Toric Geometry and the Semple-Nash modification

TL;DR

This paper develops a combinatorial framework for general toric varieties, extending the Semple-Nash modification theory to non-normal cases and linking it to blowing up monomial ideals and valuations.

Contribution

It introduces a combinatorial description of toric varieties without normality and relates the Semple-Nash modification to blowing up logarithmic jacobian ideals.

Findings

Semple-Nash modification is isomorphic to blowing up logarithmic jacobian ideals in characteristic zero.

Blowing-up of monomial ideal sheaves generalizes to non-normal toric varieties.

Iterated Semple-Nash modifications uniformize toric valuations of maximal rank.

Abstract

This paper proposes some material towards a theory of general toric varieties without the assumption of normality. Their combinatorial description involves a fan to which is attached a set of semigroups subjected to gluing-up conditions. In particular it contains a combinatorial construction of the blowing up of a sheaf of monomial ideals on a toric variety. In the second part it is shown that over an algebraically closed base field of zero characteristic the Semple-Nash modification of a general toric variety is isomorphic to the blowing up of the sheaf of logarithmic jacobian ideals and that in any characteristic this blowing-up is an isomorphism if and only if the toric variety is non singular. In the second part we prove that orders on the lattice of monomials (toric valuations) of maximal rank are uniformized by iterated Sempla-Nash modifications.

Equations

$${\gamma }_1=\frac{1}{s_1},{\gamma }_{i+1}=s_i{\gamma }_i+\frac{1}{s_1\dots s_{i+1}}.$$

$$(\sum _{\gamma }{a}_{\gamma }{t}^{\gamma })(\sum _{\delta }{b}_{\delta }{t}^{\delta })=\sum _{\zeta }(\sum _{\gamma +\delta =\zeta }{a}_{\gamma }{b}_{\delta })t^{\zeta }.$$

$$A[t^{\Gamma \times {\Gamma }^{\prime }}]\to A[u^{\Gamma }]{\otimes }_{A}A[v^{{\Gamma }^{\prime }}];t^{(\gamma ,{\gamma }^{\prime })}\mapsto u^{\gamma }{\otimes }_A{v}^{{\gamma }^{\prime }}$$

$${\mathbf{Z}}^d\to M,a=(a_1,\dots ,a_d)\mapsto \sum _{i=1,\dots d}{a}_i{m}_i$$

$$T^M(k)={\mathrm{Hom}}_{\mathrm{groups}}(M,k^{\ast })=N{\otimes }_{\mathbf{Z}}{k}^{\ast },$$

$$\{Closed\; points\; of\mathrm{Spec}k[t^{\Gamma }]\}\leftrightarrow \{semigroup\; homomorphisms\Gamma \to k\},$$

$$k[t^{\Gamma }]\to k[t^{\Gamma }]{\otimes }_{k}k[t^{\Gamma }]\to k[t^M]{\otimes }_{k}k[t^{\Gamma }]$$

$$i_{\tau }:T^{\Gamma \cap {\tau }^{\perp }}\hookrightarrow T^{\Gamma }$$

$$\gamma\mapsto\left\{\begin{array}[]{cr}u(\gamma)&\mathrm{if}\gamma\in\tau^{\bot},\\ 0&\mathrm{otherwise}.\end{array}\right.$$

$$\tau \mapsto \mathrm{orb}(\tau ,\Gamma ):=i_{\tau }(T^{M(\tau ,\Gamma )})(\text{mbox}resp.\tau \mapsto i_{\tau }(T^{\Gamma \cap {\tau }^{\perp }}))$$

$$T^{\Gamma }=\underset{\tau \le \sigma }{⨆}\mathrm{orb}(\tau ,\Gamma ).$$

$${\Gamma }_{\tau }:=\Gamma +M(\tau ,\Gamma )$$

$${\Gamma }_{\tau }=\Gamma +{\mathbf{Z}}_{\ge 0}(-m).$$

$$\begin{array}[]{ccc}{T}^{\check{\sigma}\cap M}&\hookrightarrow&{T}^{\Gamma}\\ \uparrow&&\uparrow\\ {T}^{\check{\sigma}\cap M}_{f}&\hookrightarrow&{T}^{\Gamma}_{f},\end{array}$$

$$\tau \mapsto \mathrm{orb}(\tau ,{\Gamma }_{\tau }):=i_{\tau }(T^{M(\tau ,{\Gamma }_{\tau })})$$

$$\tau \mapsto i_{\tau }(T_{\Sigma (\tau )}^{\Gamma (\tau )})$$

$${\Gamma }_{\theta }\cap {\tau }^{\perp }={\Gamma }_{\sigma }\cap {\tau }^{\perp }+M(\theta (\tau ),{\Gamma }_{\sigma (\tau )})={\Gamma }_{{\sigma }^{\prime }}\cap {\tau }^{\perp }+M(\theta (\tau ),{\Gamma }_{{\sigma }^{\prime }(\tau )}).$$

$${\mathrm{ord}}_{\mathcal{I}}:\sigma \to \mathbf{R},\nu \mapsto \underset{m\in \mathcal{I}}{\min }⟨\nu ,m⟩.$$

$$m_i\mapsto {\sigma }_i\text{mbox}ifandonlyif{\mathrm{ord}}_{\mathcal{I}}(\nu )=⟨\nu ,m_i⟩\text{mbox}forall\nu \in {\sigma }_i.$$

$${\Gamma }_i=\Gamma +⟨m_1-m_i,\dots ,m_{i-1}-m_i,m_{i+1}-m_i,\dots ,m_k-m_i⟩\subset {\check{\sigma }}_i\cap M.$$

$$\pi :B\to \mathrm{Spec}A[t^{\Gamma }]$$

$$\Gamma +\mathbf{Z}(m_2-m_1)+\sum _{j=2,\dots ,k}{\mathbf{Z}}_{\ge 0}(m_j-m_1)=\Gamma +\mathbf{Z}(m_1-m_2)+\sum _{j=2,\dots ,k}{\mathbf{Z}}_{\ge 0}(m_j-m_2),$$

$${\varphi }^{\ast }:M\to M^{\prime }\text{mbox}and{\varphi }_{\ast }:N^{\prime }\to N$$

$$k[t^M]\to k[t^{M^{\prime }}],t^m\mapsto t^{{\varphi }^{\ast }(m)},m\in M.$$

$$\begin{array}[]{ccc}T_{\Sigma^{\prime}}&\longrightarrow&T_{\Sigma}\\ \downarrow&&\downarrow\\ T_{\Sigma^{\prime}}^{\Gamma^{\prime}}&\longrightarrow&T_{\Sigma}^{\Gamma}\end{array}$$

$${\bar{\varphi }}_{{\sigma }^{\prime },\sigma }:T^{{\Gamma }_{{\sigma }^{\prime }}^{\prime }}\to T^{{\Gamma }_{\sigma }}\text{mbox}givenonclosedpointsby{\bar{\varphi }}_{{\sigma }^{\prime },\sigma }(x)=x\circ {\varphi }_{|{\Gamma }_{\sigma }}^{\ast },$$

$${\bar{\varphi }}_{{\sigma }^{\prime },\sigma }(y\cdot x)=(y\cdot x)\circ {\varphi }_{|{\Gamma }_{\sigma }}^{\ast }=(y\circ {\varphi }^{\ast })\cdot (x\circ {\varphi }_{|{\Gamma }_{\sigma }}^{\ast })=\varphi (y)\cdot {\bar{\varphi }}_{{\sigma }^{\prime },\sigma }(x).$$

$$f(\mathrm{orb}({\sigma }^{\prime },{\Gamma }_{{\sigma }^{\prime }}^{\prime }))\subset \mathrm{orb}(\sigma ,{\Gamma }_{\sigma })\text{mbox}andf(\mathrm{orb}({\tau }^{\prime },{\Gamma }_{{\tau }^{\prime }}^{\prime }))\subset \mathrm{orb}(\tau ,{\Gamma }_{\tau }).$$

$$\bar{\varphi }:T_{{\Sigma }^{\prime }}^{{\Gamma }^{\prime }}\to T_{\Sigma (\tau )}^{\Gamma (\tau )}.$$

$$k[t^{{\Gamma }_{\sigma }}]{\otimes }_{k}k[t^{{\Gamma }_{{\sigma }^{\prime }}}]\to k[t^{{\Gamma }_{\tau }}]\text{mbox},determinedby{t}^{\gamma }\otimes t^{{\gamma }^{\prime }}\mapsto t^{\gamma +{\gamma }^{\prime }}.$$

Full text

González Pérez is supported by Severo Ochoa project SEV-2011-0087 and MTM2010-21740-C02-01 grants. Teissier is partially supported by grant MTM2007-64704

To Heisuke Hironaka on the occasion of his 80th birthday

Introduction

In the first part of this paper we study abstract toric varieties without the assumption of normality. Since Sumihiro’s Theorem on the existence of a covering of a toric variety by invariant affine varieties fails without the assumption of normality, we have to set the existence of such a covering as part of the definition of a toric variety. Then an abstract toric variety has a combinatorial description: it corresponds to certain semigroups in the convex duals of the cones of a fan, which satisfy a natural gluing-up condition. This generalizes the definition of [12] which concerns toric varieties equivariantly embedded in projective space. In spirit it is also a continuation of our previous work [13] on embedded normalization and embedded toric resolution of singularities of affine toric varieties. We can then define blowing-ups of sheaves of monomial ideals as toric varieties, and describe the corresponding operations on semigroups. We also provide the combinatorial description of torus-invariant Cartier divisors on a toric variety and the general versions of the classical criteria for ampleness and very-ampleness.

In the second part of the paper we use the description of blowing-ups given in the first part to show that one can obtain a local uniformization of a monomial valuation of maximal rank dominating a toric variety, by a finite number of iterations of the blowing-up of the logarithmic jacobian ideal introduced in [14]. If the field is of characteristic zero, this blowing-up is isomorphic to the Semple-Nash modification. Recall that this is a canonical modification of a reduced equidimensional space which replaces each point by the set of limit positions of tangent spaces at nearby non-singular points. See the second part for details.

Our result is related in the special case of toric varieties to an interesting question apparently first asked by Semple in [27], that is, if the iteration of the Semple-Nash modification eventually resolves the singularities of an algebraic variety defined over a field of characteristic zero. One can present the relation of this conjectural resolution process with classical resolution of singularities as follows: two types of proper birational correspondences naturally associate a singular variety to a non-singular one: proper birational projections of an embedded non-singular algebraic variety to a smaller dimensional ambient space, and the taking of the envelope of a family of linear subspaces (of an affine or projective space) whose parameter space is a non singular algebraic variety. For example, a family of lines in the real plane parametrized by a circle, such as the Simson-Wallace lines of a triangle, parametrized by its circumscribed circle, have an envelope with singularities, generically with an odd number of cusps (see [35]). In the Simson-Wallace lines case, it is a quartic with three cusps (see [30]). The natural map from the circle to the envelope is a homeomorphism and a local isomorphism outside of the cusps. Another example is given by the discriminant hypersurface of a versal deformation of an isolated complex hypersurface singularity: it is the envelope of a family of complex hyperplanes parametrized by the critical locus, which is non-singular and is the normalization of the discriminant (see [33]) and therefore a resolution of singularities.

Hironaka’s resolution shows that in characteristic zero all singularities may be created by the first process, and Semple-Nash resolution in general would show that at least all singularities of projective varieties in characteristic zero may be created by iterating a natural generalization of the second process, if we allow singular spaces as parameter spaces. Moreover it would produce a canonical process for local uniformization of valuations in characteristic zero, as our results here do for maximal rank monomial valuations on toric varieties in a characteristic-free manner.

Acknowledgements. We are grateful to Monique Lejeune-Jalabert for introducing us to the reference [27], to Ezra Miller for bringing to our attention the work of H.M. Thompson, to Michael Thaddeus for detecting errors in previous versions of this work.

