel edges (Proposition 4.3.18) shows
qH = A1A2^H-ei-e2
+ A^{H_ei)/e2
+
= AlA2^F^e2
+ A^F/e2
+
= A1A2^F_e2
+ ^i^rUe=^3 +
= AiA2Vr_e
+ (Ai + A2)(A3Vr_e
= {AiA2 + AjA3 + A2A3)VT_e
A2VH/ei_e2
A2^F/e2
A2*r\Ae=A3
+
+ (^1 +
tfr/e)
M)^T/e-
Remark 4.3.23. Suppose I glue G\ and G2 along an edge e and then remove it, as
in Proposition 4.3.19, to produce a new graph G. If both G{ are log-divergent, i.e.,
2hi(Gi) = \E(Gi)\,
then
| E(G - e) | = | E(Gi - e) | + | E{G2 - e) | = 2h1(G1) + 2h1{G2) - 2.
The value of h\(G — e) can be computed by the following Mayer-Vietoris sequence:
0 -+Hi(Gi - e) © Hl(G2 - e)-> H^G - e) ->
HQ((Gi - e) n (G2 - e)) -+ # 0 ( G i - e) © H0(G2 - e) ^ H0(G - e) -> 0.
145
Note that {G\ - e) n (G2 - e) is a pair of disjoint vertices. Therefore,
/ i l ( G - e ) = /ii(Gi) + / i i ( G 2 ) - l ,
so G—e is log-divergent, too. The log-divergent graphs play a central role in quantum
field theory, and this operation is a simple way to produce log-divergent graphs from
log-divergent graphs. Note that the log-divergent condition 2h\(G)
= \E(G) | is a
strong condition, and finding examples becomes more difficult as /11(G) increases.
In [Blo07], the graph motive of a graph G with n edges is defined as M(G)
^n-l(Pn-l _
XGJ
q(n
_ ^y
=
ideally, one would like to understand the polynomial
formulas in the preceding propositions in terms of the corresponding geometric and
motivic objects. For example,
T h e o r e m 4.3.24 ( [BloOT]). If G is the union of two graphs G\ and G2 that are
either disjoint or intersect at a single vertex,
M(G) = M(Gi)
then
®
M(G2).
I am particularly interested in a motivic interpretation of Equation (4.6) in Proposition 4.3.19, but I do not know one. That equation,
partitions the variables into two sets based on whether the edges are in G\ or G2,
and it would be wonderful to be able to relate the motives for Gj, G^/e, and Gj — e
to the motive for G — e.
In general, suppose F and G are homogenenous polynomials of degrees / and g
in some set of variables denoted A. Let $ and T be homogeneous polynomials of
degree ip and 7 in another set of variables B. Suppose that
fi = <pg
146
so that FT + $ G is homogeneous in the set of both variables. Suppose that the
variables A are the linear forms on a vector space W, and the variables B are the
linear forms on another vector space V. The analysis of the motive for Equation (4.6)
would benefit from a description of the geometry and topology of the hypersurface
defined by FT + $ G = 0 in F(W © V).
Moreover, this general situation could be
applied to understanding the graph hypersurface of other graph operations such as
the second graph hypersurface of a disjoint union, Equation (4.5).
4.4
Quadratic Linear Series Defined by Graphs
Let G be a graph. Each edge e determines a rank at most one quadratic form Xg
on H\(G)
and H\(G,p)
Let E = { e i , . . . ,en},
by taking its dual on K
and let Xf
Qc,p similarly on F(Hi(G,p)).
Define QG : P ( # i ( G ) ) ->• F(KE)
= X\.
the map that takes a line [x] in Hi(G)
, squaring it, and restricting it.
to [Xf(x)
E
: • • • : X%(x)] in F(K ).
to be
Define
In other words, each graph comes with a naturally
defined quadratic linear series as defined in Section 2.2. The quadratic linear series
is base-point free because H\{G) and H\(G,p)
are configurations in KE.
Namely, a
nonzero element h can be written uniquely as
h — y^ hee
eeE
in the basis E with at least one nonzero coordinate he.
nonzero coordinate for Qciih])
in
Then X^ih)
= hi is a
F(KE).
