Homogeneous deformations of toric pairs

We extend the Altmann--Mavlyutov construction of homogeneous deformations of affine toric varieties to the case of toric pairs $(X, \partial X)$, where $X$ is an affine or projective toric variety and $\partial X$ is its toric boundary. As an application, we generalise a result due to Ilten to the case of Fano toric pairs.


Introduction
An important trend in modern algebraic geometry is to study pairs consisting of a variety with a divisor. Recent work by Gross-Hacking-Keel [GHK15] suggests that Mirror Symmetry, which was originally formulated for Calabi-Yau varieties, is better understood as a correspondence between log Calabi-Yau pairs, i.e. pairs (X, B) where X is a variety and B is an effective divisor such that K X + B is linearly trivial. Toric pairs -that is, pairs (X, ∂X) where X is a toric variety with toric boundary ∂X -are one of the simplest examples of log Calabi-Yau pairs, and can be understood to lie at the boundary of the moduli space of log Calabi-Yau pairs. It is therefore interesting to understand deformations of toric pairs in this setting.
The aim of this paper is to construct deformations of toric pairs via combinatorial methods of toric geometry, by generalising the constructions due to Altmann [Alt95,Alt00] and Mavlyutov [Mav]. The deformations we construct are homogeneous with respect to the action of the torus (see Remark 3.7) and unobstructed.
After surveying the work of Mavlyutov [Mav] on deformations of affine toric varieties, we extend his construction to deformations of affine toric pairs (Theorem 1.1). More precisely, if X is an affine toric variety without torus factors and ∂X is its toric boundary, then from some combinatorial input (which we call ∂-deformation datum) we construct a formal deformation of the closed embedding ∂X → X over a power series ring in finitely many variables over C. The construction of the deformation is achieved by constructing an affine toric varietyX and a closed embedding X →X and by deforming the equations of this closed embedding.
By applying Proj to this construction, we also construct deformations of projective toric pairs (Theorem 1.2). Both in the affine and in the projective case, the deformations we construct lie inside a bigger toric varietyX and are explicit in terms of Cox coordinates ofX; therefore, in specific examples, it is easy to check if we get smoothings.
Finally, we apply our construction of deformations of projective toric pairs to a particular case which arises in the study of Mirror Symmetry for Fano varieties [CCG + 13]: in this way we are able to reprove and extend an important result of Ilten [Ilt12] about families of Fano varieties coming from a combinatorial procedure on Fano polytopes called "mutation" (Theorem 1.3).
Now we give a more detailed account of what we do. = + = + + Figure 1. The two Minkowski decompositions of the hexagon 1.1. Minkowski decompositions and deformations of affine toric pairs. Let σ be a full dimensional strongly convex rational polyhedral cone inside a lattice N and let X = TV C (σ) be the affine toric variety over C associated to σ. Klaus Altmann has extensively studied the deformation theory of X. In [Alt94] he computes the tangent space T 1 X to deformations of X. In [Alt97] he describes the miniversal deformation of X when it is an isolated Gorenstein singularity. In [Alt95] he notices that Minkowski decompositions of a polyhedron inside σ, under some hypotheses, induce certain deformations of X; for example, the two Minkowski decompositions of the standard hexagon ( Figure 1) induce two different deformations of the anticanonical affine cone over the smooth del Pezzo surface of degree 6. In [Alt00] he constructs deformations of X from Minkowski decompositions of more general polyhedra inside the cone σ.
In [Mav] Anvar Mavlyutov gives a unified description of all Altmann's deformations thanks to the use of Cox coordinates. His construction has the same strategy as Altmann's: starting from a Minkowski decomposition of some polyhedron (with some assumptions) one embeds the considered affine toric variety into a larger affine toric variety and then deforms the equations of this closed embedding. More specifically, starting from a Minkowski decomposition of a polyhedron inside the cone σ one can construct a bigger coneσ in a bigger latticeÑ and embed the toric variety X associated to σ inside the toric varietyX = TV C (σ) associated toσ via binomial equations in the Cox coordinates ofX; by deforming these binomial equations with extra monomials one may produce a deformation of X. The precise statement is a theorem of Mavlyutov [Mav] and is rewritten in §3 together with a detailed proof. There the Minkowski decomposition is encoded in the notion of a deformation datum (see Definition 3.1).
We have noticed that Mavlyutov's construction can be applied also to deform the affine toric pair (X, ∂X), where ∂X is the toric boundary of X. More precisely, we construct a formal deformation of the closed embedding ∂X → X over a power series ring A in finitely many variables over C. By a formal deformation of ∂X → X over A we mean a collection made up of a commutative diagram B n / / % % X n Spec A/m n+1 A for each n ∈ N, where m A is the maximal ideal of A, B n → X n is a closed embedding, B n and X n are flat over A/m n+1 A , and all these diagrams are required to be compatible in the following way: the 0th diagram is just the embedding ∂X → X over Spec C, and the base change of the (n + 1)th diagram along the closed embedding induced by A/m n+2 A A/m n+1 A is isomorphic to the nth diagram.
Our main theorem, which significantly rests on [Mav], is the following.
Theorem 1.1. Let X be an affine toric variety without torus factors and let ∂X be its toric boundary. Given a Minkowski decomposition of a polyhedron satisfying certain conditions, one can construct a formal deformation of the pair (X, ∂X) over a power series ring in finitely many variables over C. (See Theorem 4.1 for the precise statement.) Example 4.2 shows how to use this theorem to deform the 3-fold toric cA 1 singularity Spec C[x, y, z, u]/(xy − u 2 ) together with its toric boundary.
1.2. Deformations of projective toric pairs. The deformation theory of complete toric varieties is not fully understood yet. When X is a smooth complete toric variety, Nathan Ilten [Ilt11] has computed the tangent space T 1 X to deformations of X. But, when X is a singular complete toric variety, the tangent space T 1 X and the miniversal deformation of X are unknown in general.
Nonetheless, in the literature there are some constructions of homogeneous deformations of toric varieties. Ilten and Vollmert [IV12] construct deformations of rational T -varieties of complexity 1, which are a generalisation of toric varieties [AIP + 12, AH06, AHS08]. Hochenegger and Ilten [HI13] construct deformations of a rational complexity-1 T-variety together with a T-line bundle. Mavlyutov [Mav] uses Minkowski decompositions of polyhedral complexes in order to construct homogeneous deformations. Laface and Melo [LM] construct deformations of smooth complete toric varieties by using their Cox rings.
Here, by avoiding the languages of T-varieties and of Cox rings (or more precisely by sweeping them under the carpet), we propose an explicit construction of deformations of polarised projective toric varieties together with their toric boundaries. These deformations live in an ambient projective toric varietyX and are completely explicit in terms of the Cox coordinates ofX.
Our strategy consists in deforming a projective toric variety X by deforming its affine cone C with respect to an ample torus-invariant Q-Cartier Q-divisor D on X. Deforming a polarised projective variety by deforming its affine cone have already appeared in the literature, e.g. [Pin74,PS85]; in our toric context we took the idea from a specific case in [Ilt12]. More specifically, if the fan of X is in the lattice N of rank n, then the section ring where τ is an (n+1)-dimensional strongly convex rational polyhedral cone in the lattice N 0 = N ⊕ Ze 0 such that e 0 is in the interior of τ . We refer the reader to §2.2 for more details about the relationship between the pair (X, D) and the cone τ . Starting from a Minkowski decomposition of a polyhedron inside τ and by applying Mavlyutov's construction ( [Mav] and Theorem 3.5), we can deform C = Spec C[τ ∨ ∩ M 0 ], which is the affine cone over X; by applying Proj we construct a deformation of X = Proj C[τ ∨ ∩ M 0 ]. Theorem 5.1 expresses this deformation via explicit equations in Cox coordinates of a projective varietyX.