We also mention that while this paper was in preparation, Mr. Daniel Duarte provided in [7] a proof of a result similar to our main result, a priori somewhat stronger and giving an effective bound on the number of steps required for desingularization, in the two-dimensional case. A preprint of Dima Grigoriev and Pierre Milman (see [17]) also provided an approach to the Semple-Nash desingularization problem for toric varieties with explicit results in dimension 2. In particular it contains a version of our Lemma 12.1. We thank the referee for his careful work and helpful suggestions.

We also thank Patrice Philippon for his help, David Cox for his remarks and the Institut Mathématique de Jussieu and the Dpto. Álgebra, Universidad Complutense de Madrid for their hospitality.

Part I: Toric varieties

The purpose of this part is to develop the combinatorial theory of toric varieties without the assumption of normality. We refer to [6], [11], [21], [25], and [26] for background on normal toric varieties, and to the books of Oda-Miyake ([25]), Gel*′*fand, Kapranov, and Zelevinsky ([12]) and Sturmfels ([31]) for certain classes of non necessarily normal toric varieties. We also point to previous work by H.M. Thompson towards the development of a general theory of toric varieties, see [36], [37] and, from the perspective of Log Schemes, [38]. This part is also connected with the more general theory of monoid schemes recently developped by Cortiñas et al. in [4]. We recommend [9] as a particularly accessible introduction to (normal) toric varieties, and also the recent book [5] of D. Cox, J. Little and H. Schenk on the subject.

1. Semigroups and semigroup algebras

The theory of affine toric varieties over a field $k$ is the geometric version of the theory of semigroup algebras over $k$. For part of the theory, one can omit the assumption that the semigroup is finitely generated, and replace the field $k$ by a commutative ring.

Definition 1.1**.**

A (commutative) semigroup $\Gamma$ is a set equipped with an operation $+\colon\Gamma\times\Gamma\to\Gamma$ such that $\epsilon_{1}+\epsilon_{2}=\epsilon_{2}+\epsilon_{1}$, which satisfies the associativity property and is cancellative ($\epsilon_{1}+\epsilon_{2}=\epsilon_{1}+\epsilon_{3}$ implies $\epsilon_{2}=\epsilon_{3}$). We shall assume that $\Gamma$ contains a zero element [math] such that $\epsilon+0=\epsilon$. We denote by $\mathbf{Z}\Gamma$ the group generated by $\Gamma$ (defined in a similar way as the field of fractions of an integral domain). To say that $\Gamma$ is cancellative means that the natural map of semigroups $\Gamma\to\mathbf{Z}\Gamma$ is injective. We say that $\Gamma$ is torsion free if the abelian group $\mathbf{Z}\Gamma$ is, which means that the only solution in $\mathbf{N}$ of an equation $m\gamma=m\gamma^{\prime}$ with $\gamma,\gamma^{\prime}\in\Gamma,\gamma\neq\gamma^{\prime},$ is $m=0$. Since $\Gamma$ is cancellative, it implies that the only solution in $\Gamma$ of an equation $m\gamma=n\gamma$ with $m,n\in\mathbf{N},\ m\neq n,$ is $\gamma=0$.

A system of generators of a semigroup is a subset $(\gamma_{i})$ of $\Gamma$ such that each element of $\Gamma$ is a (finite) linear combination of the $\gamma_{i}$ with non negative integral coefficients. The elements of $\mathbf{Z}\Gamma$ are finite linear combinations of the $\gamma_{i}$ with integral coefficients. If the semigroup $\Gamma$ is cancellative, torsion free and finitely generated, the group $\mathbf{Z}\Gamma$ is a lattice so that $\Gamma$ is isomorphic to a finitely generated subsemigroup of a lattice $\mathbf{Z}^{d}$.

Examples of semigroups:

$\bullet$ Given finitely many coprime integers the set of all combinations of these integers with non negative integral coefficients is a subsemigroup $\Gamma$ of the semigroup $\mathbf{N}$ of integers, and ${\mathbf{N}}\setminus\Gamma$ is finite. In fact any semigroup of integers is finitely generated.

$\bullet$ Let $(s_{i})_{i\geq 1}$ be a sequence of integers such that $s_{i}\geq 2$ for $i\geq 2$. Define a sequence of rational numbers $\gamma_{i}$ inductively by:

[TABLE]

The set of integral linear combinations of the $\gamma_{i}$ is a subsemigroup of ${\mathbf{Q}}_{\geq 0}$, which is not finitely generated. In fact the $\gamma_{i}$ form a minimal set of generators.

$\bullet$ Let $d$ be an integer and let $\check{\sigma}$ (the reason for the dual notation will appear below) be a convex cone of dimension $d$ in $\check{\mathbf{R}}^{d}$. Denote by $M$ the integral lattice of $\check{\mathbf{R}}^{d}$. Then the intersection $\check{\sigma}\cap M$ is a subsemigroup of the group $M$, which generates $M$ as a group.

By a Theorem of Gordan, if the convex cone $\check{\sigma}$ is rational in the sense that it is the intersection of finitely many half spaces determined by hyperplanes with integral coefficients, then the semigroup $\check{\sigma}\cap M$ is finitely generated.

Definition 1.2**.**

If $\Delta$ is a subsemigroup of $\Lambda$ the saturation of $\Delta$ in $\Lambda$ is the semigroup $\Theta$ consisting of those elements of $\Lambda$ which have a multiple in $\Delta$. The semigroup $\Delta$ is saturated in $\Lambda$ if $\Delta=\Theta$.

Lemma 1.3**.**

Let $\check{\tau}$ be a rational convex cone in $\check{\mathbf{R}}^{d}$ for the lattice $M$. The semigroup $\check{\tau}\cap M$ is saturated in $M$ and the saturation of a subsemigroup $\Gamma$ of $M$ is $\check{\sigma}\cap M$ where $\check{\sigma}=\mathbf{R}_{\geq 0}\Gamma$ is the closed convex cone generated by $\Gamma$.

Proof. The first statement is clear. If $\mathbf{R}_{\geq 0}\Gamma=\check{\sigma}$, any element of $\check{\sigma}\cap M$ is a combination with rational coefficients of elements of $\Gamma$. Chasing denominators shows that an integral multiple of this element is in $\Gamma$. The converse is clear. $\Box$

Definition 1.4**.**

Let $\Gamma$ be a finitely generated commutative semigroup and $A$ a commutative ring. The semigroup algebra $A[t^{\Gamma}]$ of $\Gamma$ with coefficients in $A$ is the ring consisting of finite sums $\sum_{\gamma}a_{\gamma}t^{\gamma}$ with $a_{\gamma}\in A$, endowed with the multiplication law

[TABLE]

Proposition 1.5**.**

If $\Gamma$ is a finitely generated subsemigroup of the lattice $M\subset\mathbf{R}^{d}$ such that $\mathbf{Z}\Gamma=M$ and $\check{\sigma}=\mathbf{R}_{\geq 0}\Gamma$ is the rational convex cone generated by $\Gamma$, the integral closure of $k[t^{\Gamma}]$ in its field of fractions is $k[t^{\check{\sigma}\cap M}]$.

This follows directly from Lemma 1.3.

Remark 1.6*.*

Quite generally, if $k$ is a field the Krull dimension of $k[t^{\Gamma}]$ is equal to the rational rank of the semigroup $\Gamma$, which is the integer $\hbox{\rm dim}_{\mathbf{Q}}\Gamma\otimes_{\mathbf{Z}}\mathbf{Q}$ (see [34], Proposition 3.1).

Remark 1.7*.*

If $\Gamma$ is a semigroup the ideal of $A[t^{\Gamma}]$ generated by the $(t^{\gamma})_{\gamma\in\Gamma\setminus\{0\}}$ is non trivial if and only if the cone $\mathbf{R}_{\geq 0}\Gamma$ is strictly convex. If $k$ is a field, it is then a maximal ideal. We shall mostly be interested in the local study of the spectrum of semigroup algebras in the vicinity of the origin of coordinates, which corresponds precisely to that ideal.

The semigroup algebra has the following universal property: any semigroup map from $\Gamma$ to the multiplicative semigroup of an $A$-algebra $B$ extends uniquely to an homomorphism $A[t^{\Gamma}]\to B$ of $A$-algebras.

An additive map of semigroups $\phi\colon\Gamma\to\Gamma^{\prime}$ induces a graded map of $A$-algebras $A[\phi]\colon A[t^{\Gamma}]\to A[t^{\Gamma^{\prime}}]$ which is injective (resp. surjective) if $\phi$ is. If the semigroup $\Gamma$ is torsion-free, the semigroup algebra $A[t^{\Gamma}]$ injects into $A[t^{\mathbf{Z}^{d}}]=A[t_{1}^{\pm 1},\ldots,t_{d}^{\pm 1}]$ and therefore is an integral domain if $A$ is.

Proposition 1.8**.**

Let $\Gamma,\Gamma^{\prime}$ be two semigroups. The map of $A$-algebras

[TABLE]

is an isomorphism.

Proof. This follows immediately from the universal property. $\ {\Box}$

2. Algebraic tori

Let $k$ be a field. The multiplicative group $k^{*}$ of non-zero elements of $k$ is equipped with the structure of algebraic group over $k$, usually denoted by $\mathbf{G}_{m}:=\mbox{Spec}k[t^{\pm 1}]$. A $d$-dimensional algebraic torus over $k$ is an algebraic group isomorphic to a $(k^{*})^{d}$.

If $M$ is a rank $d$ lattice then $T^{M}:=\mathrm{Spec}k[t^{M}]$ is an algebraic torus over $k$. If we fix a basis $m_{1},\dots,m_{d}$ of the lattice $M$ we get a group isomorphism

[TABLE]

and isomorphism of $k$-algebras $k[t_{1}^{\pm 1},\dots,t_{d}^{\pm 1}]\rightarrow k[t^{M}]$ which induces an isomorphism $T^{M}(k)\rightarrow(k^{*})^{d}$.

Remark 2.1*.*

More generally one can consider the scheme $\mathrm{Spec}A[t^{M}]$, which is an algebraic torus over $\mathrm{Spec}A$ for any commutative ring $A$.

A character of the torus $T(k)$ is a group homomorphism $T(k)\rightarrow k^{*}$. The set of characters $\mbox{\rm Hom }_{\mathrm{alg.groups}}({T}^{M},k^{*})$ of $T^{M}(k)$ is a multiplicative group isomorphic to the lattice $M$ by the homomorphism given by $m\mapsto t^{m}$ for $m\in M$. We identify the monomials $t^{m}$ of the semigroup algebra $k[t^{M}]$ with the characters of the torus.

By the universal property of the semigroup algebras applied to $k[t^{M}]$ we have a representation of $k$-rational points of $T^{M}$ as group homomorphisms:

[TABLE]

where $N:=\mbox{\rm Hom }(M,\mathbf{Z})$ is the dual lattice of $M$. We denote by $\langle\,,\,\rangle\colon N\times M\rightarrow\mathbf{Z}$ the duality pairing between the lattices $N$ and $M$.

A one parameter subgroup of $T^{M}(k)$ is group homomorphism $k^{*}\rightarrow T^{M}(k)$. Any vector $\nu\in N$ gives rise to a one parameter subgroup $\lambda_{\nu}$ which maps $z\in k^{*}$ to the closed point of $T^{M}(k)$ given by the homomorphism of semigroups $M\rightarrow k^{*}$, $m\mapsto z^{\langle\nu,m\rangle}$. The set of one parameter subgroups $\mbox{\rm Hom }_{\mathrm{alg.groups}}(k^{*},{T}^{M})$ forms a multiplicative group, which is isomorphic to $N$ by the homomorphism given by $\nu\mapsto\lambda_{\nu}$.

3. Affine toric varieties

In this section we consider a finitely generated subsemigroup $\Gamma$ of a free abelian group $M$ of rank $d$. We assume in addition that the group $\mathbf{Z}\Gamma$ generated by $\Gamma$ is equal to $M$. We denote by $N$ the dual lattice of $M$. We introduce some useful notations.