Intuitively, more edges will make the maps QQ and QQ p more likely to separate
points, but the combinatorics and topology of the graphs put restrictions on the
relationship between the number of edges and the first Betti number so that often
there cannot be enough edges to separate points. The main result of this section is
T h e o r e m 4 . 4 . 1 . Let G be a connected graph and p a nonzero element of
KV^',Q.
If either map QQ or Qc,p is infective, then h\{G) < 4. In particular, if either map
is an embedding, then h\(G) < 4.
147
In particular, QQ is often not an embedding. In Section 2.2, I discussed how the
incidence variety for constructing the dual of the image of QQ will provide a resolution of singularities for the dual if QQ is an embedding. Proposition 2.2.2 proves
that the dual is the graph hypersurface XQ. This approach is suggested in [BEK06].
The same is true for the second graph hypersurfaces YQP, but there is less interest
in finding a resolution of singularities for YQP because the Feynman integrals (4.3)
are periods for the complement of XQ. The implication of Theorem 4.4.1 is that
the incidence variety may not provide a resolution of singularities for the graph
hypersurfaces.
Theorem 4.4.1 and an application of a theorem of Fulton and Hansen (Theorem 2.2.7) provide a simple criterion to rule out whether QQ or QQP is an immersion.
Corollary 4.4.2. Let G be a connected graph and p a nonzero element of K ^ '^.
If h\(G) > 4 and h\(G) > \ V(G) \, then QQ is not an immersion. If h\{G) > 4 and
h\(G) > | V(G) | — 2, then QQ,P is not an immersion.
Proof. Theorem 4.4.1 proves that
QQ
and
QQJP
are not embeddings when h\ (G) > 4.
The theorem of Fulton and Hansen (Theorem 2.2.7) gives a simple criterion for
QQ and QQ^P to be embeddings assuming they are immersions satisfying certain
dimension conditions. For example, Theorem 2.2.7 applied to QQ states that if
2dimP(#i(G)) = 2 ( / i i ( G 0 - l ) > I E{G) \ - 1 = dimP
(KE^GA
and QQ is an immersion, then QQ is a closed embedding. The Euler characteristic (4.1) may be used to express the inequality more simply
2{hi{G)-\)
>
\E(G)\-1
= \V(G)\ +
hi(G) >
\V{G)\.
hi(G)-l-l,
148
Figure 4.12: Two banana graph, revisited.
Similarly, if h\{G) > \ V(G) | — 2 and Qc,p
1S a n
immersion, then QQP is an embed-
ding. Combining the dimension conditions from both theorems, it is impossible for
the maps to be immersions.
•
Before proving Theorem 4.4.1, it is useful to study some examples where QQ
is an embedding. The first exceptional case to note is when h\(G)
QQ is an embedding of a point P° in F(K^).
is 1: the map
Next I provide examples when
h\(G)
is 2, 3, and 4 for which QQ is an embedding. In the search for graphs providing
an embedding, there are two key lemmas that are useful for excluding particular
examples besides Theorem 4.4.1. Lemma 4.4.9 explains that it is unnecessary to
consider graphs containing vertices of valence one or two. Lemma 4.4.6 shows that
circuits must have at least h\{G) edges. These conditions are quite restrictive.
In the following examples, each graph has a circuit basis in which each pair of basis circuits shares an edge and the basis circuits satisfy the last part of Lemma 4.1.24.
I am not invoking the lemma; I am constructing explicit bases with those properties. After listing the examples, I will explain for all three simultaneously why the
quadratic linear series is an embedding.
Example 4.4.3 (Two Bananas, h\(G) = 2). The graph G has two vertices connected
by three edges e\, e2, and e% as in Figure 4.12. Choose the circuits i\ = e\ + e% and
^2
=
e
2 + e 3- I have used the arrows to indicate the head of each edge to compute
H\{G).
Example
They are not intrinsic data of the graph.
4AA
(A Wheel with Three Spokes, h\(G)
— 3). Let G be the complete
149
vi
Figure 4.13: A wheel with three spokes,
graph on 4 vertices as in Figure 4.13. Choose the circuit basis
h = e l + e 5 - e4^2 = e 2 + e 6 - e 5
h = e3 + e4 - e 6 .