If, in addition, the divisor D is a Z-divisor, then we are able to deform also the toric boundary of X. This is the content of the following theorem.
Theorem 1.2. Let X be a projective toric variety with toric boundary ∂X. Given an ample torus-invariant Q-Cartier Z-divisor on X and a Minkowski decomposition of a polyhedron satisfying certain conditions, one can construct a deformation of the pair (X, ∂X) over a power series ring in finitely many variables over C. (See Theorem 6.1 for the precise statement.) 1.3. Mutations and deformations of Fano toric pairs. A Fano polytope in a lattice N is a full dimensional polytope P ⊆ N R such that the origin 0 ∈ N lies in the strict interior of P and every vertex of P is a primitive lattice point. The spanning fan (also called the face fan) of a Fano polytope P ⊆ N R , i.e. the fan in N R whose cones are the cones over the faces of P , determines a toric variety X P which is Fano, i.e. its anticanonical divisor is Q-Cartier and ample. This establishes a bijective correspondence between Fano T N -toric varieties and Fano polytopes in N .
Starting from a primitive vector w ∈ M and a polytope F ⊆ w ⊥ ⊆ N R satisfying certain conditions (see Definition 7.1) with respect to the Fano polytope P ⊆ N R , it is possible to construct another Fano polytope P := mut w,F (P ) ⊆ N R (see Definition 7.2). This procedure is called mutation [ACGK12] and its motivation lies in the study of Mirror Symmetry for Fano varieties [CCG + 13, ACC + 16].
It was observed by Nathan Ilten [Ilt12] that if two Fano polytopes P and P in N R are related by a mutation then the corresponding Fano varieties X P and X P are two closed fibres of a flat family over P 1 . Ilten's construction relies on the theory of deformations of T-varieties developed in [IV12].
Here we apply our Theorem 1.2 (i.e. Theorem 6.1) to this case because the toric boundary ∂X P is an ample Q-Cartier Z-divisor on X P and the combinatorial conditions in the definition of mutation allows us to construct a ∂-deformation datum. We will show that X P and X P are two fibres of the flat family of divisors defined by a trinomial in the Cox coordinates of a projective toric variety of dimension dim X P + 1. In addition to what was done by Ilten, we can show that also the toric boundary ∂X P deforms to ∂X P . Theorem 1.3. Let P and P be two Fano polytopes related by a mutation. Let X P (resp. X P ) be the Fano toric variety associated to the spanning fan of P (resp. P ) and let ∂X P (resp. ∂X P ) be the toric boundary of X P (resp. X P ). Then there exists a commutative diagram is an open subscheme of P 2 C , the morphism B → X is a closed embedding, the morphisms B → V and X → V are projective and flat, and there are two closed points in V for which the base change of the diagram to them are the closed embeddings ∂X P → X P and ∂X P → X P over Spec C, respectively.
Very roughly speaking, Ilten's result says that mutations of Fano polytopes create a 1-dimensional skeleton in the moduli space of Fano varieties. Our theorem extends this interpretation to moduli of log Calabi-Yau pairs (X, B) where X is Fano.
The precise constructions of V , B and X in Theorem 1.3 are given in Theorem 7.3. We refer the reader to Example 7.4 for an application of this result to construct the degeneration of P 2 to the weighted projective plane P(1, 1, 4).

1.4.
Outline of the article. In §2 we discuss Cox coordinates on toric varieties and we study polarised projective toric varieties. In §3 we recall Mavlyutov's construction of deformations of affine toric varieties. In §4 we construct deformations of affine toric pairs and we prove Theorem 1.1. In §5 we construct deformations of projective toric varieties. In §6 we construct deformations of projective toric pairs and we prove Theorem 1.2. In §7 we recall the notion of mutation between Fano polytopes and we prove Theorem 1.3.
A lattice is a finitely generated free abelian group. The letters N, N 0 ,Ñ ,Ñ 0 stand for lattices and M, M 0 ,M ,M 0 for their duals, e.g. M = Hom Z (N, Z). We set N R := N ⊗ Z R and M R := M ⊗ Z R. The perfect pairing M × N → Z and its extension M R × N R → R are denoted by the symbol ·, · .
In a real vector space V of finite dimension, a cone is a non-empty subset which is closed under sum and multiplication by non-negative real numbers. The conical hull cone S of a subset S ⊆ V is the smallest cone containing S, i.e. the set made up of λ 1 s 1 + · · · + λ k s k , as k ∈ N, λ i ≥ 0, and s i ∈ S. A subset of V is called a polyhedral cone if it coincides with cone S for some finite subset S ⊆ V , or equivalently it is the intersection of a finite number of closed halfspaces passing through the origin. The convex hull of a subset S ⊆ V is denoted by conv S . A polyhedron is the intersection of a finite number of closed halfspaces, so it is always convex and closed. A compact polyhedron is called polytope. If Q is a polyhedron, vert(Q) denotes the set of vertices of Q and rec(Q) is its recession cone, i.e. the cone of the unbounded directions of Q. If Q 1 and Q 2 are polyhedra, then their Minkowski sum is Q 1 + Q 2 := {q 1 + q 2 | q 1 ∈ Q 1 , q 2 ∈ Q 2 }; in this case we say also that this is a Minkowski decomposition of Q. If Q is a polyhedron such that rec(Q) is strongly convex, then Q = conv vert(Q) + rec(Q). We refer the reader to the book [Zie95] for details.
We assume the standard terminology of commutative algebra and of algebraic geometry. By a ring we always understand a commutative ring with unit.
Acknowledgements. Parts of this article appear in the author's Ph.D. thesis at Imperial College London. The author wishes to thank his advisor Alessio Corti for suggesting the problem and Thomas Prince for fruitful conversations. Moreover, he is grateful to Alessandro Chiodo, Nathan Ilten, Alexander Kasprzyk, and Richard Thomas for helpful comments on a preliminary version of this manuscript. The author was funded by a Roth Studentship from Imperial College London, by Tom Coates' ERC Consolidator Grant 682603, and by Alexander Kasprzyk's EPSRC Fellowship EP/N022513/1.

Preliminaries on toric geometry
2.1. Cox coordinates. For generalities about toric varieties we refer the reader to [Ful93] and [CLS11]. We firstly treat toric schemes, with split tori, which are defined over arbitrary rings and consider their total coordinate rings.
