Deformations of projective toric varieties

In this section we study deformations of polarised projective toric varieties. Our strategy is to deform the corresponding affine cones thanks to Mavlyutov’s theorem (Theorem3.5) and then apply the Proj functor. We will use the lemmata in Sect.2.2.

Theorem 5.1.Let N be a lattice of rank n, let X be a projective TN-toric variety, and let D be an ample torus-invariant Q-Cartier Q-divisor on X . Let τ be the (n+1)-dimensional cone in the lattice N0 = N ⊕Ze0 associated to the pair (X,D)as in Lemma2.3. Let(Q,Q0,Q1, . . . ,Qk, w)be a deformation datum for (N0, τ)withw∈M ⊆M0. Consider the latticesN˜ =N⊕Ze1⊕ · · · ⊕Zekand N˜0 = N ⊕Ze0⊕Ze1⊕ · · · ⊕Zek. Letτ˜ ⊆(N˜0)_R andw˜ ∈ ˜M ⊆ ˜M0be as in Notation3.2. Let(X˜,D)˜ be the polarised projective toric variety associated to the coneτ˜via Lemma2.3.

(A)Then the inclusionτ → ˜τ induces a toric closed embedding X → ˜X which identifies X with the closed subscheme ofX associated to the homogeneous˜ ideal generated by the following binomials in the Cox coordinates ofX :˜

ρ∈ ˜(1): e^∗_i,ρ>0

x_ρ^ei∗,ρ−

ρ∈ ˜(1): e^∗_i,ρ<0

x⁻_ρ ^ei∗,ρ (16)

for i =1, . . . ,k, where˜ is the fan ofX in˜ N . Moreover, the k binomials in˜ (16) form a regular sequence.

(B) Let t1, . . . ,tk be the standard coordinates on A^k_C. Consider the closed subscheme X of X˜ ×SpecC A^k_C = TV_C[t₁,...,t_k]()˜ defined by the homogeneous ideal generated by the following trinomials in Cox coordinates:

ρ∈ ˜(1):

e_i^∗,ρ>0

x_ρ^e∗i,ρ−

ρ∈ ˜(1):

e_i^∗,ρ<0

x_ρ^−e∗i,ρ−ti

ρ∈ ˜(1)

x ^˜_ρ^w,ρ

ρ∈ ˜(1): e^∗_i,ρ<0

x_ρ^−e∗i,ρ (17)

for i =1, . . . ,k. Then the morphismX →A^k_Cinduces a deformation of X over C[[t1, . . . ,tk]]and over an open neighbourhood of the origin inA^k_C.

The rest of this section is devoted to the proof of Theorem5.1.

Proof of Theorem5.1(A). By Lemma3.4τ˜is a(n+1+k)-dimensional strongly convex rational polyhedral cone inN˜0. It is clear thate0∈ ˜τ.

Now we show thate0is in the interior ofτ˜: it is enough to show that, ifu˜=u+ _k

i=0hie^∗_i ∈ ˜τ^∨∩ ˜M0andh0=0, thenu˜=0. Sinceτ ⊆ ˜τ, we have thatuis non-negative onτ; bute0is in the interior ofτ, sou =0. By evaluatingu˜ =_k

i=1hie^∗_i onQ0−e1− · · · −ek,Q1+e1, …,Qk+ek, we seeh1 = · · · =hk =0. This proves thate0lies in the interior ofτ˜.

Thanks to Lemma2.4we haveX˜ =ProjC[ ˜τ^∨∩ ˜M0]andX =ProjC[τ^∨∩M0].

The ring homomorphism

C[ ˜τ^∨∩ ˜M0] −→C[τ^∨∩M0], (18)

which is induced by the inclusionτ → ˜τ and is surjective by the proof of The-orem 3.5, is homogeneous with respect to theN-grading and induces a closed embeddingι: X → ˜X. Using the isomorphisms (2) it is not difficult to write down the formulae for the actions of the toriTNandT_N˜ on the affine charts ofXandX,˜ respectively. From these formulae it is possible to see thatιis a toric morphism.

We have to prove thatXcoincides with the closed subscheme ofX˜defined by the idealJ_X˜ ⊆S_X˜generated by the binomials (16). LetJ_C˜ =J_X˜S_C˜be the extension of J_X˜ to the total coordinate ringS_C˜ of the affine coneC˜ =SpecC[ ˜τ^∨∩ ˜M0]via the ring homomorphismS_X˜ → S_C˜ defined in Lemma2.5. The ideal J_C˜ is generated generated by the binomials

ξ∈ ˜τ(1): e_i^∗,ξ>0

x_ξ^e∗i,ξ−

ξ∈ ˜τ(1):

e^∗_i,ξ<0

x_ξ^−e∗i,ξ fori=1, . . . ,k (19)

in the Cox coordinates ofC. By Theorem˜ 3.5the part of degree zero ofJ_C˜ in the ring(S_C˜)0C[ ˜τ^∨∩ ˜M0]coincides with the kernelH of the ring surjection (18).

By Lemma2.5,X =ProjC[ ˜τ^∨∩ ˜M0]/Hcoincides with the closed subscheme of X˜ defined by the ideal J_X˜.