Example 4.4.5 (Complete Bipartite Graph /C33, h\(G) = 4). Let G be the complete
bipartite graph on 3 and 3 vertices pictured in Figure 4.14. Choose the circuit basis
h = e l - e6 + e9 - e5
£2 = ^2 ~ ^7 + e 9 -
e
^3 = e3 - e 7 + e 9 - e 5
£4 = e 4 - e 6 + eg - e 8
8
To show that the maps QQ are embeddings for these examples, I note the following properties of the bases I have chosen.
(a) Each basis circuit has an edge that is in no other basis circuits, and
(b) each edge that is not unique to one circuit is in exactly two circuits, with the
exception of edge e§ in Example 4.4.5.
150
Figure 4.14: A complete bipartite graph, ^ 3 3 .
The exception in (b) prevents me from providing a one-size-fits-all explanation that
all three cases are embeddings, but the proof follows similar steps for each case.
Suppose that QQ(X) = QG(V)
m
an
y °f the examples. By property (a), I may
assume the first /11(G) quadratic forms are X^,...,
Xi
C such that xj — Cyf for all i, hence Xj = ±\/Cy^.
^ > . Thus, there is a constant
Because these are projective
coordinates, I can ignore the \/~C'.
In the first two examples, each of the other edges in the graph is shared by two
circuits as in property (b).
forces (±£/j)(±t/j) = ViVj-
If e is shared by £j and lj, then X^(x)
Thus ±'(/j and ±yj
(x'l, . . ., x^.(G)) = ± ( y i , • • • iVh-iiG))-! which
ls
=
must have the same sign.
CX%(y)
Thus
the same projective point.
In the third example, the same reasoning holds except that all circuits share e§,
which belongs to all circuits. To avoid this difficulty, use the other edges first. In
particular, use e2 to establish that ±y\ and ±2/3 have the same sign, e^ to establish
that ± y i and ±2/4 have the same sign, eg to establish that ±2/2
same sign, and eg to establish that ±2/2
an
an
d i?/3 have the
d ±2/4 have the same sign. Then use e§
to see that ±yi and ±7/2 have the same sign.
151
It may appear that some of this information is redundant, but I have assumed
for convenience that 0 has both signs. Thus, I need all of the relations above. For
example, if 2/3 = 0, knowing that ±yi and ±y2 have the same sign as ±y% is not
sufficient to conclude that ±y\ and ±y2 have the same sign.
These examples are also immersions. This fact can be checked by direction computation of the Jacobians locally. Properties (a) and (b) of the examples makes this
computation similar in each case, so I carry out the case for Example 4.4.5. First
I write the quadratic forms X? in the basis {L{\ of Hi(G)v
dual to the basis {£j}
given above:
X\ = L\
X2 = L2
Xl = L\
X^ — L 4
Xb = Lx + L3 + 2L1L3
XQ = LI + L^ + 2L\L/[
X7 = L2 + L 3 + 2L 2 L 3
Xg = L2 + £4 + 2L2L^
3
i<3
Consider the affine patch L\ ^ 0; the computation is the same for all L{ 7^ 0 by the
symmetry of the graph. Normalize to local coordinates ou = j ^ - , j 7^ 1, and let q^
152
be Xf in these local coordinates. The Jacobian is
0
0
0
2a2
0
0
0
2a3
0
0
0
2(24
0
2a3 + 2
0
0
0
2a4 + 2
2a2 + 2a3
2a2 + 2a3
0
2a2 + 2a4
2a2 + 2a4
0
\ 2a 2 + 2a 3 + 2a 4 + 2 2a 2 + 2a 3 + 2a 4 + 2 2a 2 + 2a 3 + 2a 4 + 2 /
The second, third, and fourth rows show that this is injective when 0,2,0,3, a,\ ^ 0.
Even when a 3 and 124 are zero, rows five and six maintain injectivity. There would
be a problem if 2<22 = 0, 2a2 + 2a 3 + 2*24 + 2 = 0, 2<22 + 2a 3 = 0, and 2<22 + 204 = 0,
but there is no solution to this system of equations.
Now I proceed to proving Theorem 4.4.1. To simplify the proof, I restrict to
connected graphs as each component determines a map between projective spaces.