Remark 2.1 (Toric schemes over arbitrary rings). Let A be a ring, let N be a lattice, and let Σ be a fan of strongly convex rational polyhedral cones in N R . For every cone σ ∈ Σ, we consider its dual σ ∨ ⊆ M R , the semigroup σ ∨ ∩ M , and the semigroup A-algebra A[σ ∨ ∩ M ]. We denote by TV A (Σ) the scheme obtained by gluing the affine schemes TV A (σ) = Spec A[σ ∨ ∩ M ] thanks to the structure of the fan Σ, as it is customary in toric geometry. One may prove that TV A (Σ) is a separated flat scheme of finite presentation over A with relative dimension rank N and geometrically integral fibres. When A = C, TV A (Σ) = TV C (Σ) is exactly the toric variety over C associated to the fan Σ considered in [Ful93,CLS11]. Now suppose that N R is generated as an R-vector space by the support |Σ| of Σ. In other words we assume that TV C (Σ) has no torus factors. Let Σ(1) be the set of rays of Σ. We do not distinguish a ray of Σ, which is actually a 1-dimensional cone of Σ, from its primitive generator, which is actually the lattice point on the ray that is the closest one to the origin. Generalising the definition of Cox coordinates on toric varieties (see [Cox95],[CLS11, §5.2] or [MS05]), we say that the polynomial ring S = A[x ρ | ρ ∈ Σ(1)] is the total coordinate ring of TV A (Σ). The variables x ρ are called Cox coordinates or homogeneous coordinates. The A-algebra S has a grading with respect to the divisor class group G Σ = Cl(TV C (Σ)) of the variety TV C (Σ), which is a quotient of the free abelian group Z Σ(1) according to the divisor sequence of Σ (see [CLS11,(5 For every cone σ ∈ Σ, setting xσ = ρ / ∈σ(1) x ρ ∈ S, the map defined by where S xσ is the localization of S obtained by inverting the element xσ and S (xσ) is the subring of the S xσ consisting of elements of degree 0 with respect to the G Σ -grading. Imitating [CLS11, §5.3], from a G Σ -graded S-module E one may construct a quasi-coherent sheafẼ on TV A (Σ) such that, for every cone σ ∈ Σ, the sections ofẼ over TV A (σ) are the elements of E (xσ) , i.e. the elements of degree 0 in the localization E xσ . The assignment E →Ẽ is sometimes called sheafification and is an exact functor from the category of G Σ -graded S-modules to the category of quasi-coherent sheaves on TV A (Σ). In particular, the sheafification of a G Σhomogeneous ideal J of S induces a closed subscheme of TV A (Σ), whose structure sheaf is the sheafification of S/J. Moreover, if A is noetherian and E is finitely generated graded S-module, thenẼ is coherent on TV A (Σ).
The following lemma gives a sufficient criterion to ensure the flatness of the sheafification of a graded module on a toric scheme.
Lemma 2.2. Let N be a lattice and let Σ be a fan of strongly convex rational polyhedral cones in N R such that N R is generated by |Σ| as R-vector space. Let A be a ring and let TV A (Σ) be the A-scheme constructed in Remark 2.1. Let S be the total coordinate ring of TV A (Σ) and let E be a graded S-module. If E is flat as an A-module, thenẼ ∈ QCoh(TV A (Σ)) is flat over Spec A.
Proof. It is enough to show that E (xσ) is flat over A, for every cone σ ∈ Σ. The localisation E xσ is a G Σ -graded flat A-module and the homogeneous localisation E (xσ) is its degree zero part. Therefore, E (xσ) is a direct summand of E xσ as A-modules and is flat over A.
2.2. Polarised projective toric varieties. Now we discuss projective toric varieties X polarised by an ample Q-Cartier Q-divisor which is supported on the toric boundary ∂X. This section is not necessary for §3 and §4.
The lemma below is a well known characterisation of polarised projective toric varieties.
Lemma 2.3. If N is a lattice of rank n, then the following data are naturally equivalent: (1) a pair (X, D), where X is a projective normal toric variety over C with respect to the torus T N = Spec C[M ] and D is an ample torus-invariant Q-Cartier Q-divisor on X; (2) a pair (Σ, ϕ), where Σ is a complete fan in N and ϕ is a strictly convex rational support function on Σ, i.e. ϕ : N R → R is a continuous function such that (4) a strictly convex rational polyhedral cone τ in the lattice N 0 = N ⊕Ze 0 such that the dimension of τ is n + 1 and e 0 is in the interior of τ . In the setting above there are natural bijective correspondences if in addition we require the following further conditions too: (1) D is a Q-Cartier Z-divisor on X; (2) ϕ takes integer values on the primitive generators of the rays of Σ; (3) every supporting hyperplane of P contains at least a point of the lattice M ; (4) the primitive generator of every ray of τ is of the form ρ − ae 0 for some a ∈ Z and ρ ∈ N primitive. Moreover, in the setting above there are natural bijective correspondences if we require the following more restrictive further conditions too: (1) D is a Cartier divisor on X; (2) ϕ is a strictly convex integral support function on Σ, i.e. we also require that u σ ∈ M for every σ ∈ Σ(n); (3) P is a lattice polytope; (4) every facet of τ is contained in a hyperplane of the form (u + e * 0 ) ⊥ for some u ∈ M .
The equivalence with (4) is as follows: τ is the convex hull of the graph of the function −ϕ, i.e. τ = {v + ke 0 ∈ N R ⊕ Re 0 | ϕ(v) + k ≥ 0}, or equivalently the cone with rays ρ − ϕ(ρ)e 0 as ρ ∈ Σ(1). Conversely, the cones of Σ are precisely the images of the faces of τ along the projection N ⊕ Ze 0 N and The following lemma, which is a reformulation of [CLS11, Theorem 7.1.13 and Proposition 8.2.11], describes a polarised projective toric variety X as the Proj of an N-graded ring constructed from the cone τ , where τ is the cone as in Lemma 2.3. It also gives a description of the toric boundary.
Lemma 2.4. Let N be a lattice of rank n, let τ be a (n + 1)-dimensional strongly convex rational polyhedral cone in the lattice N 0 = N ⊕ Ze 0 such that e 0 ∈ int(τ ), and let (X, D) be the pair associated to τ via Lemma 2.3. Consider the ideal which is the ideal of the toric boundary of the affine toric variety Let Σ be the fan of X and let ϕ be the support function associated to D as in Lemma 2.3. There is a bijective correspondence between cones of Σ and proper subcones of τ . For any ray ρ ∈ Σ(1), let ξ ρ ∈ τ (1) be the corresponding ray of τ . In other words, ξ ρ = b ρ ρ − a ρ e 0 where b ρ ∈ N + , a ρ ∈ Z are such that gcd(a ρ , b ρ ) = 1 and ϕ(ρ) = a ρ /b ρ .
Fix an n-dimensional cone σ ∈ Σ(n). It corresponds to an n-dimensional face of τ , namely F σ = cone ξ ρ | ρ ∈ σ(1) . Since D is Q-Cartier, there exist u σ ∈ M and h σ ∈ N + such that F σ is contained in the hyperplane (u σ + h σ e * 0 ) ⊥ . The affine open subscheme TV C (σ) of the toric variety X = TV C (Σ) is isomorphic to the affine open subscheme of Proj C[τ ∨ ∩ M 0 ] defined by the homogeneous element which is defined by In order to prove ∂X = Proj C[τ ∨ ∩ M 0 ]/L, we have to check that, for every cone σ ∈ Σ(n), the homogeneous localisation L (χ uσ +hσ e * 0 ) coincides with the ideal of the toric boundary of TV C (σ) under the ring isomorphism (2). So, let us fix a cone σ ∈ Σ(n) and elements u ∈ M and k ∈ N such that u − ku σ ∈ σ ∨ . The . In order to check this we need to pair this vector of M 0 with the rays of τ , i.e. ξ ρ = b ρ ρ − a ρ e 0 as ρ ∈ Σ(1). We distinguish two cases: • the ray ρ lies in σ; then b ρ u σ , ρ − a ρ h σ = 0; for any m ∈ N we have is positive for m big enough. This shows that the element χ u+khσe * 0 /(χ uσ+hσe * 0 ) k lies in the homogeneous localisation of L if and only if for every ρ ∈ σ(1) we have u − ku σ , ρ > 0, or equivalently if u − ku σ lies in the ideal of the toric boundary of TV C (σ).
