Introduction to Toric Geometry

Introduction to toric geometry

Ugo Bruzzo

Scuola Internazionale Superiore di Studi Avanzati and Istituto Nazionale di Fisica Nucleare Trieste ii Instructions for the reader

These are work-in-progress notes for the course “Introduction to toric geometry” that I am giving at the International School for Advanced Studies in Trieste during the academic year 2013/2014. Parts of them might be incomplete. At the moment ﬁgures are missing but I hope to include them before too long. Most of these notes are derived in an evident way from Fulton’s and Cox-Little- Schenck’s books. However I take full responsibility for possible mistakes. Missing proofs can be found in one (or both) of the two books. Thus, these notes do not pretend to be original in any way; they just serve, I hope, to trace a path through the rich and complex world of toric geometry.

Trieste, June 2014. Contents

1 Toric varieties 1 1.1 Fans and toric varieties ...... 1 1.1.1 Convex polyhedral cones ...... 1 1.1.2 Aﬃne toric varieties ...... 5 1.1.3 Constructing toric varieties from fans ...... 8 1.1.4 Torus action ...... 13 1.1.5 Limiting points ...... 16 1.1.6 The orbit-cone correspondence ...... 17 1.2 More properties of toric varieties ...... 21 1.2.1 Singularities ...... 21 1.2.2 Completeness ...... 23 1.3 Resolution of singularities ...... 26

2 Divisors and line bundles 29 2.1 Base-point free, ample and nef line bundles on normal varieties ...... 29 2.1.1 Base point free line bundles and divisors ...... 29 2.1.2 Ample and numerically eﬀective divisors ...... 30 2.1.3 Nef and Mori cones ...... 33 2.2 Polytopes ...... 34 2.2.1 Convex polytopes ...... 34 2.2.2 Canonical presentations ...... 36 2.3 Divisors in toric varieties ...... 36 2.3.1 The class group of a toric variety ...... 36 2.3.2 The Picard group of a toric variety ...... 39

v vi CONTENTS

2.3.3 Describing Cartier divisors ...... 42 2.4 Divisors versus polytopes ...... 43 2.4.1 Global sections of sheaves associated to toric divisors ...... 44 2.4.2 Base point free divisors in toric varieties ...... 45 2.4.3 Support functions ...... 46 2.4.4 Ample divisors in toric varieties ...... 49 2.5 The nef and Mori cones in toric varieties ...... 52

3 Cohomology of coherent sheaves 55 3.1 Reﬂexive sheaves and Weil divisors ...... 55 3.2 Diﬀerential forms, canonical sheaf and Serre duality ...... 57 3.2.1 Zariski forms ...... 57 3.2.2 Euler sequences ...... 58 3.2.3 Serre duality ...... 59 3.3 Cohomology of toric divisors ...... 59

References 63 Chapter 1

Toric varieties

Let Tn be the n-dimensional algebraic torus Tn = C∗ × · · · × C∗; it is an n-dimensional affine variety, with a compatible group structure (it is indeed a linear algebraic group). A toric variety will be defined as a (normal) algebraic variety over C containing a torus as an open dense subset, such that the (transitive) action of the torus on this open subset extends to the whole variety. Actually all toric varieties will have a stratification in tori of descending dimensions. As a basic example we may consider the projective plane P2. Let [z0, z1, z2] be homogeneous coordinates for it, and consider the open subset obtained by

removing the three lines {z0 = 0}, {z1 = 0}, {z2 = 0}. This set can be thought of as the aﬃne 2-plane A2 with two lines missing, so it is C∗ × C∗.1 To get P2 back, one must add three projective lines with two points removed, e.g., three copies of C∗, and three points (p1, p2, p3). So one can write

2 ∗ ∗ ∗ ∗ ∗ P = (C × C ) ∪ (C ∪ C ∪ C ) ∪ (p1, p2, p3)

Thus P2 can be stratiﬁed in smaller and smaller tori. This will be a general feature of toric varieties.

1.1 Fans and toric varieties

1.1.1 Convex polyhedral cones

The basic combinatorial objects to describe toric varieties are called fans. These are collections of convex polyhedral cones. We give here some basic notions about these objects.

1Note that A2 minus two lines is isomorphic to the cartesian product of two copies of A1 minus a point. 2 CHAPTER 1. TORIC VARIETIES

We recall that a lattice is a free, ﬁnitely generated abelian group equipped with a nondegenerate quadratic Z-valued form. The dimension of a lattice N is

dim N = dimR(N ⊗Z R).

The standard example is Zn with the restriction of the euclidean product. We also introduce the dual lattice M = Hom(N, ). Since M ⊗ = N ∗, we may think of M as a subset of Z R R N ∗. R Let us ﬁx a lattice N.