Notation 3.1**.**

We denote by $M_{\mathbf{R}}$ the $d$-dimensional real vector space $M\otimes_{\mathbf{Z}}\mathbf{R}$. The semigroup $\Gamma$, viewed in $M_{\mathbf{R}}$, spans the cone $\mathbf{R}_{\geq 0}\Gamma\subset M_{\mathbf{R}}$ which we denote also by $\check{\sigma}$. The dual cone of $\check{\sigma}$ is the cone $\sigma:=\{\nu\in N_{\mathbf{R}}\mid\langle\nu,\gamma\rangle\geq 0,\,\forall\gamma\in\check{\sigma}\}$. We use the notation $\tau\leq\sigma$ to indicate that $\tau$ is a face of $\sigma$. Any face of $\check{\sigma}$ is of the form $\check{\sigma}\cap\tau^{\bot}$ for a unique face $\tau$ of $\sigma$, where $\tau^{\bot}$ is the linear subspace $\{\gamma\in M_{\mathbf{R}}\mid\langle\nu,\gamma\rangle=0,\,\forall\nu\in\tau\}$.

Let $\gamma_{1},\ldots,\gamma_{r}$ be generators of $\Gamma$. Then the semigroup $\Gamma$ is the image of $\mathbf{N}^{r}\subset\mathbf{Z}^{r}$ by the surjective linear map $b\colon\mathbf{Z}^{r}\to M$ determined by $b(e_{i})=\gamma_{i}$ where the $e_{i},\ 1\leq i\leq{r}$ form the canonical basis of $\mathbf{N}^{r}$. The kernel ${\mathcal{L}}$ of $b$ is isomorphic to $\mathbf{Z}^{{r}-d}$.

Let us consider the map of semigroup algebras associated to the map $b|\mathbf{N}^{r}\colon\mathbf{N}^{r}\to\mathbf{Z}^{d}$, whose image is $\Gamma$. It is a map of $A$-algebras $A[U_{1},\ldots,U_{r}]\to A[t_{1}^{\pm 1},\ldots,t_{d}^{\pm 1}]$. Its image is the subalgebra $A[t^{\Gamma}]$ of $A[t_{1}^{\pm 1},\ldots,t_{d}^{\pm 1}]$.

An element $m\in\mathbf{Z}^{r}$ can be written uniquely $m=m_{+}-m_{-}$ where $m_{+}$ and $m_{-}$ have non negative entries and disjoint support.

By construction, the kernel of the surjection $A[U_{1},\ldots,U_{r}]\to A[t^{\Gamma}]$ is the ideal generated by the binomials $(U^{m_{+}}-U^{m_{-}})$ where $b(m_{+})=b(m_{-})$. It is the toric ideal associated to the map $b$. Note that it is not in general generated by the binomials associated to a basis of ${\mathcal{L}}$. Since the algebra $A[t^{\Gamma}]$ is an integral domain if $A$ is, the toric ideal is a prime ideal in that case.

Conversely, assuming now that $A$ is an algebraically closed field $k$, an ideal generated by binomials in $k[U_{1},\ldots,U_{r}]$ is called a binomial ideal. Those ideals are studied in [8], where it is shown that a prime binomial ideal $I\subset k[U_{1},\ldots,U_{r}]$ gives rise to a semigroup algebra $k[U_{1},\ldots,U_{r}]/I\simeq k[t^{\Gamma}]$, where $\Gamma=\mathbf{N}^{r}/_{\sim}$, and $\sim$ is an equivalence relation associated to the binomial relations. The affine toric variety ${T}^{\Gamma}:=\hbox{\rm Spec }k[t^{\Gamma}]$ is the subvariety of the affine space $\mathbf{A}^{r}(k)$ defined by the binomial equations generating the toric ideal. By the universal property of the semigroup algebra, there is a bijection

[TABLE]

where $k$ is considered as a semigroup with respect to multiplication (in particular $0\in\Gamma$ goes to $1\in k$).

In particular, the torus ${T}^{M}(k)=\hbox{\rm Hom}_{\mathrm{groups}}(M,k^{*})$ is embedded in ${T}^{\Gamma}$, as the principal open set where $t^{\gamma_{1}}\cdots t^{\gamma_{r}}\neq 0$.

From the description of closed points of ${T}^{\Gamma}$ in terms of homomorphisms of semigroups we have an action of the torus ${T}^{M}(k)$ on ${T}^{\Gamma}(k)$. Another way to describe this action, which shows that it is algebraic, is to say that thanks to the universal property of semigroup algebras it corresponds to the composed map of $k$-algebras

[TABLE]

where the first map is determined by $t^{\gamma}\mapsto t^{\gamma}\otimes_{k}t^{\gamma}$ and the second by the inclusion $\Gamma\subset M$. The corresponding map ${T}^{M}\times{T}^{\Gamma}\to{T}^{\Gamma}$ is the action.

Let us now seek the invariant subsets of $T^{\Gamma}$ under the torus action.

Definition 3.2**.**

Given a semigroup $\Gamma$, a subsemigroup $F\subset\Gamma$ is a face of $\Gamma$ if whenever $x,y\in\Gamma$ satisfy $x+y\in F$, then $x$ and $y$ are in $F$.

Let us remark that this condition is equivalent to the fact that the vector space of finite sums $\sum_{\delta\in\Gamma\setminus F}a_{\delta}t^{\delta}$ is in fact a prime ideal $I_{F}$ of $k[t^{\Gamma}]$. It also implies that $\Gamma\setminus F$ is a subsemigroup of $\Gamma$ (which in general is not finitely generated) and that the Minkowski sum $\Gamma+(\Gamma\setminus F)$ is contained in $\Gamma\setminus F$.

Lemma 3.3**.**

The faces of the semigroup $\Gamma$ are of the form $\Gamma\cap\tau^{\bot}$, for $\tau\leq\sigma$.

Proof. Let $F$ be a face of the semigroup $\Gamma$. Then there is a face $\check{\sigma}\cap\tau^{\bot}$ of $\check{\sigma}$ which contains $F$ and is of minimal dimension. Then $F$ is also a face of the semigroup $\Gamma\cap\tau^{\bot}$ and there is an element $\gamma_{0}\in F$ which belongs to the relative interior of the cone $\check{\sigma}\cap\tau^{\bot}$. Under these conditions is enough to prove that if $\tau=0$ then $F=\Gamma$.

Notice that if $\gamma\in\Gamma$ and if $(\gamma+\Gamma)\cap\mathbf{Z}_{\geq 0}\gamma_{0}\neq\emptyset$ then $\gamma\in F$ since $F$ is a face and $\gamma_{0}\in F$. By Theorem 1.9 [20] there is $\delta_{0}\in\Gamma\cap\mathrm{int}(\check{\sigma})$ such that $\delta_{0}+\check{\sigma}\cap M\subset\Gamma$. We deduce that the intersection $(\gamma+\delta_{0}+\check{\sigma}\cap M)\cap\mathbf{Z}_{\geq 0}\gamma_{0}$ is non-empty, for any $\gamma\in\Gamma$, since $\gamma_{0}\in\mathrm{int}(\check{\sigma})\cap\Gamma$. ${\Box}$

Notation 3.4**.**

If $\tau\leq\sigma$ the set $\Gamma\cap\tau^{\bot}$ is a subsemigroup of finite type of $\Gamma$. If $\tau\leq\sigma$ the lattice $M(\tau,\Gamma)$ spanned by $\Gamma\cap\tau^{\bot}$ is a sublattice of finite index of $M(\tau):=M\cap\tau^{\bot}$.

Remark 3.5*.*

The torus of the affine toric variety ${T}^{\Gamma\cap\tau^{\bot}}$ is $T^{M(\tau,\Gamma)}$. If $A$ is a commutative ring, the homomorphism of $A$-algebras $A[\Gamma]\rightarrow A[\Gamma\cap\tau^{\bot}]\cong A[\Gamma]/I_{\Gamma\cap\tau^{\bot}}$, is surjective and defines a closed embedding

[TABLE]

over $\mathrm{Spec}A$. If $k=A$ the image by the embedding $i_{\tau}$ of a closed point $u\in T^{\Gamma\cap\tau^{\bot}}(k)$ (or $u\in T^{M(\tau,\Gamma)}(k)$) is the semigroup homomorphism $i_{\tau}(u):\Gamma\rightarrow k$ given by

[TABLE]

Proposition 3.6**.**

The map

[TABLE]

defines a bijection (resp. inclusion-reversing bijection) between the faces of $\sigma$ and the orbits (resp. the closures of the orbits) of the torus action on ${T}^{\Gamma}$.

Proof. Let $u:\Gamma\rightarrow k$ be a semigroup homomorphism. Then $u^{-1}(k^{*})$ is a face of $\Gamma$, hence of the form $\Gamma\cap\tau^{\bot}$ for some face $\tau$ of $\sigma$. Any such $u$ extends in a unique manner to a group homomorphism $M(\tau,\Gamma)\rightarrow k^{*}$ defining an element of the torus ${T}^{M(\tau,\Gamma)}$ of the affine toric variety ${T}^{\Gamma\cap\tau^{\bot}}$. Conversely, given a group homomorphism $u\colon M(\tau,\Gamma)\rightarrow k^{*}$ we define a semigroup homomorphism $i_{\tau}(u):\Gamma\rightarrow k$ as indicated above.

It follows that the orbit of the point defined by $u$ by the action of $T^{M}$ coincides with the image by $i_{\tau}$ of the orbit ${T}^{M(\tau,\Gamma)}$ of the point $u_{|\Gamma\cap\tau^{\bot}}\colon\Gamma\cap\tau^{\bot}\rightarrow k^{*}$ on the toric variety ${T}^{\Gamma\cap\tau^{\bot}}$. The rest of the assertion follows from Remark 3.5. $\ {\Box}$

The partition induced by the orbits of the torus action on ${T}^{\Gamma}$ is of the form:

[TABLE]

Proposition 3.7**.**

If $X$ is an affine toric variety with torus ${T}^{M}$ then $X$ is ${T}^{M}$-equivariantly isomorphic to ${T}^{\Gamma}$, where $\Gamma\subset M$ a semigroup of finite type such that $\mathbf{Z}\Gamma=M$.

Proof. This is well-known (see Proposition 2.4, Chapter 5 of [12]). $\ {\Box}$

We characterize the affine $T^{M}$-invariant open subsets of $T^{\Gamma}$.

Definition 3.8**.**

For any face $\tau$ of $\sigma$ the set

[TABLE]

is a semigroup of finite type generating the lattice $M$.

Notice that the cone $\mathbf{R}_{\geq 0}\Gamma_{\tau}$ is equal to $\check{\tau}$ and if $\tau\leq\sigma$ the set $\mathrm{int}(\check{\sigma}\cap\tau^{\bot})\cap\Gamma$ is non empty ($\mathrm{int}$ denotes relative interior).

Lemma 3.9**.**

**

1. i.

The minimal face of the semigroup $\Gamma$ is a sublattice of $M$ equal to $\Gamma\cap\sigma^{\bot}$. 2. ii.

For any $m\in\Gamma$ in the relative interior of $(\check{\sigma}\cap\tau^{\bot})$ we have that

[TABLE] 3. iii.

If $\tau\leq\theta\leq\sigma$ we have that $M(\tau,\Gamma_{\theta})=M(\tau,\Gamma_{\tau})$ and $\Gamma_{\tau}=\Gamma_{\theta}+M(\tau,\Gamma_{\theta})$.