In the following lemma we compare the homogeneous coordinate rings of a polarised toric variety and of its affine cone. We deduce an alternate description of closed subschemes of a polarised toric variety.
Lemma 2.5. Let N be a lattice of rank n, let τ be a (n + 1)-dimensional strongly convex rational polyhedral cone in the lattice N 0 = N ⊕ Ze 0 such that e 0 ∈ int(τ ), and let (X, D) and (Σ, ϕ) be the pairs associated to τ via Lemma 2.3. Consider the affine toric variety C = Spec C[τ ∨ ∩ M 0 ]. Let S X and S C be the homogeneous coordinate rings of X and C, respectively.
For every ray ρ ∈ Σ(1), where r Σ is the ray map of X, r τ is the ray map of C, pr is the natural projection, and b is the diagonal matrix with entries b ρ . Consider the dual maps r * Σ and r * τ and the following commutative diagram with exact rows, where G Σ is the divisor class group of X and G τ is the divisor class group of C.
The ring homomorphism S X → S C is homogeneous with respect to the group homomorphism G Σ → G τ . In particular, the ideal J X S C ⊆ S C is G τ -homogeneous. Fix a full dimensional cone σ ∈ Σ(n) and let u σ ∈ M and h σ ∈ N + be such that the hyperplane (u σ + h σ e * 0 ) ⊥ contains the corresponding face F σ of τ , as in the proof of Lemma 2.4. We setū σ = u σ + h σ e * 0 ∈ M 0 for brevity. We have to show that the ideal (J Sinceū σ is zero on the face F σ and strictly positive on τ \ F σ , a Cox coordinate x ξ of C appear in the monomial xū σ ∈ S C if and only if ξ / ∈ F σ . This implies that there is a ring homomorphism that is the localisation of S X → S C defined above. At this point it is not difficult to show that there is a commutative diagram of rings where the equality symbols stand for isomorphisms. Now consider the ideal Since S C is a finite free S X -module, S C is faithfully flat over S X . Therefore, also the localised homomorphism (3) is faithfully flat. By [Mat89, Theorem 7.5(ii)] the contraction of K to (S X ) xσ is the extension of J X . This implies that (J X ) (xσ) is the contraction of K to (S X ) (xσ) along the homomorphisms in the diagram above.
On the other hand, it is clear that K is the extension of J X S C to (S C ) xūσ . Since xū σ has degree zero with respect to the G τ -grading of S C , it is not difficult to check that the extension of Since the two ideals that must be checked to coincide are both contractions of the same ideal K, we are done.

Deformations of affine toric varieties after A. Mavlyutov
In this section we recall the work [Mav] by Anvar Mavlyutov on the deformations of affine toric varieties. We have rewritten a detailed proof, as it will be useful for our generalisations, and we have taken this opportunity to fill in details missing from Mavlyutov's original paper. In so doing we have reformulated many of his statements in terms of deformation datum.
Definition 3.1. Let N be a lattice and σ ⊆ N R be a strongly convex rational polyhedral cone with dimension rank N . A deformation datum for (N, σ) is a tuple (Q, Q 0 , Q 1 , . . . , Q k , w) where w ∈ M and Q, Q 0 , Q 1 , . . . , Q k are non-empty rational polyhedra in N R such that the following conditions are satisfied: (v) the minimum of w on Q exists and is not smaller than −1; for (N, σ) such that the following further condition is satisfied: (iv) Q 1 , . . . , Q k are lattice polyhedra.
It is immediate to see that (iv) implies (iv').
Lemma 3.4. Let N be a lattice of rank n, let σ ⊆ N R be a strongly convex rational polyhedral cone of dimension n, let (Q, Q 0 , Q 1 , . . . , Q k , w) be a deformation datum for (N, σ), and letÑ andσ be as in Notation 3.2. Thenσ is a strongly convex rational polyhedral cone inÑ of dimension n + k such that σ =σ ∩ N R .
We will postpone the proof of Lemma 3.4 to page 15.
Theorem 3.5 (Mavlyutov [Mav]). Let N be a lattice, let σ ⊆ N R be a strongly convex rational polyhedral cone with dimension rank N , let (Q, Q 0 , Q 1 , . . . , Q k , w) be a deformation datum for (N, σ), and letÑ ,σ andw be as in Notation 3.2. Let X be the affine toric variety associated to σ and letX be the affine toric variety associated toσ.
(A) Then the toric morphism X →X, induced by the inclusion N →Ñ , is a closed embedding and identifies X with the closed subscheme ofX associated to the homogeneous ideal generated by the following binomials in the Cox coordinates of X: x − e * i ,ξ ξ for i = 1, . . . , k. Moreover, these binomials form a regular sequence.
(B) Let t 1 , . . . , t k be the standard coordinates on A k C . Consider the closed subscheme X ofX × Spec C A k C = TV C[t1,...,t k ] (σ) defined by the homogeneous ideal generated by the following trinomials in Cox coordinates: Remark 3.6. We will clarify what we mean when we say that the aforementioned closed subscheme induces a formal deformation of TV C is a flat morphism, but it is "formally flat" over the origin in the following sense: for every (t 1 , . . . , t k )-primary ideal q of C[t 1 , . . . , t k ], the fibre product X × A k C Spec C[t 1 , . . . , t k ]/q is flat over Spec C[t 1 , . . . , t k ]/q. Since the inverse limit of the rings C[t 1 , . . . , t k ]/q is C[[t 1 , . . . , t k ]], we say that we have a formal deformation over C[[t 1 , . . . , t k ]] by usingà la Schlessinger terminology.
As we will see in §5, if we had been dealing with deformations of complete varieties there would have been no need to specify the adverb "formally" thanks to Lemma 5.2 Remark 3.7. Here we explain the meaning of the adjective homogeneous in the title of this paper. Let us assume that we are using notation from Theorem 3.5. The torus T N = Spec C[M ] acts on the affine toric variety X and consequently on T 1 X = Ext 1 (Ω X , O X ), which is the tangent space of the deformation functor of X. Therefore T 1 X is an M -graded vector space over the field C. Every formal deformation of X over a complete local noetherian C-algebra (A, m A ) with residue field C induces a C-linear map which is called the Kodaira-Spencer map of the deformation (see [TV13, §6.1.2]). By [Mav,Theorem 2.14] the image of the Kodaira-Spencer map of the deformation of X constructed in Theorem 3.5 is contained in T 1 X (w), which is the homogeneous component of T 1 X with degree w ∈ M . The rest of this section is devoted to the proof of Theorem 3.5 and relies entirely on [Mav].
The following lemma is a very particular case of a result by K. G. Fischer and J. Shapiro [FS96] that gives a necessary and sufficient criterion for a sequence of binomials to be a regular sequence. For every a ∈ Z, define a + := max{a, 0} and a − := max{−a, 0}.
If the rank of M is k and every column of M has at most one positive entry, then f 1 , . . . , f k is a regular sequence in C[x 1 , . . . , x n ].
When we have a cone in a latticeÑ , it is possible to intersect it with a saturated sublattice N ofÑ and get a toric morphism. The following lemma describes the scheme-theoretic image of this toric morphism under some hypotheses.