The matricesM_˜ = (e^∗_i, ρ)_{1≤i≤k,ρ∈ ˜(1)} andM_τ˜ = (e^∗_i, ξ)1≤i≤k,ξ∈ ˜τ(1)

differ just by multiplication by a positive integer on each column, namely the numbersb_ρdefined in Lemma2.5. From the proof of Lemma3.9we see thatM_τ˜ has rankkand each of its columns has at most one positive entry. Therefore also the matrixM_˜ has these two properties. By Lemma3.8or Remark4.5, the binomials (16) form a regular sequence.

This concludes the proof of Theorem5.1(A).

The following two lemmata should be well known, but we have not been able to find an adequate reference for them.

Lemma 5.2.Let(A,m)be a noetherian local ring and letπ:Y →SpecA be a proper morphism of schemes such that Y×SpecASpecA/mⁿ→SpecA/mⁿis flat for every n∈N. Thenπis flat.

Proof. This proof relies on an argument that appears in the proof of [28, Propo-sition 6.51]. We want to show that the set Z = {y ∈ Y |OY,yis not flat over A}

is empty. By coveringY with open affine subschemes and by using [23, Theorem 24.3], one can see thatZis closed inY.

Assume by contradiction thatZ is non-empty. Sinceπ is closed, the setπ(Z) is a closed non-empty subset of SpecA. Thereforem ∈π(Z). Hence there exists y0∈ Zsuch thatπ(y0)=m. Let SpecRbe an affine open neighbourhood ofy0in Y and letB=OY,y0be the local ring ofY aty0. We know thatA/mⁿ→ R/mⁿR is flat for everyn ∈ N. Therefore the local homomorphism A → B is such that A/mⁿ → B/mⁿB is flat for every n ∈ N. By the local flatness criterion [23, Theorem 22.3]A→ Bis flat. But this is absurd becausey0∈ Z.

Lemma 5.3.Let S be a noetherian scheme and let Y → S be a scheme morphism of finite type such that Y ×SSpecOS,s →SpecOS,s is flat for some point s∈ S.

Then there exists an open neighbourhood U of s in S such that Y ×SU →U is flat.

Proof. Since the problem is local andY → Sis quasi-compact, we may assume S =SpecA,Y = SpecB ands =mfor some noetherian ring A, some finitely generated A-algebra Band some prime idealmof A. We know thatB⊗A A_mis flat overA_m. Let us consider the set

V = {P ∈SpecB| BPis flat overAP∩A} = {P ∈SpecB| BPis flat overA}, which is open in SpecB by [23, Theorem 24.3]. The equality above holds by transitivity of flatness and [23, Theorem 7.1].

We identify Spec(B ⊗A A_m)with the set of primes P ∈ SpecB such that P∩A⊆m. IfP∈SpecBis such thatP∩A⊆m, then by [23, Theorem 7.1] from the flatness ofB⊗A A_moverA_mwe deduce thatBPis flat over(A_m)(P∩A)A_m=

AP∩A. This shows that Spec(B⊗A A_m)is contained inV.

Consider the set A\mendowed with the order relation≤such that f ≤g if and only ifg ∈ √

A f. If f ≤ g, there is the localisation map Af → Ag, given by the restriction of the structure sheaf of SpecAfrom the principal open subset defined by f to the principal open subset defined byg. As f runs inA\m, the rings Af form a direct system and the local ring A_m is the direct limit of this system.

Since tensor products and direct limits commute, B⊗A A_mis the limit ofBf as f ∈ A\m. We are in the situation of inverse limits of affine schemes studied in [17, Section 8], i.e. Spec(B⊗A A_m)is the projective limit of the affine schemes SpecBf as f runs inA\m.

For every f ∈ A\m, consider the set Ef =V ∩SpecBf, which is open in SpecBf becauseV is open in SpecB. Since Spec(B⊗A A_m)is contained inV, the setE =V ∩Spec(B⊗A A_m)coincides with Spec(B⊗A A_m). SinceEis the limit of the Ef’s, by [17, Corollaire 8.3.5] we have that there exists f0 ∈ A\m such thatEf0 =SpecBf0. This implies thatBf0is flat overAf0. Therefore we may

takeU =SpecAf0.

Proof of Theorem5.1(B). The proof of the fact that the trinomials (17) are elements ofC[t1, . . . ,tk][x_ρ |ρ∈ ˜(1)]is completely analogous to what is done in the proof of Theorem3.5(B) and will be omitted.

Let X be the closed subscheme of X˜ ×SpecC A^k_C defined by the homoge-neous ideal generated by the trinomials (17). By (A) the fibre of X → A^k_C over the origin is X. The fibred product X ×_A^k

C SpecC[t1, . . . ,tk]/q is flat overC[t1, . . . ,tk]/qfor every(t1, . . . ,tk)-primary idealqofC[t1, . . . ,tk], thanks to Lemma 3.10 and Lemma 2.2, as in the proof of Theorem 3.5(B). If A = C[t1, . . . ,tk]₍t1,...,t_k)is the local ring ofA^k_Cat the originO, by Lemma5.2the mor-phismX×_A^k

CSpecA→SpecAis flat, and consequently it induces a deformation ofXoverAˆ =C[[t1, . . . ,tk]]. By Lemma5.3we may find an open neighbourhood U ⊆A^k_CofOsuch thatX×_A^k

CUis flat overU.