Proof. i. By Lemma 3.3 the correspondence $\tau\mapsto\Gamma\cap\tau^{\bot}$ is a bijection between the faces of the cone $\sigma$ and the faces of the semigroup $\Gamma$. By duality the minimal face of $\Gamma$ is equal to $\Gamma\cap\sigma^{\bot}$. It is enough to prove that if $\Gamma$ is a semigroup such that $\mathbf{Z}\Gamma=M$ and $\mathbf{R}_{\geq 0}\Gamma=M_{\mathbf{R}}$ then $\Gamma=M$. Since $M$ is the saturation of $\Gamma$ the assertion reduces to the case of rank one semigroups, for which it is elementary by Bezout identity.

ii. If $m\in\mathrm{int}(\check{\sigma}\cap\tau^{\bot})\cap\Gamma$ then the semigroup $\Gamma+\mathbf{Z}_{\geq 0}(-m)\subset M$ spans the cone $\check{\tau}=\check{\sigma}+\tau^{\bot}\subset M_{\mathbf{R}}$. By i. the minimal face of this semigroup is the lattice $(\Gamma+\mathbf{Z}_{\geq 0}(-m))\cap\tau^{\bot}$ which coincides by definition with the lattice $M(\tau,\Gamma)$.

iii. The lattices $M(\tau,\Gamma_{\theta})$ and $M(\tau,\Gamma)$ are both generated by $\Gamma\cap\tau^{\bot}$ hence are equal. We have that $\Gamma_{\tau}=\Gamma_{\theta}+M(\tau,\Gamma_{\theta})$ since $\theta^{\bot}\subset\tau^{\bot}$. $\ {\Box}$

Lemma 3.10**.**

If $\tau\leq\sigma$ the inclusion of semigroups $\Gamma\subset\Gamma_{\tau}$ determines a ${T}^{M}$-equivariant embedding ${T}^{\Gamma_{\tau}}\subset{T}^{\Gamma}$ as an affine open set. Conversely, if $X\subset T^{\Gamma}$ is a ${T}^{M}$-equivariant embedding of an affine open set then there is a unique $\tau\leq\sigma$ such that $X$ is ${T}^{M}$-equivariantly isomorphic to $T^{\Gamma_{\tau}}$.

Proof. By Lemma 3.9 we have that $\Gamma_{\tau}=\Gamma+\mathbf{Z}_{\geq 0}(-m)$. More generally if $\gamma\in\Gamma$ and $f=t^{\gamma}$, the localization ${T}^{\Gamma}_{f}=\mbox{\rm Spec \, }k[\Gamma]_{f}$ is equal to ${T}^{\Gamma+(-\gamma)\mathbf{Z}_{\geq 0}}$ and it is embedded in ${T}^{\Gamma}$ as a principal open set.

Conversely, an affine ${T}^{M}$-invariant open subset of ${T}^{\Gamma}$ is an affine toric variety for the torus ${T}^{M}$ hence it is of the form ${T}^{\Lambda}$, for $\Lambda\subset M$ a subsemigroup of finite type, such that $\mathbf{Z}\Lambda=M$ (see Proposition 3.7). We denote the cone $\mathbf{R}_{\geq 0}\Lambda$ by $\check{\theta}$. Since the embedding ${T}^{\Lambda}\subset{T}^{\Gamma}$ is ${T}^{M}$ equivariant it is defined by the inclusion of algebras $k[t^{\Gamma}]\rightarrow k[t^{\Lambda}]$ corresponding to the inclusion of semigroups $\Gamma\subset\Lambda$. We deduce that $\check{\sigma}\subset\check{\theta}$ and hence that $\theta\subset\sigma$ by duality. We prove that if $\tau$ is the smallest face of $\sigma$ which contains $\theta$ then $\Lambda=\Gamma_{\tau}$. It is enough to prove that if $\mathrm{int}{(\theta)}\cap\mathrm{int}(\sigma)\neq\emptyset$ then $\Lambda=\Gamma$.

Notice that the lattice $F=\sigma^{\bot}\cap M$ is the minimal face of $\Gamma$ and the prime ideal $I_{F}$ of $k[t^{\Gamma}]$ defines the orbit $\mathrm{orb}(\sigma,\Gamma)$, which is embedded as a closed subset of ${T}^{\Gamma}$. Let us consider a vector $\nu$ such that $\nu\in\mathrm{int}(\theta)\cap\mathrm{int}(\sigma)$. Then we get that $\sigma^{\bot}\cap M=\check{\sigma}\cap\nu^{\bot}\cap M$ is contained in $\check{\theta}\cap\nu^{\bot}\cap M=\theta^{\bot}\cap M$ hence $\Gamma\setminus(\sigma^{\bot}\cap M)$ is contained in $\Lambda\setminus(\theta^{\bot}\cap M)$ and therefore $1\notin I_{F}k[t^{\Lambda}]$. Since ${T}^{\Lambda}\subset{T}^{\Gamma}$ is an open immersion $\mathrm{orb}(\sigma,\Gamma)$ is contained in ${T}^{\Lambda}$. By (1) and Proposition 3.6 the closure of any orbit contained ${T}^{\Gamma}$ contains $\mathrm{orb}(\sigma,\Gamma)$ thus ${T}^{\Gamma}\subset{T}^{\Lambda}$. $\ {\Box}$

Remark 3.11*.*

The immersion of ${T}^{M}$-invariant affine open subsets is compatible with normalization. By Lemma 3.10 any $T^{M}$-invariant affine open set of ${T}^{\Gamma}$ is of the form $T^{\Gamma}_{f}$ for $f=t^{\gamma}$, $\gamma\in\Gamma$. Then the following diagram commutes:

[TABLE]

since $\Gamma+(-\gamma)\mathbf{Z}_{\geq 0}$ is saturated in $\check{\sigma}\cap M+(-\gamma)\mathbf{Z}_{\geq 0}$. The vertical arrows are embeddings as principal open sets while the horizontal arrows are normalization maps (see Proposition 1.5).

4. Toric varieties

Given a finite dimensional lattice $N$, recall that a fan is a finite set $\Sigma$ of strictly convex polyhedral cones of the real vector space $N_{\mathbf{R}}$ which are rational for the lattice $N$, such that if $\sigma\in\Sigma$ any face $\tau$ of $\sigma$ belongs to $\Sigma$ and if $\sigma,\sigma^{\prime}\in\Sigma$ the cone $\tau=\sigma\cap\sigma^{\prime}$ is in $\Sigma$. If $j\geq 0$ is an integer the subset of $\Sigma(j)$ of $j$-dimensional cones of $\Sigma$ is called the $j$-skeleton of the fan. The support of the fan $\Sigma$ is the set $|\Sigma|=\cup_{\sigma\in\Sigma}\sigma\subset N_{\mathbf{R}}$.

We give first a combinatorial definition of toric varieties.

Definition 4.1**.**

A toric variety is given by the datum of a triple $(N,\Sigma,\Gamma)$ consisting of lattice $N$, a fan $\Sigma$ in $N_{\mathbf{R}}$ and a family of finitely generated subsemigroups $\Gamma=\{\Gamma_{\sigma}\subset\check{\sigma}\cap M\}_{\sigma\in\Sigma}$ of a lattice $M=\mbox{Hom}(N,\mathbf{Z})$ such that:

1. i.

$\mathbf{Z}\Gamma_{\sigma}=M$ and $\mathbf{R}_{\geq 0}\Gamma_{\sigma}=\check{\sigma}$, for $\sigma\in\Sigma$. 2. ii.

$\Gamma_{\tau}=\Gamma_{\sigma}+M(\tau,\Gamma_{\sigma})$, for a each $\sigma\in\Sigma$ and any face $\tau$ of $\sigma$.

The corresponding toric variety ${T}_{\Sigma}^{\Gamma}$ is the union of the affine varieties ${T}^{\Gamma_{\sigma}}$ for $\sigma\in\Sigma$ where for any pair $\sigma,\sigma^{\prime}$ in $\Sigma$ we glue up ${T}^{\Gamma_{\sigma}}$ and ${T}^{\Gamma_{\sigma^{\prime}}}$ along their common open affine variety ${T}^{\Gamma_{\sigma\cap\sigma^{\prime}}}$.

Remark 4.2*.*

The lattice $N$ in the triple $(N,\Sigma,\Gamma)$ is determined by $\Gamma$. We recall it by convenience. We omit the reference to the lattice $N$ in the notation $T^{\Gamma}_{\Sigma}$.

Remark 4.3*.*

This definition is consistent with the case of affine toric varieties. Let ${T}^{\Gamma}$ be an affine toric variety in the sense of Section 3. If $\sigma^{\prime}:=\{\tau\mid\tau\leq\sigma\}$ and $\Gamma^{\prime}:=\{\Gamma_{\tau}\mid\tau\leq\sigma\}$, where $\Gamma_{\tau}$ is the semigroup defined by (2) for $\tau\leq\sigma$ then the conditions i. and ii. are satisfied by Lemma 3.9. Then ${T}^{\Gamma}$ is $T^{M}$-equivariantly isomorphic to ${T}^{\Gamma^{\prime}}_{\sigma^{\prime}}$.

Remark 4.4*.*

A triple $(N,\Sigma,\Gamma)$ determines similarly a toric scheme over $\mathrm{Spec}A$, for any commutative ring $A$.

Lemma 4.5**.**

Let $(\Sigma,\Gamma)$ as in Definition 4.1 define a toric variety ${T}_{\Sigma}^{\Gamma}$. Then we have:

1. i.

If $\sigma,\theta\in\Sigma$ and if $\tau=\sigma\cap\theta$ then $\Gamma_{\tau}=\Gamma_{\sigma}+\Gamma_{\theta}$. 2. ii.

The variety ${T}_{\Sigma}^{\Gamma}$ is separated.

Proof. The intersection $\tau=\sigma\cap\theta$ is a face of both $\sigma$ and $\theta$. By Lemma 3.9 we have that $M(\tau,\Gamma_{\tau})=M(\tau,\Gamma_{\sigma})=M(\tau,\Gamma_{\theta})$. By axiom i. in the Definition 4.1 we get $\Gamma_{\theta},\Gamma_{\sigma}\subset\Gamma_{\tau}$ and $\Gamma_{\theta}+\Gamma_{\sigma}\subset\Gamma_{\tau}$. Conversely, by the separation lemma for polyhedral cones, for any $u\in\mathrm{int}(\check{\sigma}\cap(-\check{\theta}))$ we have that $\tau=\sigma\cap u^{\bot}=\theta\cap u^{\bot}$. Notice that we can assume that $u\in\Gamma_{\sigma}\cap(-\Gamma_{\theta})\cap\mathrm{int}(\check{\sigma}\cap(-\check{\theta}))\neq\emptyset$. Then by Lemma 3.9 we obtain $\Gamma_{\tau}=\Gamma_{\sigma}+\mathbf{Z}_{\geq 0}(-u)$. Hence $\Gamma_{\tau}$ is contained in $\Gamma_{\sigma}+\Gamma_{\theta}$ since $-u\in\Gamma_{\theta}$.

The homomorphism $k[t^{\Gamma_{\theta}}]\otimes_{k}k[t^{\Gamma_{\sigma}}]\rightarrow k[t^{\Gamma_{\tau}}]$ which sends $t^{\gamma}\otimes t^{\gamma^{\prime}}\mapsto t^{\gamma+\gamma^{\prime}}$ is surjective since $\Gamma_{\sigma}+\Gamma_{\theta}=\Gamma_{\tau}$. In geometric terms this implies that the diagonal map ${T}^{\Gamma_{\tau}}\rightarrow{T}^{\Gamma_{\theta}}\times{T}^{\Gamma_{\sigma}}$ is a closed embedding for any $\theta,\sigma\in\Sigma$ with $\tau=\theta\cap\tau$, hence the variety ${T}^{\Gamma}_{\Sigma}$ is separated (see Chapter 2 of [18]). $\ {\Box}$

Remark 4.6*.*

The morphisms corresponding to the inclusions $k[t^{\Gamma_{\sigma}}]\to k[t^{\Gamma_{\sigma}+\Gamma_{\sigma^{\prime}}}]$ are open embeddings compatible with the normalization maps. The normalization of the toric variety ${T}_{\Sigma}^{\Gamma}$ is the toric variety ${T}_{\Sigma}$ corresponding to the fan $\Sigma$ and the normalization map is obtained by gluing-up normalizations ${T}_{\sigma}:=T^{\check{\sigma}\cap M}$ of the charts ${T}^{\Gamma_{\sigma}}$, for $\Gamma_{\sigma}\in\Gamma$ and $\sigma\in\Sigma$.