Lemma 3.9. Let N be a lattice and letÑ = N ⊕ Z k . Denote by e 1 , . . . , e k the standard basis of Z k . Letσ ⊆Ñ R be a (rankÑ )-dimensional strongly convex rational polyhedral cone that satisfies the following condition: the Z k -component of every ray ofσ has at most one positive entry, i.e.
If σ is the coneσ ∩ N R inside N R , then the scheme-theoretic image of the toric morphism TV C (σ) → TV C (σ) is the closed subscheme of TV C (σ) defined by the homogeneous ideal generated by the following binomials in the Cox coordinates of TV C (σ): x − e * i ,ξ ξ for i = 1, . . . , k. Moreover, these binomials form a regular sequence.
Proof. The toric morphism TV C (σ) → TV C (σ) is associated to the ring homomorphism that maps χũ to χ φ(ũ) , where φ :σ ∨ ∩M → σ ∨ ∩M is the semigroup homomorphism given by u + a 1 e * 1 + · · · + a k e * k → u. Let I ⊆ C[σ ∨ ∩M ] be the kernel of (6). The scheme-theoretic image of TV C (σ) → TV C (σ) is the closed subscheme of TV C (σ) defined by the ideal I.
We consider the Cox ring of TV C (σ): S = C[x ξ | ξ ∈σ(1)], with its Gσ-grading. Consider the following monomials in Cox coordinates: x − e * i ,ξ ξ , for i = 1, . . . , k. Let J ⊆ S be the ideal generated by y 1 − z 1 , . . . , y k − z k . It is obviously homogeneous. In order to prove the thesis, we need to show that, under the Cox isomorphism between C[σ ∨ ∩ M ] and S 0 , the ideal I equals the degree zero part of the ideal J, i.e. We now prove the containment ⊆ in (7). Since I is the kernel of (6), it is not difficult to show that I is generated by the elements χ r − χ s whenever r, s ∈σ ∨ ∩M are such that φ(r) = φ(s).
Let us show that q ∈σ ∨ . We need to show that q is non-negative on the rays ofσ. By (5), we distinguish two cases: • v = n − b 1 e 1 − · · · − b k e k ∈σ(1), for some n ∈ N and b 1 , . . . , b k ∈ N; then • v = n + be i ∈σ(1), for some n ∈ N , 1 ≤ i ≤ k and b ∈ N + ; then For each ξ ∈σ(1) we have Therefore, in the ring S we have the equality By (5) every Cox variable appearing in y 1 · · · y k does not appear in z 1 · · · z k . There- is clearly in the ideal J. In a completely analogous way we prove that Cox(χ s − χ q ) is in J. Therefore, by taking the difference, we have that Cox(χ r − χ s ) is in J.
We now prove the containment ⊇ in (7). Let f ∈ J ∩ S 0 . We may write for some f i ∈ S. Let β i ∈ Gσ be the degree of y i − z i . By taking the homogeneous components with respect to the Gσ-grading, we may assume that f i is homogeneous of degree −β i . By decomposing f i into the sum of its monomials, in order to show the containment ⊇ in (7), it is enough to show that p(y i − z i ) ∈ Cox(I), whenever i ∈ {1, . . . , k} and p ∈ S is a monomial of degree −β i . Since py i and pz i are monomials of degree 0 in S, there exist r, s ∈σ ∨ ∩M such that py i = Cox(χ r ) and pz i = Cox(χ s ). Since p(y i − z i ) = Cox(χ r − χ s ), we must show that φ(r) = φ(s). Let us assume that For each ξ ∈σ(1), we have that b ξ + e * i , ξ + = r, ξ b ξ + e * i , ξ − = s, ξ , therefore e * i , ξ = r − s, ξ . Sinceσ is full dimensional, we have r − s = e * i ; this proves that φ(r) = φ(s) and χ r − χ s ∈ I. Now we prove that y 1 − z 1 , . . . , y k − z k is a regular sequence. By Lemma 3.8 it is enough to show that the matrix M = ( e * i , ξ ) 1≤i≤k, ξ∈σ(1) has rank k and every column of M has at most one positive entry. The latter condition is satisfied by (5).
The linear map associated to the matrix M is the composite of the ray map ρ : Z |σ(1)| →Ñ = N ⊕ Z k of TV C (σ) and the projection π :Ñ = N ⊕ Z k → Z k . Sinceσ is full-dimensional, ρ ⊗ Z id R is surjective. This implies that (π • ρ) ⊗ Z id R is surjective and that M has rank k.
This implies that the coneσ is a rational convex polyhedral cone inÑ . Moreover, the rays ofσ are among the following rays: • rays passing through the vertices of Q 0 − e 1 − · · · − e k ; • rays passing through the vertices of Q i + e i , as i = 1, . . . , k; • rays of σ that are not in the cone generated by the previous rays. Now we prove that σ =σ ∩ N R . The containment ⊆ is obvious. We need to show the containment ⊇. Letṽ ∈σ ∩ N R . By the convexity of Q 0 , Q 1 , . . . , Q k , which implies that cone Q i + e i = R ≥0 (Q i + e i ) and an analogous statement for Q 0 , we may assume that v = v + λ 0 (q 0 − e 1 − · · · − e k ) + λ 1 (q 1 + e 1 ) + · · · + λ k (q k + e k ) = v + λ 0 q 0 + λ 1 q 1 + · · · + λ k q k + (λ 1 − λ 0 )e 1 + · · · + (λ k − λ 0 )e k for some v ∈ σ, q i ∈ Q i and λ i ≥ 0. Sinceṽ ∈ N R , λ 0 = λ i for every i. Thereforẽ v = v + λ 0 (q 0 + q 1 + · · · + q k ). By (iii) and (i), q 0 + q 1 + · · · + q k ∈ Q ⊆ σ and we conclude thatṽ ∈ σ. Now we show thatσ is strongly convex. Since σ is strongly convex and 0 / ∈ Q by (ii), we may find u ∈ int(σ ∨ ) such that min Q u > 0. Since the recession cones of Q, Q 0 , Q 1 , . . . , Q k are contained in σ, the minimum of u on each of these polyhedra exists. Considerũ In order to show thatσ is strictly convex, we prove thatũ is positive on the rays ofσ. We may distinguish three cases as follows: • the ray passes through v − e 1 − · · · − e k , for some v ∈ vert(Q 0 ); then • the ray passes through v + e i , for some v ∈ vert(Q i ) and 1 ≤ i ≤ k; then • the ray is a ray of σ through v ∈ N \ {0}; then ũ, v = u, v > 0, because u ∈ int(σ ∨ ); This concludes the proof of the strong convexity ofσ. We now show thatσ has dimension rankÑ . Equivalently we see that zero is the unique linear functional onÑ that vanishes overσ. Letũ = u + k i=1 a i e * i ∈M be such that it vanishes overσ. In particular it vanishes over σ, hence u = 0 because σ is full-dimensional. By evaluatingũ on Q i + e i we see that a i must be zero. This implies thatũ = 0.
This concludes the proof of Lemma 3.4.
Proof of Theorem 3.5(A). By Lemma 3.4 and Lemma 3.9 it is enough to show that the toric morphism TV C (σ) → TV C (σ) is a closed embedding. Before proving this, we shall prove the following claim: Firstly we show that the minimum of u on Q is attained on a vertex of Q; this comes from the strong convexity of σ as follows. By (i) rec(Q) is contained in σ and so is a strongly convex cone. We have (10) Q = conv vert(Q) + rec(Q).