Some of the facts that I prove below require h\{G) > 1, which is strong enough for
Theorem 4.4.1
The proof requires some features of the combinatorics of the graph G. To connect
the combinatorics of the graph with the configuration H\{G),
I use circuit bases as
defined in Lemma 4.1.24.
The girth of a graph is the smallest number of edges in a circuit.
L e m m a 4.4.6 (hi(G)
Bounds Girth.). Consider QG : P ( F i ( G ) ) -> ¥{KE)
a field K of characteristic
injective,
not two. If h\{G)
is grea,ter than 1 and the map QQ is
then the girth of G must be at least h\{G).
a line in H\{G,p),
so if Qc,p
must be at least h\(G)
zs
injective,
when h\[G) > 1.
over
Every line in H\(G)
is also
then so is QQ, and again the girth of G
153
Remark
4.4.7. Using Lemma 4.1.24, it is easy to find examples in characteristic
two that are injective but do not satisfy the girth condition. For the proof, I need
b\ 7^ —b\ for nonzero b\ G K. For the remainder of this section, assume char(K) ^ 2.
The idea for the proof is that if a basis circuit t\ has fewer than h\{G) edges,
then I can find [b2 : • • • : b^ /£)] £ P ( i / i ( G — ei)) such that
QG (\h
: b2 : • • • : bhl(G)))
= QG ( h & l • b2 : • • • : bhl(G)})
for any b\. In particular, since not all b2, • • . , ^ W G ) are zero, any nonzero b\ will
provide two points mapping to the same point via QQ.
Proof. Suppose £\ is a circuit with fewer than h\{G) edges. I show that QQ is not injective. Complete {£\} to a circuit basis {£\,...
}£fl,/Q\}
for H\(G) by Lemma 4.1.24.
For each edge e, consider when
X2e (bi,b2,...,bhl(G))
=X2e (-bi,b2,---,bhl(G))
•
Let
hi(G)
2
X
e
{bX,b2, • • • , ^ ( G ) ) = J ]
a
e,ifo? + Yl Ce,i]bibJ-
i=\
i<j
and ae_j = X e (£ v ;) 2 . Then
In particular, each c e ^ = Xe(£i)Xe(£j)
Xl [bi,b2,...,bhl{G)j
=X^-b1,b2,...,bhi(G^
if and only if
hx(G)
2 Yl
Factoring out the b\, this is 2b\ Y2j=2
c
e,ljblbj
c
e,ljbj
= 0.
= 0. Let
MG)
£ e (&2,---,&fc1(G)) = 5 Z c e,lj fe jJ'=2
154
Let He be the hyperplane in W(H\(G — e)) defined by the linear form Le. I will
show that the girth assumption implies that there is a projective point [fy? : • • • :
fyii(G)] m ^he intersection f]eejr;He.
If e ^ £1, then Xe{i\)
= 0, so ce^j = 0 for all
j and Le(b) = 0. Thus
f]He=
eeE
f]He.
ee£\
If e G £i, then Le(6) may be nonzero. However, according to the form of the circuit
basis in the last part of Lemma 4.1.24, I may assume X^(b) = bf, so Lei(b) = 0.
If the circuit l\ has at most h\{G) — 1 edges, then there are at most h\(G) — 2
linear forms Le(b) that are nonzero. In particular, Heefi ^e *s ^ n e intersection of at
most h\{G) — 2 hyperplanes in ¥(H\(G — e\)). Therefore, the dimension of Heefi -^e
is at least 0; that is, there is a point [62 : • • • : b^ IQ\] e Pleefi ^e<
Pick any nonzero 61. Then
QG {[h : 62 = • • ' = ^ ( G ) ] ) = <?G ( h b l : &2 = ' ' ' = ^ ^ G ) ] ) >
and [61 : 62 : " '" : fyji (G)l ^s n o t
ec ua
l l t o [—&1 : ^2 : "' " : bh iG\}.