Lemma 4.7**.**

Let $\lambda_{v}$ be a one-parameter subgroup of the torus $T^{M}$ for some $v\in N$. Then $\lim_{z\rightarrow 0}\lambda_{v}(z)$ exists in the toric variety ${T}^{\Gamma}_{\Sigma}$ if and only if $v$ belongs to $|\Sigma|\cap N$.

Proof. The statement is well-known in the normal case (see Proposition 1.6 [26]). The normalization map $n\colon{T}_{\Sigma}\rightarrow{T}^{\Gamma}_{\Sigma}$ is an isomorphism over the torus $T^{M}$. If $\lambda_{v}\colon k^{*}\rightarrow T^{M}\subset{T}^{\Gamma}_{\Sigma}$ is a one-parameter subgroup defined by $v\in N$ it lifts to the normalization, i.e., there is a morphism $\bar{\lambda}_{v}\colon k^{*}\rightarrow T^{M}\subset{T}_{\Sigma}$ in such a way that $n\circ\bar{\lambda}_{v}=\lambda_{v}$. Since the normalization is a proper morphism we get by the valuative criterion of properness that $\lim_{z\rightarrow 0}\lambda_{v}(z)$ exists in the toric variety ${T}^{\Gamma}_{\Sigma}$ if and only if $\lim_{z\rightarrow 0}\bar{\lambda}_{v}(z)$ exists in ${T}_{\Sigma}$. $\ {\Box}$

Lemma 4.8**.**

Let $T^{\Gamma}_{\Sigma}$ be a toric variety. Then the map

[TABLE]

defines a bijection between the faces of $\Sigma$ and the orbits of the torus action on ${T}^{\Gamma}_{\Sigma}$.

Proof. This is consequence of the definitions and Lemma 3.6. $\ {\Box}$

In order to illustrate the combinatorial definition of a toric variety we describe the orbit closures as toric varieties.

Notation 4.9**.**

If $\tau\in\Sigma$ we denote by $N_{\tau}$ the sublattice of $N$ spanned by $\tau\cap N$ and by $N(\tau)$ the quotient $N/N_{\tau}$. The lattice $N(\tau)$ is the dual lattice of $M(\tau)=M\cap\tau^{\bot}$. Since $M(\tau,\Gamma_{\tau})$ is a sublattice of finite index $i(\tau,\Gamma_{\tau})$ of $M(\tau)$ then the dual lattice $N(\tau,\Gamma_{\tau})$ of $M(\tau,\Gamma_{\tau})$ contains $N(\tau)$ as a sublattice of finite index equal to $i(\tau,\Gamma_{\tau})$.

If $\sigma\in\Sigma$ and $\tau\leq\sigma$ the image $\sigma(\tau)$ of $\sigma$ in $N(\tau)_{\mathbf{R}}=N_{\mathbf{R}}/(N_{\tau})_{\mathbf{R}}$ is a polyhedral cone, rational for the lattice $N(\tau,\Gamma_{\tau})$. The set $\Sigma(\tau):=\{\sigma(\tau)\mid\sigma\in\Sigma,\tau\leq\sigma\}$ is a fan in $N(\tau)_{\mathbf{R}}$. If $\sigma(\tau)\in\Sigma(\tau)$ we set $\Gamma_{\sigma(\tau)}:=\Gamma_{\sigma}\cap\tau^{\bot}$. The set $\sigma(\tau)\subset N(\tau)_{\mathbf{R}}$ is the dual cone of the cone spanned by $\Gamma_{\sigma}\cap\tau^{\bot}$ in $M(\tau)_{\mathbf{R}}$. Let us denote by $\Gamma(\tau)$ the set $\{\Gamma_{\sigma(\tau)}\mid\sigma(\tau)\in\Sigma(\tau)\}$.

Lemma 4.10**.**

Let $T^{\Gamma}_{\Sigma}$ be a toric variety. If $\tau\in\Sigma$ the triple $(N(\tau,\Gamma_{\tau}),\Sigma(\tau),\Gamma(\tau))$ defines a toric variety $T_{\Sigma(\tau)}^{\Gamma(\tau)}$. We have a closed embedding $i_{\tau}:T_{\Sigma(\tau)}^{\Gamma(\tau)}\to T^{\Gamma}_{\Sigma}$. The map

[TABLE]

defines a bijection between the faces of $\Sigma$ and orbit closures of the action of ${T}^{M}$ on ${T}^{\Gamma}_{\Sigma}$.

Proof. If $\tau$ is not a face of $\sigma$, for $\sigma\in\Sigma$ then $\mathrm{orb}(\tau,\Gamma_{\tau})$ is does not intersect the affine invariant open set ${T}^{\Gamma_{\sigma}}$; if $\tau\leq\sigma$, for $\sigma\in\Sigma$ the closure of the orbit $\mathrm{orb}(\tau,\Gamma_{\tau})$ in the affine open set ${T}^{\Gamma_{\sigma}}$ is equal to ${T}^{\Gamma_{\sigma}\cap\tau^{\bot}}$ (see Lemma 3.6).

If $\tau\leq\theta\leq\sigma$ then $\theta(\tau)\leq\sigma(\tau)$ and $\theta^{\bot}\subset\tau^{\bot}$ hence $M(\theta,\Gamma_{\sigma})=M(\theta(\tau),\Gamma_{\sigma}\cap\tau^{\bot})$ is the sublattice spanned by $\Gamma_{\sigma}\cap\theta^{\bot}$.

If $\tau\leq\sigma,\sigma^{\prime}$ and if $\theta=\sigma\cap\sigma^{\prime}$ then we deduce from condition ii. in Definition 4.1 that:

[TABLE]

We obtain that the triple $(N(\tau,\Gamma_{\tau}),\Sigma(\tau),\Gamma(\tau))$ satisfies the axioms in Definition 4.1 with respect to the torus $T^{M(\tau,\Gamma_{\tau})}$.

We have also described an embedding ${T}_{\Sigma(\tau)}^{\Gamma(\tau)}\hookrightarrow{T}^{\Gamma}_{\Sigma}$ in such a way that the intersection of this variety with any affine chart containing $\mathrm{orb}(\tau,\Gamma)$ is the closure of the orbit $\mathrm{orb}(\tau,\Gamma)$ in the chart. The conclusion follows from Lemma 4.8. $\ {\Box}$

Remark 4.11*.*

The non-singular locus of the toric variety ${T}^{\Gamma}_{\Sigma}$ is the union of the orbits $\mathrm{orb}(\tau,\Gamma)$ corresponding to regular cones $\tau\in\Sigma$ such their index $i(\tau,\Gamma_{\tau})$ is equal to $1$.

5. Blowing ups

The theory of normal toric varieties deals with normalized equivariant blowing ups, i.e., blowing ups of equivariant ideals followed by normalization. In this section we build blowing ups of equivariant ideals in toric varieties.

Let $\sigma$ be a strictly convex rational cone in $N_{\mathbf{R}}$ and $\Gamma$ a subsemigroup of finite type of the lattice $M$ such that $\mathbf{Z}\Gamma=M$ and the saturation of $\Gamma$ in $M$ is equal to $\check{\sigma}\cap M$. For simplicity we assume that the cone $\sigma$ is of dimension $d$ hence $\check{\sigma}$ is strictly convex.

Let us consider a graded ideal ${\mathcal{I}}$ in $A[t^{\Gamma}]$, which is necessarily generated by monomials $t^{m_{1}},\ldots,t^{m_{k}}$. We build the corresponding Newton polyhedron ${\mathcal{N}}_{\sigma}({\mathcal{I}})$, by definition the convex hull in $M_{\mathbf{R}}$ of the $m_{i}+\check{\sigma}$, which is also the convex hull of the set $|{\mathcal{I}}|$ of exponents of monomials belonging to the ideal ${\mathcal{I}}$ of $A[t^{\Gamma}]$. It is quite convenient to denote with the same letter $\mathcal{I}$ the set $\{m_{1},\dots m_{k}\}$.

The set $\mathcal{I}$ determines the order function:

[TABLE]

The order function $\mathrm{ord}_{\mathcal{I}}$ coincides with the support function à la Minkowski of the polyhedron ${\mathcal{N}}={\mathcal{N}}_{\sigma}(\mathcal{I})$ defined as the function $H\colon N_{\mathbf{R}}\to{\mathbf{R}}$ given by $H(\nu)={\rm min}_{m\in{\mathcal{N}}}\langle\nu,m\rangle$. It is a gauge ($\mathrm{ord}_{\mathcal{I}}(\lambda u)=\lambda\mathrm{ord}_{\mathcal{I}}(u)\ \ \hbox{\rm for }\lambda>0$) which is piecewise linear. The maximal cones of linearity of the function $\mathrm{ord}_{\mathcal{I}}$ form the $d$-skeleton of the fan $\Sigma(\mathcal{I})$ subdividing $\sigma$. Each such cone $\sigma_{i}$ in the $d$-skeleton of $\Sigma(\mathcal{I})$ is the convex dual of the convex rational cone generated by the vectors $(m-m_{i})_{m\in{\mathcal{N}}_{\sigma}(\mathcal{I})}$, where $m_{i}$ is a vertex of ${\mathcal{N}}_{\sigma}(\mathcal{I})$. The correspondence $m_{i}\mapsto\sigma_{i}$ is a bijection between the set of vertices $\{m_{1},\dots,m_{s}\}\subset\mathcal{I}$ of the polyhedron ${\mathcal{N}}_{\sigma}(\mathcal{I})$ and the $d$-skeleton of $\Sigma(\mathcal{I})$, such that

[TABLE]

Note that $\Gamma\subset\check{\sigma}\cap M\subset\check{\sigma}_{i}\cap M$. In each of the cones $\check{\sigma}_{i}$ we consider the semigroup

[TABLE]

By Lemma 1.3, the saturation in $M$ of this semigroup is equal to $\check{\sigma}_{i}\cap M$. We denote by $\Gamma(\mathcal{I})$ the set consisting of the semigroups $\Gamma_{i}$, together with $\Gamma_{i,\tau}$ (defined by equation (2)) for $\tau\leq\sigma_{i}$, $i=1,\dots,s$.

Proposition 5.1**.**

The triple $(N,\Sigma(\mathcal{I}),\Gamma(\mathcal{I}))$ defines a toric scheme $B$ over $\hbox{\rm Spec }A$. The inclusions $\Gamma\subset\Gamma_{i}$, $i=1,\dots,s$, determine a map of schemes

[TABLE]

over $\hbox{\rm Spec }A$, which is the blowing up of the ideal $\mathcal{I}$.

Proof. We prove first that the triple $(N,\Sigma(\mathcal{I}),\Gamma(\mathcal{I}))$ satisfies the compatibility conditions stated in Definition 4.1. By Lemma 3.9 it is enough to check them for the affine open sets corresponding to two vertices, say $m_{1}$ and $m_{2}$, of ${\mathcal{N}}_{\sigma}(\mathcal{I})$. Then, if $\tau=\sigma_{1}\cap\sigma_{2}$ the condition we have to prove is that $\Gamma_{1,\tau}=\Gamma_{2,\tau}$.

Notice that the vector $m:=m_{2}-m_{1}\in\Gamma_{1}$ belongs to the interior of $\check{\sigma}_{1}\cap\tau^{\perp}$. By Lemma 3.9 and the definitions we get $\Gamma_{1,\tau}=\Gamma_{1}+\mathbf{Z}_{\geq 0}(-m)$ and similarly $\Gamma_{2,\tau}=\Gamma_{2}+\mathbf{Z}_{\geq 0}m$. Then the assertion follows since $\Gamma_{1,\tau}$, which is equal to

[TABLE]

is the same semigroup as $\Gamma_{2,\tau}$.