Since u ∈ σ ∨ , u is non-negative on rec(Q). Therefore there exists a vertex v of Q such that min Q u = u, v . Now we prove the claim (9). By (iv') we may find vertices v i ∈ vert(Q i ), i = 0, 1, . . . , k, such that v = v 0 + v 1 + · · · + v k and they are all integral with at most one exception. This implies that the numbers u, v 0 , u, v 1 , . . . , u, v k are all integral with at most one exception. Therefore But min Q u = u, v and it is clear that min Qi u = u, v i for i = 0, 1, . . . , k. Therefore we have proved (9). Now we prove that the toric morphism TV C (σ) → TV C (σ) is a closed embedding. Equivalently, we have to show that the semigroup homomorphism if we prove thatũ ∈σ ∨ we have finished because the equality φ(ũ) = u obviously holds true. It is clear thatũ is non-negative on σ and it is very easy to show that u is non-negative on Q i + e i , for each i = 1, . . . , k. So it remains to show thatũ is non-negative on Q 0 − e 1 − · · · − e k . If q ∈ Q 0 , then where the last equality is (9) and the last inequality holds because of (i). This concludes the proof of Theorem 3.5(A). Proof of Theorem 3.5(B). From (10) and (v), we have that w is non-negative on rec(Q) and min Q w = w, v for some vertex v of Q. By (iv') we may find vertices v i ∈ vert(Q i ), i = 0, 1, . . . , k, such that v = v 0 + v 1 + · · · + v k and they are all integral with at most one exception. This implies that the numbers w, v 0 , w, v 1 , . . . , w, v k are all integral with at most one exception. Therefore But min Q w = w, v and it is clear that min Qi w = w, v i for i = 0, 1, . . . , k. Therefore we have proved the equality Now we show that the trinomials (4) are elements of the polynomial ring which is the homogeneous coordinate ring of TV C[t1,...,t k ] (σ). It is enough to show that every Cox coordinate appearing in the third monomial in (4) has a non-negative exponent. Fix a ray ξ ofσ. We may distinguish three cases as follows.
• ξ passes through a vertex of Q i +e i , for some 1 ≤ i ≤ k. Then ξ = λ(v +e i ), for some λ ∈ N + and v ∈ vert(Q i ). Then w, ξ = λ w, v − λ min Qi w ≥ 0. • ξ is a ray of σ too. We need to show that w, ξ = w, ξ is non-negative.
For a contradiction assume that w, ξ < 0. Therefore a positive multiple of ξ lies in the polyhedron P := σ ∩ {n ∈ N R | w, n = −1}. Since rec(P ) is strongly convex, P = conv vert(P ) + rec(P ). By (vi) we obtain that ξ = λq + r, for some λ > 0, q ∈ Q, r ∈ rec(P ). Since λq and r are both in σ and ξ is a ray of σ, we have that ξ = µq for some µ ≥ 0. From (iii) we have that ξ is in cone Q 0 − e 1 − · · · − e k , Q 1 + e 1 , . . . , Q k + e k . This contradicts the fact that ξ is a ray of both σ andσ. Now the closed subscheme X is well defined. We need to show that the restriction of X → A k C to any infinitesimal neighbourhood of O ∈ A k C is flat. Fix a (t 1 , . . . , t k )-primary ideal q. Consider the local artinian C-algebra A = C[t 1 , . . . , t k ]/q. We need to show that X × A k C Spec A → Spec A is flat. The homogeneous coordinate ring of TV A (σ) is the polynomial A-algebra B = A[x ξ | ξ ∈σ(1)]. By (A) the trinomials (4) form a (B ⊗ A C)-regular sequence. By Lemma 3.10 the homogeneous ideal J ⊆ B generated by the trinomials (4) is such that B/J is flat over A. By Lemma 2.2 the sheafification of the Gσ-graded B-module B/J is a coherent sheaf on TV A (σ) which is flat over Spec A. This sheaf is the structure sheaf of the closed subscheme X × A k C Spec A of TV A (σ). Therefore we have proved that X × A k C Spec A is flat over Spec A. This concludes the proof of Theorem 3.5(B).

Deformations of affine toric pairs
If in Theorem 3.5 we assume that (Q, Q 0 , Q 1 , . . . , Q k , w) is a ∂-deformation datum, then Mavlyutov's construction of deformations of affine toric varieties, which appears in [Mav] and is rewritten in §3, actually gives deformations of their toric boundary too.
More precisely, in the setting of Theorem 3.5 with the additional hypothesis (iv), we construct a reduced divisor D in the toric varietyX = TV C (σ) such that D ∩X is the toric boundary ∂X of X. Theorem 3.5 constructs a formal deformation X → A k C of X as a closed subscheme in the trivial familyX × C A k C ; then one can see that the closed subscheme X ∩(D× C A k C ) gives a deformation of ∂X. In other words, induces a formal deformation of the toric pair (X, ∂X). This is the content of the following theorem, which is the precise formulation of Theorem 1.1.
Theorem 4.1. Let N be a lattice, let σ ⊆ N R be a strongly convex rational polyhedral cone with dimension rank N , let (Q, Q 0 , Q 1 , . . . , Q k , w) be a ∂-deformation datum for (N, σ), and letÑ ,σ andw be as in Notation 3.2.
Let X be the affine toric variety associated to σ, let ∂X be the toric boundary of X, and letX be the affine toric variety associated toσ. Consider the reduced effective divisor D onX defined by the homogeneous ideal generated by the following monomial in the Cox coordinates ofX: Let t 1 , . . . , t k be the standard coordinates on A k C . Consider the closed subscheme X ofX × Spec C A k C = TV C[t1,...,t k ] (σ) defined by the homogeneous ideal generated by the following trinomials in Cox coordinates: x − e * i ,ξ ξ for i = 1, . . . , k.
The trinomial (13) is xy − u 2 − tz p and the monomial (12) is zu. By Theorem 4.1 we consider the following closed subschemes ofX × C Spec C[t]: induces a formal deformation of the pair (X, ∂X) over C [[t]], by base changing to Spec C[t]/(t n+1 ) for each n ∈ N.
The rest of this section is devoted to the proof of Theorem 4.1.
Lemma 4.3. Let S be a polynomial ring over C in finitely many indeterminates. Let m 1 , . . . , m t ∈ S \ {1} be some monomials such that the t sets of indeterminates appearing in these monomials have empty pairwise intersections. Let R be the C-subalgebra of S generated by m 1 , . . . , m t . Then m 1 , . . . , m t are algebraically independent over C and S is a free R-module.
Proof. It is clear that the monomials m 1 , . . . , m t are algebraically independent over C. Another way to see this is to notice that they form a regular sequence in S and then use [Mat89, Exercise 16.6].
Now we want to prove that S is a free R-module. For each i = 1, . . . , t, let S i be the polynomial ring over C in the indeterminates that appear in m i and let R i ⊆ S i be the C-subalgebra generated by m i . Let S 0 be the polynomial ring over C in the indeterminates of S that do not appear in any m i 's. If we prove that S i is a free R i -module for each i = 1, . . . , t, then S = S 0 ⊗ C S 1 ⊗ C · · · ⊗ C S t will be free over Therefore we may assume that t = 1 and that all the indeterminates of S appear in m := m 1 , i.e. S = C[x 1 , . . . , x n ] and m = x a1 1 · · · x an n with a 1 , . . . , a n ∈ N + . The set is a free basis of S as R-module.