•
Lemma 4.4.6 shows that injectivity of QQ puts a restriction on the girth of the
graph. The next theorem uses the girth and valence of the graph to give a lower
bound on the number of vertices. The valence of a vertex is the number of incident
edges (an edge with a single vertex, i.e., a loop, counts twice). The valence of a
graph is the minimum valence of its vertices. This theorem is in many texts that
discuss extremal graph theory, for example [Lov93] from which the proof below is
adapted. The result originally is due to Erdos and Sachs [ES63], I provide the proof
for the readers who are not graph theorists.
Theorem 4.4.8 (Erdos-Sachs). / / G has girth g and every vertex has valence at
least k, then
I V(G) \>l
+ k + k(k-l)
+ --- + k{k~l)^r
for g odd
155
and
| V(G) | > 2 ( l + (k - 1) + • • • + (k - l ) ^ " 1 )
for g even.
Proof. In the following proof, I use the word connected in the graph-theoretic sense
that two vertices in a graph are connected if there is an edge in the graph with those
vertices. In particular, I am not using it in a topological sense.
First suppose g is odd. Pick a vertex v in V(G). Let Si be the number of vertices
a distance i from v. Note that | SQ \ = 1 and \S\\ > k.
I claim that each vertex in Sj_i connects to at least (k — 1) vertices in Si for
i = 2,...,
y~Y~ J a n d thus
|3|>(fc-l).|$_l|
for i = 2 , . . . , ^—. Each vertex in S j - i is connected to at least one vertex in Sj_2
for i > 2 by definition of Sj-i- If a vertex x in S?;_i is connected to two vertices in
5^-2, then there will be two paths from v to x of length 2 — 1, giving a circuit of
length at most 2z — 2. In particular, each vertex in Sj_i must only be connected to
only one vertex in Sj„2 f° r ^ < f + !• If two vertices in Sj_i are connected, then
there will be a circuit of length at most 2z — 1. In particular, the vertices in Sj-i
are not connected if i < ^—. Therefore, when
o+ l
o
each vertex in S^_i is connected to one vertex in Sj_2
an
d no vertices in Sj_i.
Because g is odd, the maximum value of i in that range is ^—. To satisfy the
valence condition, each vertex in Sj„i must connect to (k — 1) vertices in Si for
i = 2,...,
^2", and
|$|>(fc-l)l$-i|
for such z. Therefore,
^ ( C ) | > | So | + | Si | + • • • +
> l + k + k(k-!)
+ •••+ k(k-
1)V.
156
If g is even, start by picking a pair of adjacent vertices v and w. Let S{ be the
number of vertices a distance i from the the pair {v, w}. Note that | SQ \ — 2.
I claim that each vertex in 5"j_i connects to at least (k — 1) vertices in Sj for
i = l , . . . , | — 1, and thus
I St | > (k - 1) • | 5 , _ ! |
for i = 2 , . . . , | — 1. Each vertex in <S^_i is connected to at least one vertex in S*j_2
for i > 2 by definition of S^-i- If a vertex x in Sj_i is connected to two vertices
in Si-2 for i > 2, then there will be two paths from {v,w} to x giving a circuit of
length at most 2i — 1. In particular, each vertex in 5j_i is connected to exactly one
vertex in S{^2 f° r 2 < i < ^—. If a two vertices in 5j_i are connected for i > 2,
then there will be a circuit of length at most 2z. In particular, the vertices in S{-i
are not connected if 2 < z < | . Therefore, when
o
9+ 1
2< i< - <
,
2
2 '
each vertex in S^-i is connected to one vertex in Si-2
ari(
i
n0
vertices in 5^-1.
Because (7 is even, the maximum value of i in that range is 5-2
^
Si_l = ^o is the pair of connected vertices {v,w}.
1. When i = 1,
To satisfy the valence condition,
each vertex in Si-i must connect to (k — 1) vertices in Si for i = 1 , . . . , | — 1, and
ISil^fc-i)!^!
for such i. Therefore,
V(G) I > I
SQ I
+ I Si I •
> 2 ( l + (fc-l)
(fc
Si
1)2 + -
+ (fc-i)§-i;
n
Next I show that determining whether QQ is injective only needs to be answered
for graphs whose vertices have valence at least 3. First, I mention some terminology.
157
Contraction
of an edge of a graph is defined to be the removal of an edge with distinct
endpoints followed by identification of its endpoints. That is, contract the edge in a
topological sense. A contraction
of a graph is a series of contractions of edges in the
graph (order does not matter).