It follows that the scheme $B$ is covered by the affine sets $\hbox{\rm Spec }A[t^{\Gamma_{i}}]$ for $i=1,\dots,s$. Since each $\Gamma_{i}$ contains $\Gamma$, there is a natural map $\pi\colon\hbox{\rm Spec }A[t^{\Gamma_{i}}]\to\hbox{\rm Spec }A[t^{\Gamma}]$. The sheaf of ideals on ${B}$ determined by the compositions with $\pi$ of the generators of ${\mathcal{I}}$ is generated by $t^{m_{i}}\circ\pi$ in the chart $\hbox{\rm Spec}A[t^{\Gamma_{i}}]$.

It is not difficult to prove that any semigroup $\Gamma_{i}$ defined by (4), for $i>s$, that is when $m_{i}$ is not a vertex of ${\mathcal{N}}_{\sigma}(\mathcal{I})$, is of the form $\Gamma_{j,\tau}$ for some $1\leq j\leq s$ and $\tau\leq\sigma_{j}$. This means that the corresponding affine chart $\hbox{\rm Spec }A[t^{\Gamma_{i}}]$ of the blowing up of $\mathcal{I}$ is in fact an affine open subset of $\hbox{\rm Spec }A[t^{\Gamma_{j}}]$, where $m_{j}$ is a vertex of ${\mathcal{N}}_{\sigma}(\mathcal{I})$. $\ {\Box}$

Remark 5.2*.*

With the above notations let us consider the Rees algebra of $\mathcal{I}$, defined as $R[\mathcal{I}]=\bigoplus_{l\geq 0}\mathcal{I}^{l}s^{l}$. Since each power $\mathcal{I}^{l}$ is a monomial ideal, the term $\mathcal{I}^{l}s^{l}$ is of the form $\mathcal{I}^{l}s^{l}=\oplus_{\gamma\in|\mathcal{I}^{l}|}t^{\gamma}s^{l}$. Consider also the semigroup $\Gamma_{\mathcal{I}}$ of $M\times\mathbf{Z}$ generated by $(\Gamma\times\{0\})\cup(|\mathcal{I}|\times\{1\})$. By using the map of semigroups $\Gamma\to\Gamma_{\mathcal{I}}$, defined by $\gamma\mapsto(\gamma,0)$, the semigroup algebra $A[t^{\Gamma_{\mathcal{I}}}]$ has the structure of $A[t^{\Gamma}]$-algebra. There is a unique isomorphism of semigroups algebras $R[\mathcal{I}]\to A[t^{\Gamma_{\mathcal{I}}}]$ over $A$, such that $t^{\gamma}s^{l}\mapsto t^{(\gamma,l)}$. This is also an isomorphism of $A[t^{\Gamma}]$- graded algebras, when the grading of a monomial $t^{(\gamma,l)}$ is defined to be equal to $l$. The canonical map $\textrm{Proj}(R[\mathcal{I}])\to\textrm{Spec}A[t^{\Gamma}]$ is the blowing up of the ideal $\mathcal{I}$. We have given in Proposition 5.1 the combinatorial description of this map as a toric morphism as defined in Section 6. See also Section 11.3 [5].

Corollary 5.3**.**

The blowing-up of an equivariant sheaf of ideals on a toric variety $T_{\Sigma}^{\Gamma}$ is a toric variety. Its description above each equivariant open affine chart of $T_{\Sigma}^{\Gamma}$ is given by Proposition 5.1.

6. Toric morphisms

Recall that a morphism $\phi\colon T^{M^{\prime}}\to T^{M}$ of algebraic tori gives rise to two group homomorphisms

[TABLE]

between the corresponding lattices of characters and between the corresponding lattices of one-parameter subgroups. The homomorphisms $\phi^{*}$ and $\phi_{*}$ are mutually dual and determine the morphism $\phi\colon T^{M^{\prime}}\to T^{M}$ of algebraic tori. Note that $\phi$ is defined algebraically by

[TABLE]

Now suppose that we have two toric varieties $T^{\Gamma}_{\Sigma}$ and $T^{\Gamma^{\prime}}_{\Sigma^{\prime}}$ with respective tori $T^{M}$ and $T^{M^{\prime}}$ defined by the combinatorial data given by the triples $(N,\Sigma,\Gamma)$ and $(N^{\prime},\Sigma^{\prime},\Gamma^{\prime})$ (see Definition 4.1).

Definition 6.1**.**

The homomorphism $\phi_{*}$ is a map of fans with attached semigroups $(N,\Sigma,\Gamma)\to(N^{\prime},\Sigma^{\prime},\Gamma^{\prime})$ if for any $\sigma^{\prime}\in\Sigma^{\prime}$ there exists $\sigma\in\Sigma$ such that $\phi^{*}(\Gamma_{\sigma})\subset\Gamma_{\sigma^{\prime}}^{\prime}$.

Note then that $\phi_{*}$ is a map of fans, that is, for any $\sigma^{\prime}\in\Sigma^{\prime}$ there is a cone $\sigma\in\Sigma$ such that image by $\sigma^{\prime}$ by the $\mathbf{R}$-linear extension of $\phi_{*}$ is contained in $\sigma$. See Section 1.5 [26].

Proposition 6.2**.**

Let $\phi\colon T^{M^{\prime}}\to T^{M}$ be a morphism of algebraic tori. If $\phi_{*}$ defines a map of fans with attached semigroups $(N,\Sigma,\Gamma)\to(N^{\prime},\Sigma^{\prime},\Gamma^{\prime})$ then it gives rise to a morphism: $\bar{\phi}\colon T_{\Sigma^{\prime}}^{\Gamma^{\prime}}\to T_{\Sigma}^{\Gamma}$ which extends $\phi\colon T^{M^{\prime}}\to T^{M}$ and is equivariant with respect to $\phi$. Conversely, if $f\colon T_{\Sigma^{\prime}}^{\Gamma^{\prime}}\to T_{\Sigma}^{\Gamma}$ is an equivariant morphism with respect to $\phi$ then $\phi_{*}$ defines a map of fans with attached semigroups $(N^{\prime},\Sigma^{\prime},\Gamma^{\prime})\to(N,\Sigma,\Gamma)$ and $f=\bar{\phi}$. In addition we have a commutative diagram

[TABLE]

where the vertical arrows are normalizations and the horizontal ones are the toric morphisms which extend $\phi:T^{M^{\prime}}\rightarrow T^{M}$.

Proof. For any $\sigma^{\prime}\in\Sigma^{\prime}$ there exists a cone $\sigma\in\Sigma$ such that the restriction of $\phi^{*}$ determines a semigroup homomorphism $\Gamma_{\sigma}\to\Gamma_{\sigma^{\prime}}^{\prime}$. The corresponding homomorphism of $k$-algebras $k[\Gamma_{\sigma}]\to k[\Gamma_{\sigma^{\prime}}^{\prime}]$ defines a morphism:

[TABLE]

where $x\in T^{\Gamma_{\sigma^{\prime}}^{\prime}};x\colon\Gamma_{\sigma^{\prime}}^{\prime}\to k$ is a homomorphism of semigroups. The morphism $\bar{\phi}_{\sigma^{\prime},\sigma}$ is equivariant through $\phi$ since for any $y\in T^{M^{\prime}}$, $y\colon M^{\prime}\rightarrow k^{*}$ group homomorphism and any $x\in T^{\Gamma_{\sigma^{\prime}}^{\prime}}$ we get:

[TABLE]

By gluing-up the affine pieces together we get a morphism $\bar{\phi}\colon T_{\Sigma^{\prime}}^{\Gamma^{\prime}}\to T_{\Sigma}^{\Gamma}$ which is equivariant with respect to $\phi$.

For the converse, since $f$ is assumed to be equivariant through $\phi$ the image by $f$ of each orbit of the action of $T^{M^{\prime}}$ on $T^{\Sigma^{\prime}}_{\Gamma^{\prime}}$ is contained in one orbit of the action of $T^{M}$ on $T_{\Sigma}^{\Gamma}$. If $\tau^{\prime}\leq\sigma^{\prime}$ and $\sigma^{\prime}\in\Sigma^{\prime}$ then the orbit $\mathrm{orb}(\sigma^{\prime},\Gamma_{\sigma^{\prime}}^{\prime})$ is contained in the closure of $\mathrm{orb}(\tau^{\prime},\Gamma_{\tau^{\prime}}^{\prime})$ by Proposition 3.6. Then there exist $\sigma,\tau\in\Sigma$ such that

[TABLE]

Since $f$ is continuous $\mathrm{orb}(\sigma,\Gamma_{\sigma})$ must be contained in the closure of $\mathrm{orb}(\tau,\Gamma_{\tau})$, hence $\tau$ is a face of $\sigma$ by Proposition 3.6 and Lemma 3.5. By (1) it follows that $f(T^{\Gamma_{\sigma^{\prime}}^{\prime}})\subset T^{\Gamma_{\sigma}}$. The restriction $f_{|T^{\Gamma_{\sigma^{\prime}}^{\prime}}}\colon T^{\Gamma_{\sigma^{\prime}}^{\prime}}\to T^{\Gamma_{\sigma}}$ is equivariant with respect to $\phi:T^{M^{\prime}}\rightarrow T^{M}$. Hence $f_{|T^{\Gamma_{\sigma^{\prime}}^{\prime}}}\colon T^{\Gamma_{\sigma^{\prime}}^{\prime}}\to T^{\Gamma_{\sigma}}$ is defined algebraically by the homomorphism of $k$-algebras $k[t^{\Gamma_{\sigma}}]\rightarrow k[t^{\Gamma_{\sigma^{\prime}}^{\prime}}]$, which is obtained by restriction from the homomorphism of $k$-algebras $k[t^{M}]\rightarrow k[t^{M^{\prime}}]$ which maps $t^{m}\mapsto t^{\phi^{*}(m)}$ for $m\in M$. This implies that $\phi^{*}(\Gamma_{\sigma})\subset\Gamma_{\sigma^{\prime}}^{\prime}$ and also that $f=\bar{\phi}$.

Since $\phi_{*}$ is a map of fans it defines a toric morphism between the normalizations of $T^{\Gamma^{\prime}}_{\Sigma^{\prime}}$ and $T^{\Gamma}_{\Sigma}$. Finally, it is easy to check that the diagram above is commutative. $\ {\Box}$

It is sometimes useful to consider morphisms of toric varieties which send the torus of the source into a non dense orbit of the target: Let $(N,\Sigma,\Gamma)$ and $(N^{\prime},\Sigma^{\prime},\Gamma^{\prime})$ be two triples defining toric varieties $T^{\Gamma}_{\Sigma}$ and $T^{\Gamma^{\prime}}_{\Sigma^{\prime}}$. Let $\tau$ be a cone of $\Sigma$. Suppose that we have a morphism of algebraic tori $\phi:T^{M^{\prime}}\to T^{M(\tau,\Gamma_{\tau})}$ such that $\phi_{*}:N^{\prime}\to N(\tau,\Gamma_{\tau})$ defines a map of fans with attached semigroups $(N^{\prime},\Sigma^{\prime},\Gamma^{\prime})\to(N(\tau,\Gamma_{\tau}),\Sigma(\tau),\Gamma(\tau))$. Then by Proposition 6.2 and Lemma 3.5 we have a toric morphism

[TABLE]

Let us denote by $n\colon T_{\Sigma}\rightarrow T^{\Gamma}_{\Sigma}$ the normalization map and by $\bar{i}_{\tau}\colon T_{\Sigma(\tau)}\to T_{\Sigma}$ the closed embedding of the closure of $\mathrm{orb}(\tau)$ in $T_{\Sigma}$. The following Proposition is consequence of Proposition 6.2 and Lemma 3.5.