Proof. Consider the polynomial ring R = C[Y 1 , . . . , Y k , Z 1 , . . . , Z r ] where the Y i 's and the Z j 's are indeterminates. Consider the C-algebra homomorphism ϕ : R → S defined by Y i → y i and Z j → z j . By Lemma 4.3, ϕ is injective and flat. Consider the monomial Z 0 = Z c1 1 · · · Z cr r . Consider the C-algebra automorphism θ of R that fixes the Z j 's and maps Y i to Y i − Z 0 . By applying θ to the regular sequence Y 1 , . . . , Y k , Z 1 · · · Z r we get the regular sequence Now, by applying the flat map ϕ to this regular sequence, we get that the sequence (14) is regular.
Remark 4.5. A slight generalisation of Lemma 4.4 can be used to prove that the binomials in Theorem 3.5(A) form a regular sequence, without using [FS96] and Lemma 3.8.
Proof of Theorem 4.1. By Theorem 3.5, it is enough to deal with the toric boundary. Here we adopt some notations used in the proof of Lemma 3.9. Let I be the kernel of the surjective ring homomorphism ψ : that is associated to the surjective semigroup homomorphism φ :σ ∨ ∩M → σ ∨ ∩ M given by u + a 1 e * 1 + · · · + a k e * k → u. The ideal of the toric boundary ∂X in X is Therefore the ideal of ∂X inX is Now we consider the Cox ring ofX: S = C[x ξ | ξ ∈σ(1)] with its Gσ-grading. In the proof of Theorem 3.5(A) we had the following description of the rays ofσ.
• Rays passing through the vertices of Q 0 − e 1 − · · · − e k . We denote by z 0,1 , . . . , z 0,s0 the corresponding Cox coordinates. • Rays passing through the vertices of Q i + e i , as i = 1, . . . , k. We denote by y i,1 , . . . , y i,si the corresponding Cox coordinates. • Rays of σ that are not in the cone generated by the previous rays. We denote by z σ,1 , . . . , z σ,sσ the corresponding Cox coordinates.
The exponents c 0 , . . . , c s0 are the minimal positive integers by which we have to multiply the vertices of Q 0 to get lattice points. Here we have used (iv) in Definition 3.1 to deduce that y i are reduced monomials. We see that y i are exactly the ones used in the proof of Lemma 3.9, whereas the monomials z 1 , . . . , z k there coincides with z 0 in our case. We see that y i − z 0 is the binomial obtained from the trinomial (13) by setting t i = 0 and z is the monomial in (12). Let J ⊆ S be the ideal generated by y 1 − z 0 , . . . , y k − z 0 and let J = J + Sz. We already know, from Lemma 3.9 or Theorem 3.5, that the Cox isomorphism between C[σ ∨ ∩M ] and S 0 ⊆ S maps the ideal I onto the degree zero part of the ideal J, i.e. Cox(I) = J ∩S 0 . We have to prove that This equality will imply that the scheme-theoretic intersection X ∩D coincides with ∂X.
This concludes the proof of the equality (15) and, consequently, of the fact that X ∩ D = ∂X.
By Lemma 4.4 we have that y 1 −z 0 , . . . , y k −z 0 , z is a regular sequence. Adapting the proof of Theorem 3.5(B) we conclude.

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 (Theorem 3.5) and then apply the Proj functor. We will use the lemmata in §2.2.
(A) Then the inclusion τ →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: x − e * i ,ρ ρ for i = 1, . . . , k, whereΣ is the fan ofX inÑ . Moreover, the k binomials in (16) form a regular sequence.
(B) Let t 1 , . . . , t k be the standard coordinates on A k C . Consider the closed subscheme X ofX × Spec C A k C = TV C[t1,...,t k ] (Σ) defined by the homogeneous ideal generated by the following trinomials in Cox coordinates: x − e * i ,ρ ρ for i = 1, . . . , k. Then the morphism X → A k C induces a deformation of X over C[[t 1 , . . . , t k ]] and over an open neighbourhood of the origin in A k C . The rest of this section is devoted to the proof of Theorem 5.1.
Proof of Theorem 5.1(A). By Lemma 3.4τ is a (n + 1 + k)-dimensional strongly convex rational polyhedral cone inÑ 0 . It is clear that e 0 ∈τ . Now we show that e 0 is in the interior ofτ : it is enough to show that, ifũ = u+ k i=0 h i e * i ∈τ ∨ ∩M 0 and h 0 = 0, thenũ = 0. Since τ ⊆τ , we have that u is nonnegative on τ ; but e 0 is in the interior of τ , so u = 0. By evaluatingũ = k i=1 h i e * i on Q 0 − e 1 − · · · − e k , Q 1 + e 1 , . . . , Q k + e k , we see h 1 = · · · = h k = 0. This proves that e 0 lies in the interior ofτ .
Thanks to Lemma 2.4 we haveX = Proj C[τ ∨ ∩M 0 ] and X = Proj C[τ ∨ ∩ M 0 ]. The ring homomorphism , which is induced by the inclusion τ →τ and is surjective by the proof of Theorem 3.5, is homogeneous with respect to the N-grading and induces a closed embedding ι : X →X. Using the isomorphisms (2) it is not difficult to write down the formulae for the actions of the tori T N and TÑ on the affine charts of X andX, respectively. From these formulae it is possible to see that ι is a toric morphism.
We have to prove that X coincides with the closed subscheme ofX defined by the ideal JX ⊆ SX generated by the binomials (16). Let JC = JX SC be the extension of JX to the total coordinate ring SC of the affine coneC = Spec C[τ ∨ ∩M 0 ] via the ring homomorphism SX → SC defined in Lemma 2.5. The ideal JC is generated generated by the binomials The matrices MΣ = ( e * i , ρ ) 1≤i≤k,ρ∈Σ(1) and Mτ = ( e * i , ξ ) 1≤i≤k,ξ∈τ (1) differ just by multiplication by a positive integer on each column, namely the numbers b ρ defined in Lemma 2.5. From the proof of Lemma 3.9 we see that Mτ has rank k and each of its columns has at most one positive entry. Therefore also the matrix MΣ has these two properties. By Lemma 3.8 or Remark 4.5, the binomials (16) form a regular sequence.
This concludes the proof of Theorem 5.1(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 let π : Y → Spec A be a proper morphism of schemes such that Y × Spec A Spec A/m n → Spec A/m n is flat for every n ∈ N. Then π is flat.
Proof. This proof relies on an argument that appears in the proof of [TV13, Proposition 6.51]. We want to show that the set Z = {y ∈ Y | O Y,y is not flat over A} is empty. By covering Y with open affine subschemes and by using [Mat89,Theorem 24.3], one can see that Z is closed in Y .
