on the conjecture of cox and von renesse in dimension 3

ON THE CONJECTURE OF COX AND VON RENESSE IN
DIMENSION 3
Robin Guilbot
07/03/2013
In their joint paper [CvR09], D. Cox and C. von Renesse give the first complete proof
of the following result :
Theorem 1 (Cox, von Renesse - 2009) If Σ is a quasi-projective fan of maximal dimension then the Mori cone of the toric variety XΣ is generated by primitive classes :
X
NE(XΣ ) =
R+ [CP ].
P ∈PΣ
At the end of the article they discuss the necessity of the quasi-projective hypothesis
and make the following conjecture :
Conjecture 2 (Cox, von Renesse - 2009) Let Σ be a simplicial fan in NR ' Rn . Then
the Mori cone of the toric variety XΣ is generated by primitives classes :
X
NE(XΣ ) =
R+ [CP ]
P ∈PΣ
In my thesis [Gui12] I gave counterexamples to this conjecture all of which are complete
simplicial fans of dimension 3 with at least one singularity. Then, after having introduced
locally contractible curves (Definition 2.2.11), I formulated the following variant of the
conjecture :
Conjecture 3 (G. - 2012) Let Σ be a smooth complete fan in NR ' Rn . Then the Mori
cone of the toric variety XΣ is generated by the classes of locally contractible curves :
X
NE(XΣ ) =
R+ [CP ]
P ∈PΣ
However I found an example of smooth complete XΣ with Picard number 15 for which
NE(XΣ ) is generated by primitive classes but not by classes of locally contractible curves
(see the Appendix).
My goal here is to give the broad lines of what I believe to be an important step towards
the following conjecture in dimension 3 :
Conjecture 4 Let Σ be a smooth complete fan in NR ' Rn . Then the Mori cone of the
toric variety XΣ is generated by primitives classes :
X
NE(XΣ ) =
R+ [CP ]
P ∈PΣ
First recall that by the Toric Cone Theorem it suffices to show that the class of every
toric curve can be written as a linear combination with positive coefficients of primitive
classes :
X
∀τ ∈ Σ(2), [Cτ ] ∈
R+ [CP ].
P ∈PΣ
I will use definitions and notations from my thesis [Gui12] plus the following :
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Definition 5 A d -corridor or corridor of dimension d is a finite sequence of d-cones
C = (α1 , . . . , αr ) of the simplicial fan Σ such that
(a) The αk are all distincts.
(b) For all 1 6 k 6 r − 1, βk = αk ∩ αk+1 ∈ Σ(d − 1).
We write C(d) = {α1 , . . . , αr } and for all 1 6 i 6 d − 1 we denote by C(i) the set of all
i-cones of Σ which is a face of an αk . We call r the length of the corridor.
For all 1 6 k 6 r − 1, we denote by bk the only element of αk (1) \ βk (1) and hk the only
element of αk+1 (1) \ βk (1).
A corridor is said to be simple if b1 ∈
/ αr (1), we then say it is a corridor from b1 to αr .
If instead b1 ∈ αr (1) and αr ∩ α1 ∈ Σ(n − 1) then the corridor is said to be circular. In
this case we always consider the numbering of the αk and the βk modulo the length r.
Notation 6 The set of all locally contractible curves of the toric variety XΣ will be denoted
by LC(XΣ ). The cone generated by the classes of these curves will be denoted by NELC (XΣ ).
Similarly, the cone generated by the classes of primitive curves will be denoted by NEP (XΣ ).
Every toric variety considered here will be complete and Q-factorial.
Let us consider the following algorithm, which is an adaptation of the one used by C.
Casagrande to prove the main theorem (Th. 4.1) of [Cas03] :
Algorithm 7 (1) Take a toric curve Cτ which is not locally contractible.
(2) Consider a decomposition of [Cτ ] involving all other toric curves of a toric surface
S = V(ζ), ζ ∈ Σ(n − 2) containing Cτ :
X
[Cτ ] =
λτ 0 [Cτ 0 ] with λτ 0 ∈ Q∗+ for all τ 0 ∈ Σ(n − 1) \ {τ }, ζ < τ 0 . (∗)
ζ<τ 0 ,τ 0 6=τ
(3) If every Cτ 0 is locally contractible then return the 1-cycle
each Cτ 0 not locally contractible, go to (2).