L e m m a 4.4.9 (Contract to Valence at least 3). If every component T of a graph
G has hi(F)
> 1 (in particular
contraction G' with the
if G is connected with h\(G)
> I), then G has a
properties
(a) every vertex of G' has valence at least 3,
(b) QQ is mjective if and only if QQ/ is infective
(c) h1(G) =
Remark
h1(G')
4.4.10. The proof below is simple: contract edges with "valence" less than
3 (obviously, vertices, not edges, have valence; this is clear in the proof). The key
conclusions of the lemma are parts (a) and (b); part (c) is true for any contraction.
The hypothesis that h\(G)
> 1 is necessary to conclude (a). As mentioned at the
beginning, in case h\{G) — 1, QQ is always injective.
Proof. If a vertex v in G has valence one, contract its incident edge. If a vertex v
has valence two and two distinct incident edges, contract one of its incident edges.
There will not be valence two vertices with one incident edge (i.e., loops) because
I have assumed that no components T of G have h\(T)
= 1.
Carrying out all
such contractions yields a graph G' whose vertices have valence at least 3, so (a) is
satisfied.
An edge e incident to a valence one vertex will not be in a circuit (Lemma 4.1.19),
so Xe induces the zero functional on H\(G)
(Lemma 4.3.2). If edges e\ and e2
meet at a valence two vertex, then every circuit containing one of these edges must
contain the other (Lemma 4.1.19).
X\\JJ
/Q\
— ±X2\JJ,(Q\,
so X\
Depending on the orientation of the edges,
= Ar| as quadratic forms on H\(G).
In particu-
lar, the contractions to get G' do not change the injectivity of QQ, SO (b) is satisfied.
As already noted, every contraction G' of G will have h\{G) = h\(G').
•
158
Now that I can restrict to connected graphs whose vertices have valence at least
3, the Euler characteristic (4.1) gives a useful upper bound on the number of vertices
of the graph.
Lemma 4.4.11 (Bound on V(G)). A connected graph with valence at least 3 must
have
\V{G)\ < 2 M G ) - 2 .
Proof. For any graph G,
val u
2 | E(G) | = Yl
( )
veV(G)
where val(v) is the valence of v. Because the vertices have valence at least three,
2\E(G)\
If the graph is connected, then
X(G)
>3\V{G)\.
= 1, so the Euler characteristic (4.1) gives
IIQ(G)
= l-h1(G)
=
\V(G)\-\E(G)\
<\V(G)\-^\V(G)\
=
-\\V{G)\
Therefore,
\V{G)\
<2fci(G)-2.
D
The next proposition puts together the bounds for | V(G) | given the assumptions
that I can make about the graph without losing generality as in Lemma 4.4.9. The
proof of Theorem 4.4.1 follows the proposition by checking when it is possible to find
a number between these bounds.
159
Proposition 4.4.12 (Requirement for Injectivity). A connected graph G with valence at least 3, with h\{G) > 1,and for which QQ or Qc,p ig infective must satisfy
h](G)-3
1 + 3 + 3(2) + • • • + 3(2)—2
< | V(G) | < 2hi(G) - 2
for hx{G) odd
and
2(1 + 2 + • • • + 2 " ^
l
) < | V(G) | < 2h1(G) - 2
for hx{G) even-
Proof. The right-hand inequality follows from the valence of the graph as proven in
Lemma 4.4.11. The left-hand inequality follows from the Erdos-Sachs Theorem 4.4.8
because the graph has valence at least 3 and must have girth at least h\{G) by
Lemma 4.4.6.
•
Proof of Theorem 4-4-1- When h\{G) = 3 or 4, the upper and lower bounds agree
in the appropriate case. Simple calculus shows that these inequalities cannot be
satisfied for h\{G) > 4.
In particular, the calculation goes as follows for the case when h\{G) is odd. Use
the geometric sum formula to simplify the lower bound on | V(G) |:
h1(G)-3
1 + 3 + 3(2) + • • • + 3(2)
2
/
h^G)-!