Assume by contradiction that Z is non-empty. Since π is closed, the set π(Z) is a closed non-empty subset of Spec A. Therefore m ∈ π(Z). Hence there exists y 0 ∈ Z such that π(y 0 ) = m. Let Spec R be an affine open neighbourhood of y 0 in Y and let B = O Y,y0 be the local ring of Y at y 0 . We know that A/m n → R/m n R is flat for every n ∈ N. Therefore the local homomorphism A → B is such that A/m n → B/m n B is flat for every n ∈ N. By the local flatness criterion [Mat89,Theorem 22.3] A → B is flat. But this is absurd because y 0 ∈ Z. Proof. Since the problem is local and Y → S is quasi-compact, we may assume S = Spec A, Y = Spec B and s = m for some noetherian ring A, some finitely generated A-algebra B and some prime ideal m of A. We know that B ⊗ A A m is flat over A m . Let us consider the set We identify Spec(B ⊗ A A m ) with the set of primes P ∈ Spec B such that P ∩A ⊆ m. If P ∈ Spec B is such that P ∩ A ⊆ m, then by [Mat89, Theorem 7.1] from the flatness of B ⊗ A A m over A m we deduce that B P is flat over (A m ) (P ∩A)Am = A P ∩A . This shows that Spec Consider the set A \ m endowed with the order relation ≤ such that f ≤ g if and only if g ∈ √ Af . If f ≤ g, there is the localisation map A f → A g , given by the restriction of the structure sheaf of Spec A from the principal open subset defined by f to the principal open subset defined by g. As f runs in A \ m, the rings A f 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 m is the limit of B f as f ∈ A\m. We are in the situation of inverse limits of affine schemes studied in [Gro66,§8], i.e. Spec(B ⊗ A A m ) is the projective limit of the affine schemes Spec B f as f runs in A \ m.
For every f ∈ A \ m, consider the set . Since E is the limit of the E f 's, by [Gro66,Corollaire 8.3.5] we have that there exists f 0 ∈ A \ m such that E f0 = Spec B f0 . This implies that B f0 is flat over A f0 . Therefore we may take U = Spec A f0 .
Proof of Theorem 5.1(B). The proof of the fact that the trinomials (17) are elements of C[t 1 , . . . , t k ][x ρ | ρ ∈Σ(1)] is completely analogous to what is done in the proof of Theorem 3.5(B) and will be omitted.
Let X be the closed subscheme ofX × Spec C A k C defined by the homogeneous 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 Spec C[t 1 , . . . , t k ]/q is flat over C[t 1 , . . . , t k ]/q for every (t 1 , . . . , t k )-primary ideal q of C[t 1 , . . . , t k ], thanks to Lemma 3.10 and Lemma 2.2, as in the proof of Theorem 3.5(B). If A = C[t 1 , . . . , t k ] (t1,...,t k ) is the local ring of A k C at the origin O, by Lemma 5.2 the morphism X × A k C Spec A → Spec A is flat, and consequently it induces a deformation of X overÂ = C[[t 1 , . . . , t k ]]. By Lemma 5.3 we may find an open neighbourhood U ⊆ A k C of O such that X × A k C U is flat over U .

Deformations of projective toric pairs
In Theorem 5.1, from a projective toric variety X with an ample Q-Cartier Q-divisor D and a Minkowski decomposition of a certain polyhedron with some properties we constructed a deformation of X. Here we show that if D is a Zdivisor then we can construct a deformation of the toric pair (X, ∂X). This is the content of the following theorem.
Let ∂X be the toric boundary of X. Let (X,D) be the polarised projective toric variety associated to the coneτ via Lemma 2.3. Consider the reduced effective divisor D onX defined by the homogeneous ideal generated by the following monomial in the Cox coordinates ofX: whereΣ is the fan ofX inÑ . Let t 1 , . . . , t k be the standard coordinates on A k C . Consider the closed subscheme X ofX × Spec C A k C = TV C[t1,...,t k ] (Σ) defined by the homogeneous ideal generated by the following trinomials in Cox coordinates: x − e * i ,ρ ρ for i = 1, . . . , k.
Then the diagram and over some open neighbourhood of the origin in A k C . We mean that the base change of the diagram above to the origin of A k C is the closed embedding ∂X → X over Spec C and that we get flat families when we base change the morphisms X ∩ (D × Spec C A k C ) → A k C and X → A k C to Spec C[[t 1 , . . . , t k ]] and to some open neighbourhood of the origin in A k C . Proof of Theorem 6.1. By Theorem 5.1 it is enough to deal with the toric boundary. Let C andC be the affine cones over X andX as in the proof of Theorem 5.1. Let JX be the ideal in the Cox ring ofX generated by the binomials (16) and the monomial (20). We need to show that the closed subscheme ofX defined by JX coincides with ∂X.
Since X is polarised by a Z-divisor, by Lemma 2.3 the primitive generator of every ray of the cone τ is of the form ρ − a ρ e 0 for some a ρ ∈ Z and ρ ∈ N primitive. From the definition ofτ in Notation 3.2 it is easy to see that alsoτ has the same property, i.e. the primitive generator of every ray ofτ is of the form ρ − a ρ e 0 for some a ρ ∈ Z and ρ ∈Ñ primitive. Therefore the homomorphism SX → SC, defined in Lemma 2.5, is the identity. In particular the extended ideal JC := JX SC ⊆ SC is generated by the binomials (19) and the monomial ξ∈τ (1) : ∀i∈{1,...,k}, e * i ,ξ ≤0

Mutations of Fano polytopes and deformations of Fano toric pairs
Here we recall the definition of mutations between Fano polytopes from [ACGK12] and we prove Theorem 7.3. The definition of Fano polytope has been given at the beginning of §1.3.
If N is a lattice, w ∈ M R \ {0} and h ∈ R, then we denote by H w,h the set of all points of N R lying at height h with respect to w, i.e. the affine hyperplane H w,h := {v ∈ N R | w, v = h}. In particular w ⊥ = H w,0 .
Definition 7.1. Let P ⊆ N R be a Fano polytope. A mutation datum for P is a pair (w, F ) where w ∈ M is a primitive vector and F ⊆ w ⊥ ⊆ N R is a lattice polytope satisfying the following condition: for every h ∈ Z such that min P w ≤ h < 0, there exists a (possibly empty) lattice polytope G h ⊆ N R such that Note that, for given Fano polytope P ⊆ N R and primitive vector w ∈ M , a polytope F such that (w, F ) is a mutation datum for P need not exist. From a mutation datum we make the following construction.
Definition 7.2 ([ACGK12, Definition 5]). Let P ⊆ N R be a Fano polytope and let (w, F ) be a mutation datum for P . Assume that {G h } min P w≤h<0 is a collection of lattice polytopes satisfying (22). We define the corresponding mutation to be the lattice polytope mut w,F (P ) := conv −1 The polytope mut w,F (P ) does not depend on the choice of {G h }. Moreover, mut w,F (P ) is a Fano polytope. See [ACGK12,§3] or [Akh15, §2.5] for the proofs of these statements.
Roughly speaking, mut w,F (P ) is obtained from P by adding hF at height h (with respect to w) for h > 0 and by removing (−h)F at height h for h < 0. The pair (w, F ) is a mutation datum precisely when it is possible to remove from P multiples of F at negative heights. For an example of mutation of Fano polytopes see the beginning of Example 7.4.
The following theorem is the precise version of Theorem 1.3.
Theorem 7.3. Let P ⊆ N R be a Fano polytope, let (w, F ) be a mutation datum for P , and let P = mut w,F (P ) be the mutated polytope. Let X P (resp. X P ) be the Fano toric variety associated to the spanning fan of P (resp. P ) and let ∂X P (resp. ∂X P ) be the toric boundary of X P (resp. X P ). Set vert(P ) ≥0 = vert(P ) ∩ {v ∈ N | w, v ≥ 0}, vert(P ) <0 = vert(P ) ∩ {v ∈ N | w, v < 0}.