P
ζ<τ 0 ,τ 0 6=τ
λτ 0 Cτ 0 . Else for
Definition 8 In the situation of (∗) above, we say that for every τ 0 ∈ Σ(n − 1) \ {τ } such
that ζ < τ 0 , [Cτ 0 ] belongs to the decomposition of [Cτ ] in the surface S and we write
decV(ζ) ([Cτ ]) = {[Cτ 0 ] | τ 0 ∈ Σ(n − 1) \ {τ }, ζ < τ 0 }.
If Algorithm 7 stops for every starting curve Cτ ( which always happens when the variety
XΣ is projective) then we have
NE(XΣ ) = NELC (XΣ ) = NEP (XΣ ).
When this is not the case then we have the following :
Proposition-Definition 9 Let XΣ be a complete Q-factorial toric variety of dimension 3.
If NE(XΣ ) is not generated by the classes of locally contractible curves, then there exist a
circular (n − 1)-corridor (α1 , . . . , αr ) such that for all 1 6 k 6 r we have
Card(JC+α ), Card(JC−α ) = (3, 1) and Cαk+1 ∈ decV(βk ) (Cαk ).
k
k
Such a corridor is called a decomposition circle.
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Here is the result I think to be the most important, partly because it is related to
the problem of the combinatorial characterization of non projective fans (Question 2.5.6 of
[Gui12]), and mostly because it gives non trivial information on the structure of the Mori
cone of the concerned varieties :
Proposition 10 Let XΣ be a smooth complete toric threefold and let α ∈ Σ(2) be a 2cone of Σ such that [Cα ] ∈
/ NELC (XΣ ). Then there exists a unique decomposition circle
A = (α1 , . . . , αr ) such that α1 = α.
Sketch of proof — The point here is that no smooth toric surface may contain two toric
curves having positive self-intersection. This implies that two decomposition circles cannot
meet in any smooth toric divisor of XΣ and therefore, by Proposition 9, a decomposition
circle can neither split into two circles. This argument depends crucially on the fact that
we consider locally contractible curves instead of just contractible curves.
The main consequence of the fact that decomposition circles of Definition 9 are “isolated”
of each other is the following :
Corollary 11 Let XΣ be a smooth complete toric threefold and A = (α1 , . . . , αr ) be a
decomposition circle. Then for every 1 6 k 6 r we have Cone(αk , αk+1 ) ∈
/ Σ(3) and
{[Cτ ] | τ ∈ Σ(2), Card (τ (1) ∩ A(1)) = 1} ⊂ NELC (XΣ ).
Remark – In every counterexample to the non smooth case I gave in [Gui12], there is a
circle of decomposition composed of 2-cones contained in a unique singular 3-cone. Corollary
11 shows that this cannot happen in the smooth case.
For the purpose of proving Conjecture 4, we can deduce from Corollary 11 the following
statement :
Corollary 12 Let XΣ be a smooth complete toric threefold and A = (α = α1 , . . . , αr ) be a
decomposition circle. Denote by β the only element of JC−α and put
I = {ρ ∈ Σ(1) | Cone(ρ, β) ∈ Σ(2)}
Let C be a 1-cycle and suppose that the two following conditions are verified :
(a) JC ⊂ I,
(b) there exists a connected component H of NR \
Sr
k=1
αk such that JC ⊂ H ∪ A(1).
Then [C] ∈ NELC (XΣ ).
Sketch of proof — This can be easily proved using straight n-corridors like in Prop. 2.3.5
of [Gui12]. The point is that we can manage to decompose [C] only with classes of curves
Cτ with τ ∈ Σ(2) such that Card (τ (1) ∩ A(1)) = 1.