=1 +3 2
h1(G)-l
Differentiating with respect to h\(G), I have 3 • 2
j
2
2
-1
,2)
°^ '. The derivative of the
upper bound is 2. The lower bound increases faster than the upper bound as long
as
3-2^
^ >2.
2
Solving for hi(G), the lower bound increases faster than the upper bound when
log
hl{G)
>
-
32
log(2)
~2-37'
160
Because the lower bound equals the upper bound for h\(G) = 3 and the lower bound
increases faster than the upper bound for h\(G) > 2.37, there is no odd h\(G) > 3
for which QQ is injective.
The calculation goes similarly for the case when /11(G) is even. Use the geometric
sum formula to simplify the lower bound on | V(G) |:
h^G)
2 1 + 2 + 2 2 + --- + 2 " ^
\
l
/
=2
hx{G)
2~~T~ - 1
Differentiating with respect to /11(G), I have 2 • 2 2
q} ', The derivative of the
upper bound is 2. The lower bound increases faster than the upper bound as long
as
MG) log(2)
2 - 2 ^ T " ^ ^ > 2.
Solving for /11(G), the lower bound increases faster than the upper bound when
2 log
h
^ >
log ( °2 S ) ( 2 ) / - 3 - 0 6 -
Because the lower bound equals the upper bound for h\{G) = 4 and the lower bound
increases faster than the upper bound for h\(G) > 3.06, there is no even h\(G) > 4
for which QQ is injective.
•
REFERENCES
[AM08]
P. Aluffi and M. Marcolli, Feynman
motives
of banana graphs, preprint
arxiv:0807.1690, 2008.
[AM09]
, Parametric
Feynman
integrals and determinant
hyper surf aces,
preprint arxiv:0901.2107, 2009.
[BB03]
P. Belkale and P. Brosnan, Matroids, motives, and a conjecture of Kontsevich, Duke Math. J. 116 (2003), no. 1, 147-188.
[BEK06] S. Bloch, H. Esnault, and D. Kreimer, On motives
polynomials,
[BK95]
associated to graph
Commun. Math. Phys. 267 (2006), no. 1, 181-225.
D. J. Broadhurst and D. Kreimer, Knots and numbers in 0
theory to 7
loops and beyond, Internat. J. Modern Phys. C 6 (1995), no. 4, 519-524.
[BK97]
, Association
of multiple zeta values with positive knots via Feynman
diagrams up to 9 loops, Phys. Lett. B 393 (1997), no. 3-4, 403-412.
[BK08]
S. Bloch and D. Kreimer, Mixed Hodge structures
and renorrnalization
in
physics, Commun. Num. Theor. Phys. 2 (2008), no. 4, 637-718.
[Blo07]
S. Bloch, Motives associated to graphs, Jpn. J. Mathematics 2 (2007), no. 1,
165-196.
[Bor77]
A. Borel, Cohomologie de sl n et valeurs de fonctions zeta aux points
entiers,
Ann. Scuola Norm. Sup. Pisa CI. Sci. (4) 4 (1977), no. 4, 613-636.
[Bou07]
N. Bourbaki, Elements
de mathematique.
Algebre. Chapitre 9., Springer-
Verlag, Berlin, 2007, Reprint of the 1959 original.
[Bro09]
F. Brown, The rnassless higher-loop two-point function,
Phys. 287 (2009), 925-958.
161
Commun. Math.
162
[Del73]
P. Deligne, Quadriques, Groupes de Monodromie en Geometrie Algebrique.
II. (SGA 7 II). Expose XII, Lecture Notes in Mathematics, vol. 340,
Springer-Verlag, Berlin-New York, 1973, pp. 62-81.
[DG05]
P. Deligne and A. Goncharov, Groupes fondamentaux
motiviques
de Tate
mixte, Ann. Sci. Ecole Norm. Sup. (4) 38 (2005), no. 1, 1-56.
[ES63]
P. Erdos and H. Sachs, Reguldre Graphen gegebener Taillenweite rait ramimaler Knotenzahl,
Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-
Natur. Reihe 12 (1963), 251-257.
[FH79]
W. Fulton and J. Hansen, A connectedness
with applications
to intersections
theorem for projective
and singularities
varieties,
of mappings, Ann. of
Math. (2) 110 (1979), no. 1, 159-166.