Proposition 6.3**.**

The composite of $\bar{\phi}$ with the closed embedding $i_{\tau}:T_{\Sigma(\tau)}^{\Gamma(\tau)}\hookrightarrow T_{\Sigma}^{\Gamma}$ lifts to the normalization of $T_{\Sigma}^{\Gamma}$, i.e., there exists a toric morphism $\psi:T_{\Sigma^{\prime}}^{\Gamma^{\prime}}\to T_{\Sigma(\tau)}$ such that $i_{\tau}\circ\bar{\phi}=n\circ\bar{i}_{\tau}\circ\psi$ if and only if there is a lattice homomorphism $\varphi^{*}:M(\tau)\rightarrow M^{\prime}$ such that $\varphi^{*}_{|M(\tau,\Gamma_{\tau})}=\phi^{*}$ and then $\psi=\bar{\varphi}$.

Example 6.4**.**

By Proposition 6.3 the map $u\mapsto(u,0,0)$, which parametrizes the singular locus of the Whitney umbrella $\{x_{1}^{2}x_{2}-x_{3}^{2}=0\}$ does not lift to the normalization while $u\mapsto(u^{2},0,0)$ does.

7. Abstract toric varieties

We recall the usual definition of toric variety.

Definition 7.1**.**

A toric variety $X$ is an irreducible (separated) algebraic variety equipped with an action of an algebraic torus $T$ embedded in $X$ as a Zariski open set such that the action of $T$ on $X$ is morphism which extends the action of $T$ over itself by multiplication.

As stated in Proposition 3.7 any affine toric variety is the spectrum of certain semigroup algebra. Gel*′*fand, Kapranov, and Zelevinsky have defined and studied those projective toric varieties which are equivariantly embedded in the projective space, which is viewed as a toric variety, see [12], Chapter 5.

The following Theorem, which is consequence of a more general result of Sumihiro, provides the key to establish a combinatorial description of normal toric varieties.

Theorem 7.2**.**

(see [32]) Any normal toric variety $X$ has a finite covering by $T$-invariant affine normal toric varieties.

The statement of Theorem 7.2 does not hold if the normality assumption is dropped.

Example 7.3**.**

Let $C\subset{\mathbf{P}}^{2}_{\mathbf{C}}$ be the projective nodal cubic with equation $y^{2}z-x^{2}(x+z)$. It is a rational curve with a node singularity at $P=(0:0:1)$ and only one point $Q=(0:1:0)$ at the line of infinity $z=0$. The curve $C$ is rational and has a parametrization $\pi:{\mathbf{P}}^{1}_{\mathbf{C}}\rightarrow C$ such that $\pi(0)=\pi(\infty)=P$ and $\pi(1)=Q$. Then we have that $\pi_{|\mathbf{C}^{*}}\colon\mathbf{C}^{*}\rightarrow C\setminus\{P\}$ is an isomorphism. The multiplicative action of $\mathbf{C}^{*}$ on ${\mathbf{P}}^{1}_{\mathbf{C}}$ corresponds by $\pi$ to the group law action on the cubic hence it is algebraic. It follows that $C$ is a toric variety with respect to Definition 7.1. Notice that $C$ is the only open set containing $P$ which is invariant by the action of $\mathbf{C}^{*}$. This example is also a projective toric curve which does not admit any equivariant embedding in the projective space (see [25] page 4 and [12] Chapter 5, Remark 1.6).

Definition 7.4**.**

An action of a group on an algebraic variety $X$ is good if $X$ is covered by a finite number of affine open subsets which are invariant by the action.

We modify the abstract definition of toric varieties as follows:

Definition 7.5**.**

A toric variety $X$ is an irreducible separated algebraic variety equipped with a good action of an algebraic torus $T$ embedded in $X$ as a Zariski open set such that the action of $T$ on $X$ extends the action of $T$ on itself by multiplication.

Theorem 7.6**.**

If $X$ is a toric variety in the sense of Definition 7.5 with torus $T$, then there exists a triple $(N,\Sigma,\Gamma)$ as in Definition 4.1 and an isomorphism $\varphi\colon T\to T^{M}$ such that the pair $(T,X)$ is equivariantly isomorphic to $(T^{M},T^{\Gamma}_{\Sigma})$ with respect to $\varphi$.

Proof. We denote by $M$ the lattice of characters of the torus $T$ hence $T=T^{M}$ and $N$ is the dual lattice of $M$.

By Proposition 3.7 an affine $T^{M}$-invariant open subset is of the form $T^{\Gamma_{\sigma}}$ where ${\Gamma_{\sigma}}$ is a subsemigroup of finite type of $M$ such that $\mathbf{Z}{\Gamma_{\sigma}}=M$, and $\sigma\subset N_{\mathbf{R}}$ is the dual cone of $\check{\sigma}=\mathbf{R}_{\geq 0}{\Gamma_{\sigma}}\subset M_{\mathbf{R}}$. By Lemma 3.10 the open affine $T^{M}$-invariant subsets of $T^{\Gamma_{\sigma}}$ are $T^{\Gamma_{\tau}}$, for $\tau\leq\sigma$, where $\Gamma_{\tau}=\Gamma_{\sigma}+M(\tau,\Gamma_{\sigma})$.

By definition $X$ is covered by a finite number of $T^{M}$-invariant affine open subsets of the form $\{T^{\Gamma_{\sigma}}\}_{\sigma\in\Sigma}$. We can assume that if $\sigma\in\Sigma$ and if $\tau\leq\sigma$ then $\tau\in\Sigma$. We are going to show that $\Sigma$ is a fan in $N_{\mathbf{R}}$, hence $T^{\Gamma_{\sigma}}\neq T^{\Gamma_{\sigma^{\prime}}}$ if $\sigma\neq\sigma^{\prime}$.

We have that for any $\sigma,\sigma^{\prime}\in\Sigma$ the intersection $T^{\Gamma_{\sigma}}\cap T^{\Gamma_{\sigma^{\prime}}}$ is an affine open subset of the separated variety $X$ (see Chapter 2 of [18]). It is also a $T^{M}$-invariant affine subset of both $T^{\Gamma_{\sigma}}$ and $T^{\Gamma_{\sigma^{\prime}}}$, hence it is of the form $T^{\Gamma_{\tau}}$. By Lemma 3.10 we obtain two inclusion of semigroups $\Gamma_{\sigma}\to\Gamma_{\tau}$ and $\Gamma_{\sigma^{\prime}}\to\Gamma_{\tau}$. Since $X$ is separated the diagonal map $T^{\Gamma_{\tau}}\to T^{\Gamma_{\sigma}}\times T^{\Gamma_{\sigma^{\prime}}}$ is a closed embedding (see Chapter 2 of [18]). Algebraically, this implies the surjectivity of the homomorphism

[TABLE]

It follows that the homomorphism of semigroups ${\Gamma_{\sigma}}\times{\Gamma_{\sigma^{\prime}}}\to{\Gamma_{\tau}}$, $(\gamma,{\gamma^{\prime}})\mapsto{\gamma+\gamma^{\prime}}$ is surjective. This proves that $\Gamma_{\tau}={\Gamma_{\sigma}}+{\Gamma_{\sigma^{\prime}}}$ thus

[TABLE]

By duality we deduce that $\tau=\sigma\cap\sigma^{\prime}$. By Proposition 3.10 we obtain

[TABLE]

In conclusion, $\Sigma$ is a fan in $N_{\mathbf{R}}$ and if $\Gamma:=\{\Gamma_{\sigma}\mid\sigma\in\Sigma\}$ the triple $(N,\Sigma,\Gamma)$ satisfies the compatibility properties of Definition 4.1 and the variety $T^{\Gamma}_{\Sigma}$ is $T^{M}$-equivariantly isomorphic to $X$. $\ {\Box}$

The following corollary is consequence of Proposition 6.2 and Theorem 7.6.

Corollary 7.7**.**

The category whose objects are the triples $(N,\Sigma,\Gamma)$ of Definition 4.1 and whose morphisms are the maps of fans with attached semigroups of Definition 6.1 is equivalent to the category whose objects are the toric varieties of Definition 7.5 and whose morphisms are those equivariant morphism which extend morphisms of the corresponding algebraic tori; see Proposition 6.2.

8. Invertible sheaves on toric varieties

In this section we describe how some of the classical results in the study of equivariant invertible sheaves on a normal toric variety extend to the general case.

Let $T_{\Sigma}^{\Gamma}$ denote a toric variety defined by the triple $(N,\Sigma,\Gamma)$. Recall that if $\sigma\in\Sigma$ we denote by $T_{\sigma}=T^{\check{\sigma}\cap M}$ the normalization of the chart $T^{\Gamma_{\sigma}}$ and by $T_{\Sigma}$ the normalization of $T^{\Gamma}_{\Sigma}$.

A support function $h:|\Sigma|\to\mathbf{R}$ is a continuous function such that for each $\sigma\in\Sigma$ the restriction $h_{|\sigma}\colon\sigma\to\mathbf{R}$ is linear. We say that $h$ is integral with respect to $N$ if $h(|\Sigma|\cap N)\subset\mathbf{Z}$. We denote by $\mbox{SF}(N,\Sigma)$ the set of support functions integral with respect to $N$. If $h$ is a support function integral with respect to $N$ then for any $\sigma\in\Sigma$ there exists $m_{\sigma}\in M$ such that

[TABLE]

Notice that by continuity we have that

[TABLE]

The set $\{m_{\sigma}\mid\sigma\in\Sigma\}$ determines $h$ but may not be uniquely determined.

Definition 8.1**.**

A support function for the triple $(N,\Sigma,\Gamma)$ is a support function $h:|\Sigma|\to\mathbf{R}$ integral with respect to $N$ which in addition has the compatibility property

[TABLE]

We denote by $\mbox{SF}(N,\Sigma,\Gamma)$ the additive group of support functions for the triple $(N,\Sigma,\Gamma)$. It is a subgroup of $\mbox{SF}(N,\Sigma)$. A vector $m\in M$ defines an element of $\mbox{SF}(N,\Sigma,\Gamma)$ hence we have a homomorphism $M\to\mbox{SF}(N,\Sigma,\Gamma)$, which is injective if the support of $\Sigma$ spans $N_{\mathbf{R}}$ as a real vector space.

Any $h\in\mbox{SF}(N,\Sigma)$ determines $T^{M}$-invariant * Cartier divisor* $D_{h}$ on $T_{\Sigma}$ by

[TABLE]

where $\mbox{{\rm div}}(g)$ denotes the principal Cartier divisor of the rational function $g$ on an irreducible variety. Notice that $D_{h}$ is independent of the possible choices of different Cartier data $\{m_{\sigma}\mid\sigma\in\Sigma\}$ defining $h$. If $\sigma,\sigma^{\prime}\in\Sigma$, $\tau=\sigma\cap\sigma^{\prime}$ then $T_{\tau}=T_{\sigma}\cap T_{\sigma^{\prime}}$ and (5) guarantees that $t^{-m_{\sigma}+m_{\sigma^{\prime}}}$ and $t^{m_{\sigma}-m_{\sigma^{\prime}}}$ are both regular functions on $T_{\tau}$. Any $T^{M}$-invariant Cartier divisor on $T_{\Sigma}$ is of the form $D_{h}$ for $h\in\mbox{SF}(N,\Sigma)$, i.e., it is defined by Cartier data.

Lemma 8.2**.**

If $h\in\mbox{SF}(N,\Sigma)$ is defined by the Cartier data $\{m_{\sigma}\mid\sigma\in\Sigma\}$ then it defines a $T^{M}$-invariant Cartier divisor on $T^{\Gamma}_{\Sigma}$ if and only if (6) holds, that is, if and only if $h\in\mbox{SF}(N,\Sigma,\Gamma)$.