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I believe we can reach conditions (a) and (b) by subtracting suitable primitive
curves
P
to a given toric curve Cα , α ∈ A(2). This would yield a 1-cycle C 0 = Cα − li=1 CPi whose
class belongs to NELC (XΣ ) by Corollary 12 and we then could conclude that
[Cα ] = C 0 +
l
X
[CPi ] ∈ NEP (XΣ ).
i=1
To be more precise, it seems that an algorithm like the following should provide such a
1-cycle C 0 :
Algorithm 13 (1) Take a toric curve Cα with α ∈ A(2) and initialize C to Cα −CP1 where
P1 is the unique primitive collection contained in JCα (we have [C] ∈ A1 (XΣ )∩NE(XΣ )
by the remark following Lemma 2.2.5 of [Gui12]).
(2) If C satisfies conditions (a) and (b) then return C.
(3) Else if JC ⊂ I then choose a primitive S
collection P = {ρ1 , ρ2 } ⊂ JC+ with one element
in each connected component of NR \ rk=1 αk , numbered such that C · Dρ1 6 C · Dρ2 .
Then go to (2) with the 1-cycle C − (C · Dρ1 )CP .
(4) Else if JC ⊂
6 I then choose ρ1 ∈ JC \I and β ∈ JC−α such that P = {β, ρ1 } is a primitive
relation. Then go to (2) with the 1-cycle C − (C · Dρ1 )CP .
Remark – At each step we have JC− ⊂ JC−α = {β} so that [C] ∈ A1 (XΣ ) ∩ NE(XΣ ).
+
.
Furthermore, by construction, we have each time ρ1 ∈ JC+ \ JC−(C·D
ρ )CP
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Final comments
Proving Conjecture 4 for higher dimensions should be more difficult because when each
cone of codimension 1 contains n − 1 > 3 cones of codimension 2, there seems to be no
reason to get such a simple structure as in Proposition 10 : decomposition circles may meet.
Complications of the same type may in fact arise in dimension 3 if we keep the algorithm
used in the projective case, with only contractible curves. When contractible curves do not
generate the Mori cone, then we do have some kind of decomposition circles but if there are
several of them, they may meet each other. This should be interesting to study because it
may yield some general result about the structure of the Mori cone of smooth (or perhaps
even Q-factorial) complete non projective toric varieties.
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APPENDIX :
Example of a smooth complete toric variety XΣ whose Mori cone is not
generated by locally contractible curves.
Coordinates of the minimal generators of Σ in N :
u1 = (−1, −1, 0), u2 = (1, −1, 0), u3 = (1, 1, 0), u4 = (−1, 1, 0),
u5 = (0, 0, 1), u6 = (−1, −1, 1), u7 = (0, −1, 1), u8 = (1, 1, 1),
u9 = (0, 1, 1), u10 = (−1, 0, −1), u11 = (1, −1, −1), u12 = (1, 0, −1),
u13 = (−1, 1, −1), u14 = (0, 0, −1), u15 = (0, −1, 0), u16 = (1, 0, 0),
u17 = (−1, 0, 0), u18 = (0, 1, 0).
Set of maximal cones ([i,j,k] stands for Cone(ui , uj , uk )) :
Σ(3) = {[1, 6, 9], [1, 6, 15], [1, 9, 17], [1, 10, 13], [1, 10, 15], [1, 13, 17],
[2, 6, 7], [2, 6, 15], [2, 7, 16], [2, 10, 11], [2, 10, 15], [2, 11, 16], [3, 7, 8],
[3, 7, 16], [3, 8, 18], [3, 11, 12], [3, 11, 16], [3, 12, 18], [4, 8, 9], [4, 8, 18],
[4, 9, 17], [4, 12, 13], [4, 12, 18], [4, 13, 17], [5, 6, 9], [5, 6, 7], [5, 7, 8],
[5, 8, 9], [10, 11, 14], [10, 13, 14], [11, 12, 14], [12, 13, 14]}.
Non convex polytope with vertices u1 , . . . , u18 and faces corresponding one to one to cones
of Σ :
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References
[Cas03] Cinzia Casagrande. Contractible classes in toric varieties. Math. Z., 243:99–126,
2003.
[CvR09] David A. Cox and Christine von Renesse. Primitive collections and toric varieties.
Tohoku Math. J., 61:309–332, 2009.
[Gui12] Robin Guilbot. Quelques aspects combinatoires et arithmétiques des variétés
toriques complètes. PhD thesis, Université de Toulouse, 2012.
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