[FL81]
W. Fulton and R. Lazarsfeld, On the connectedness
of degeneracy loci and
special divisors, Acta Math. 146 (1981), no. 3-4, 271-283.
[Ful98]
W. Fulton, Intersection
theory, second ed., Ergebnisse der Mathematik und
ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics,
no. 2, Springer-Verlag, Berlin, 1998.
[GKZ94] I. M. Gelfand, M. M. Kapranov, and A. V. Zelevinsky,
resultants,
and multidimensional
determinants,
Discriminants,
Mathematics: Theory &
Applications, Birkhauser Boston, Inc., Boston, 1994.
[Gra05]
W. Graham, Nonemptiness
of symmetric
degeneracy loci, Amer. J. Math.
127 (2005), no. 2, 261-292.
[Har62]
R. Hartshorne, Complete intersections
and connectedness, Amer. J. Math.
84 (1962), 497-508.
[Har77]
, Algebraic
geometry,
Graduate Texts in Mathematics, no. 52,
Spring-Verlag, New York, 1977.
163
[Har92]
J. Harris, Algebraic geometry:
a first course, Graduate Texts in Mathe-
matics, no. 133, Springer-Verlag, New York, 1992.
[HT90]
J. Harris and L. W. Tu, The connectedness
of symmetric
degeneracy loci:
odd ranks, Topics in algebra, Part 2 (Warsaw) (S. Balcerzyk, ed.), Banach
Center Publ., vol. 26, PWN, 1990, Appendix to: The connectedness
of
degeneracy loci, pp. 249-256.
[IL99]
B. Ilic and J. M. Landsberg, On symmetric
symmetric
degeneracy loci, spaces of
matrices of constant rank and dual varieties, Math. Ann. 314
(1999), no. 1, 159-174.
[IZ80]
C. Itzykson and J.-B. Zuber, Quantum field theory, McGraw-Hill, New
York, 1980.
[JLP82]
T. Jozefiak, A. Lascoux, and P. Pragacz, Classes of deterrninantal
associated with symmetric
and skew-symmetric
varieties
matrices, Math. USSR-Izv.
18 (1982), no. 3, 575-586.
[Kat73]
N. Katz, Pinceaux de Lefschetz:
theoreme
d 'existence, Groupes de Mon-
odromie en Geometrie Algebrique. II. (SGA 7 II). Expose XVII, Lecture
Notes in Mathematics, vol. 340, Springer-Verlag, Berlin-New York, 1973,
pp. 212-253.
[Laz04]
R. Lazarsfeld, Positivity
in algebraic geometry.
I. Classical setting:
Line
bundles and linear series, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics, no. 48, SpringerVerlag, Berlin, 2004.
[Lov93]
L. Lovasz,
Combinatorial
problems
and exercises,
second ed., North-
Holland Publishing Co., Amsterdam, 1993.
[Mas91]
W. S. Massey, A basic course in algebraic topology, Graduate Texts in
Mathematics, no. 127, Springer-Verlag, New York, 1991.
164
[Nak71] N. Nakanishi, Graph theory and Feynman integrals, Gordon and Breach,
New York, 1971.
[Rim72] D. S. Rim, Formal deformation theory, Groupes de Monodromie en
Geometrie Algebrique. I. (SGA 7 I). Expose VI, Lecture Notes in Mathematics, vol. 288, Springer-Verlag, Berlin-New York, 1972, pp. 32-132.
[Ser73]
J.-P. Serre, A course in arithmetic, Graduate Texts in Mathematics, no. 7,
Springer-Verlag, New York-Heidelberg, 1973, translated from the French.
[Sou86]
C. Soule, Regulateurs, Asterisque Seminar Bourbaki, Vol. 1984/1985
(1986), no. 133-134, 237-253.
[Tu89]
L. W. Tu, The connectedness of symmetric and skew-symmetric degeneracy
loci: even ranks, Trans. Amer. Math. Soc. 313 (1989), no. 1, 381-392.
[Zak93]
F. L. Zak, Tangents and secants of algebraic varieties, Translations of
Mathematical Monographs, vol. 127, American Mathematical Society,
Providence, 1993.