Proof. The condition to determine a Cartier divisor is that for any $\sigma,\sigma^{\prime}\in\Sigma$, $\tau=\sigma\cap\sigma^{\prime}$ the transition function $t^{-m_{\sigma}+m_{\sigma^{\prime}}}$ is an invertible regular function on $T^{\Gamma_{\tau}}=T^{\Gamma_{\sigma}}\cap T^{\Gamma_{\sigma^{\prime}}}$. By Lemma 3.9 this is equivalent to (6). $\ {\Box}$

We have shown that the group $\mbox{\rm CDiv}_{T^{M}}(T^{\Gamma}_{\Sigma})$ of $T^{M}$-invariant Cartier divisors on $T^{\Gamma}_{\Sigma}$ can be seen as a subset of $\mbox{\rm CDiv}_{T^{M}}(T_{\Sigma})$. The set $\{\mathrm{div}(t^{m})\}_{m\in M}$ is a subgroup of $\mbox{\rm CDiv}_{T^{M}}(T^{\Gamma}_{\Sigma})$ consisting of principal Cartier divisors.

The map

[TABLE]

is a group isomorphism. The inverse map sends a Cartier divisor $D$ on $T_{\Sigma}^{\Gamma}$, given by the Cartier data $\{m_{\sigma}\mid\sigma\in\Sigma\}$, to the function

[TABLE]

A Cartier divisor on $T_{\Sigma}$ determines an invertible sheaf $\mathcal{O}_{T_{\Sigma}}(D)$. If $U$ is an affine open set in which $D=\mbox{{\rm div}}(g_{U})$ for some rational function $g_{U}$ then the set of sections $H^{0}(U,\mathcal{O}_{T_{\Sigma}}(D))$ consists of those rational functions $f$ such that $fg_{U}$ is a regular function on $U$.

We denote by $\mathcal{O}_{T^{\Gamma}_{\Sigma}}$ the structure sheaf on the toric variety ${T^{\Gamma}_{\Sigma}}$. The invertible sheaf of a $T^{M}$-invariant Cartier divisor $D$ on $T_{\Sigma}^{\Gamma}$ is the sheaf of $\mathcal{O}_{T^{\Gamma}_{\Sigma}}$-modules $\mathcal{O}_{T_{\Sigma}^{\Gamma}}(D)$. By (7) the set of sections of this sheaf on $T^{\Gamma_{\sigma}}$ is

[TABLE]

We denote by $P_{D}^{\Gamma}$ the following subset of $M$:

[TABLE]

The set of global sections of the sheaf $\mathcal{O}_{T^{\Gamma}_{\Sigma}}{(D)}$ is equal to

[TABLE]

Remark 8.3*.*

As in the normal case, a $T^{M}$-invariant Cartier divisor $D$ defines an equivariant line bundle $\mathcal{L}_{D}$ whose sections coincide with those of the invertible sheaf $\mathcal{O}_{T^{\Gamma}_{\Sigma}}(D)$. See [26], Chapter 2.

The Picard group $\mathrm{Pic}(X)$ of a variety $X$ consists of the isomorphism classes of invertible sheaves in $X$.

Lemma 8.4**.**

Suppose that $|\Sigma|=N_{\mathbf{R}}$. For any Cartier divisor $D$ on the toric variety $T^{\Gamma}_{\Sigma}$ we have an $\mathcal{O}_{T^{\Gamma}_{\Sigma}}$-module isomorphism $\mathcal{O}_{T^{\Gamma}_{\Sigma}}(D)\cong\mathcal{O}_{T^{\Gamma}_{\Sigma}}(D_{h})$ for some $h\in\mathrm{SF}(N,\Sigma,\Gamma)$. The following are equivalent for $h\in\mathrm{SF}(N,\Sigma,\Gamma)$.

1. i.

$h\in M$** 2. ii.

$D_{h}$* is a principal Cartier divisor.* 3. iii.

$\mathcal{L}_{D_{h}}$* is a trivial line bundle.* 4. iv.

The sheaf $\mathcal{O}_{T^{\Gamma}_{\Sigma}}(D_{h})$ is isomorphic to $\mathcal{O}_{T^{\Gamma}_{\Sigma}}$ as $\mathcal{O}_{T^{\Gamma}_{\Sigma}}$-module.

Proof. See Proposition 2.4 of [26]. ${\Box}$

Proposition 8.5**.**

Suppose that $|\Sigma|=N_{\mathbf{R}}$. Then we have canonical isomorphisms

[TABLE]

from which we deduce a canonical injection $\mathrm{Pic}(T^{\Gamma}_{\Sigma})\rightarrow\mathrm{Pic}(T_{\Sigma})$.

Proof. This follows by using the same arguments as in Corollary 2.5 [26]. ${\Box}$

If $\rho$ belongs to the $1$-skeleton $\Sigma(1)$ of the fan $\Sigma$ we denote by $\nu_{\rho}$ the primitive integral vector for the lattice $N$ in the ray $\rho$, that is the generator of the semigroup $\rho\cap N$. We associate to $h\in\mbox{SF}(N,\Sigma)$ the polyhedron

[TABLE]

Recall that

[TABLE]

for any integer $l\geq 1$ and $h,h^{\prime}\in\mbox{SF}(N,\Sigma)$.

Proposition 8.6**.**

Suppose that $|\Sigma|=N_{\mathbf{R}}$. The following are equivalent for $h\in\mbox{SF}(N,\Sigma,\Gamma)$ defining a Cartier divisor $D=D_{h}$.

1. i.

The $\mathcal{O}_{T^{\Gamma}_{\Sigma}}$-module $\mathcal{O}_{T^{\Gamma}_{\Sigma}}(D)$ is generated by its global sections. 2. ii.

$h$* is upper convex, i.e., $h(\nu)+h(\nu^{\prime})\leq h(\nu+\nu^{\prime})$ for all $\nu,\nu^{\prime}\in N_{\mathbf{R}}$.* 3. iii.

The polytope $P_{h}$ has vertices $\{m_{\sigma}\mid\sigma\in\Sigma\}$.

If these conditions hold the convex hull of the set $P_{D}^{\Gamma}$ is the polytope $P_{h}$ and $h$ is the support function of the polytope $P_{h}$.

Proof. The proof follows as in the normal case (see Theorem 2.7 [26]). ${\Box}$

If $|\Sigma|=N_{\mathbf{R}}$ the support function $h\in\mbox{SF}(N,\Sigma,\Gamma)$, defined by the Cartier data $\{m_{\sigma}\mid\sigma\in\Sigma\}$, is strictly upper convex if it is upper convex and in addition

[TABLE]

Suppose that $h\in\mbox{SF}(N,\Sigma,\Gamma)$ satisfies the equivalent conditions of Proposition 8.6. Set $D=D_{h}$. If $P_{D}^{\Gamma}=\{u_{1},\dots,u_{s}\}$ we have a morphism

[TABLE]

(defined in homogeneous coordinates of ${\mathbf{P}}^{s-1}_{k}$). The morphism $\Phi_{D}$ is equivariant with respect the map of tori $\Phi_{|T^{M}}\colon T^{M}\to T^{M^{\prime}}$, where $T^{M^{\prime}}$ denotes the torus of ${\mathbf{P}}^{s-1}_{k}$ with respect to the fixed coordinates.

Proposition 8.7**.**

Suppose that $|\Sigma|=N_{\mathbf{R}}$. The following are equivalent for $h\in\mbox{SF}(N,\Sigma,\Gamma)$ defining a Cartier divisor $D=D_{h}$.

1. i.

$D$* is very ample.* 2. ii.

$h$* is strictly upper convex and for all $\sigma\in\Sigma(d)$ the set $\{m-m_{\sigma}\mid m\in P_{D}^{\Gamma}\}$ generates the semigroup $\Gamma_{\sigma}$.*

Proof. Suppose that $h$ is not strictly upper convex. Then there exists $d$-dimensional cones $\sigma,\sigma^{\prime}\in\Sigma$ such that $\tau=\sigma\cap\sigma^{\prime}$ is of dimension $d-1$ and $m_{\sigma}=m_{\sigma^{\prime}}$. This implies that the section defined by $t^{m_{\sigma}}$ in the open set $U=T^{\Gamma_{\sigma}}\cup T^{\Gamma_{\sigma^{\prime}}}$ is nowhere vanishing.

By definition there exists $1\leq i\leq s$ such that $m_{\sigma}=u_{i}$.

The restriction of $\Phi_{D}$ to $U$ factors through the affine open set $\mathbf{C}^{s-1}$, where the $i$-th homogeneous coordinate does not vanish. It is of the form:

[TABLE]

By Lemma 3.5 the closure of the orbit $\mathrm{orb}(\tau,\Gamma)$ is a complete one-dimensional toric variety contained in $U$. The restriction $\Phi_{|\overline{\mathrm{orb}(\tau,\Gamma)}}$ must be constant hence $\Phi$ is not an embedding. This implies that if $D_{h}$ is very ample $h$ is strictly upper convex.

Suppose that $h$ is strictly upper convex. If $\sigma\in\Sigma$ is a $d$-dimensional cone then $m_{\sigma}$ belongs to $\{u_{i}\}_{i=1}^{s}$, say $m_{\sigma}=u_{s}$. The restriction of $\Phi$ to $T^{\Gamma_{\sigma}}$ factors though the affine open set of ${\mathbf{P}}^{s-1}_{k}$ where the last homogeneous coordinate does not vanish. It is described algebraically by the homomorphism of $k$-algebras:

[TABLE]

This maps defines a closed immersion if and only if it is surjective. This happens if and only if the set of vectors $\{u_{i}-m_{\sigma}\}_{1\leq i\leq s-1}$ generate the semigroup $\Gamma_{\sigma}$. ${\Box}$

Proposition 8.8**.**

Suppose that $|\Sigma|=N_{\mathbf{R}}$. The following are equivalent for $h\in\mbox{SF}(N,\Sigma,\Gamma)$.

1. i.

$D_{h}$* is ample* 2. ii.

$h$* is strictly upper convex.*

Proof. If $D$ is ample then $lD$ is very ample for $l\gg 0$. Since $lD=D_{lh}$ it follows that $h$ is strictly upper convex if $lh$ is and the assertion holds by Proposition 8.7.

Conversely suppose that $h$ is strictly upper convex. We prove that $lD_{h}$ is very ample for $l\gg 0$. By Proposition 8.7 it is sufficient to prove that there exists an integer $l\gg 0$ such that for each $d$-dimensional cone $\sigma\in\Sigma$ the semigroup $\Gamma_{\sigma}$ is generated by $\{m-lm_{\sigma}\mid m\in P^{\Gamma}_{D_{lh}}\}$.

If $\sigma^{\prime}\in\Sigma$, $\dim\sigma^{\prime}=d$, $\tau=\sigma^{\prime}\cap\sigma$ we have that $\Gamma_{\tau}=\Gamma_{\sigma}+\mathbf{Z}_{\geq 0}(-u)$ for any $u\in\Gamma_{\sigma}$ in the relative interior of the cone $\tau^{\perp}\cap\check{\sigma}$ (see Lemma 3.9). For instance we take $u=m_{\sigma^{\prime}}-m_{\sigma}$. We obtain similarly that $\Gamma_{\tau}=\Gamma_{\sigma^{\prime}}+\mathbf{Z}_{\geq 0}(u).$

If $\gamma\in\Gamma_{\sigma}$ then $\gamma$ belongs to $\Gamma_{\tau}$ and there exists $\gamma^{\prime}\in\Gamma_{\sigma^{\prime}}$ and an integer $p\geq 0$ such that $\gamma=\gamma^{\prime}+pu$. If $l\geq p$ we obtain:

[TABLE]

If $l$ is big enough, a formula of the form (14) holds for any $\gamma$ in a finite set $G_{\sigma}$ of generators of $\Gamma_{\sigma}$ (where $p$ and $\gamma^{\prime}$ vary with $\gamma$) and for any cone $\sigma^{\prime}\in\Sigma(d)$. Since $\gamma^{\prime}$ and $m_{\sigma}-m_{\sigma^{\prime}}$ belong to $\Gamma_{\sigma^{\prime}}$ this implies $t^{lm_{\sigma}+\gamma}$ defines a section in $H^0(T^{{}_{}}$