Deﬁnition 1.1. A convex polyhedral cone (cpc for short) is a subset of NR = N ⊗Z R of the form ( s ) X σ = ri vi | ri ∈ R≥0 ⊂ NR i=1 where v1, . . . , vs are ﬁxed vectors in NR, called the generators of σ. The dimension of σ is the dimension of the vector space generated by v1, . . . , vs. σ is said to be rational if it is generated by vectors in N. A 1-dimensional cpc is called a ray. 4

Deﬁnition 1.2. The dual of a cpc is the subset of ∗ deﬁned as NR

∨ ∗ σ = {u ∈ NR | u(v) ≥ 0 for all v ∈ σ}.

Any nonzero u ∈ N ∗ deﬁnes a normal hyperplane H , and a closed half-space R u

+ Hu = {v ∈ NR | u(v) ≥ 0}.

+ Given a polyhedral cone σ lying in Hu , we say that Hu is a supporting hyperplane of σ. A face of a cpc is its intersection with a supporting hyperplane. The following facts are quite easy to prove.

1. The face of a cpc is a cpc. 2. If a cpc is rational, all its faces are rational as well. 3. An intersection of faces is a face. 4. The face of a face is a face.

Examples 1.3. 4 1.1. FANS AND TORIC VARIETIES 3

As it will be the case for many easy results, we do not provide a proof for the following Proposition.

Proposition 1.4. Let σ be a cpc.

1. If σ is rational, then σ∨ is rational as well, i.e., it is generated by elements in M.

2. σ∨∨ = σ.

∨ Given a cone σ, we deﬁne a subset of M as Sσ = σ ∩ M. The latter has a natural semigroup structure.

Proposition 1.5 (Gordon’s lemma). If σ is a rational cpc, then Sσ is a ﬁnitely generated semigroup.

∨ ∨ Proof. Since σ is rational, we can choose generators u1, . . . , us ∈ M for σ . The set

( s ) X K = ti ui | 0 ≤ ti ≤ 1 i=1

∨ Ps is a compact subset of σ , and since M is discrete, K ∩ M is ﬁnite. If u = i=1 ri ui ∈ Sσ, split ri = mi + ti with mi integers, and ti ∈ [0, 1). Then

s s X X u = miui + tiui i=1 i=1

Ps Ps Ps and since u and i=1 miui are in M, also i=1 tiui is, hence i=1 tiui ∈ M ∩ K. This says that Sσ is generated by M ∩ K (note indeed that also ui ∈ M ∩ K).

An immediate consequence is that the semigroup algebra C[Sσ] (a fundamental object in the theory we are developing) is ﬁnitely generated (as a C-algebra).

Let σ be a rational cpc (for short, rcpc), and let u ∈ Sσ.

Proposition 1.6. σ ∩ u⊥ = τ is an rcpc, and all faces of σ are of this type. Moreover,

Sτ = Sσ + Z≥0 · (−u). (1.1) 4 CHAPTER 1. TORIC VARIETIES

Proof. Only equation (1.1) needs a proof. Figure 1.1 shows this result in the example were N = Z2 with the standard scalar product and σ is the convex cone generated by (1, 3) and (1, 1), with τ the face generated by (1.3). This actually generalizes to a proof valid in all cases.

At this point we need a basic result in the theory of convex bodies which basically says that the two convex bodies can be separated by a hyperplane. This fact is quite evident in a way and in any case is quite easy to prove (see e.g. [?]). We need some preliminary deﬁnition.

Deﬁnition 1.7. The span of a cone σ is the smallest vector subspace of NR containing σ. The relative interior Relint(σ) of σ is the interior of σ in its span. 4

Note that if σ generates NR, then its relative interior coincides with its interior. Lemma 1.8 (Separation lemma). If σ and σ0 are cpc’s having a common face τ, there is for any u ∈ Relint(σ∨ ∩ (−σ0∨)

0 τ = σ ∩ Hu = σ ∩ Hu.

Corollary 1.9. If σ and σ0 are rcpc that intersect along a common face τ, then

Sτ = Sσ + Sσ0 .

0 Proof. Since τ lies in both σ and σ , then Sτ contains both Sσ and Sσ0 , hence Sτ ⊃

Sσ + Sσ0 . To show the opposite inclusion we note that by Lemma 1.8 we can take a u ∈ σ∨ ∩ (−σ0)∨ ∩ M such that τ = σ ∩ u⊥ = σ0 ∩ u⊥. By Proposition 1.6 we have

Sτ = Sσ + Z≥0 · (−u) ⊂ Sσ + Sσ0 since −u ∈ Sσ0 .

2 ∨ Example 1.10. Let N = Z and σ = h(1, 2), (2, 1)i. Then σ = h(−1, 2), (2, −1)i, and Sσ is the set of integral points of σ∨. If u = (2, −1), then τ = σ ∩ u⊥ is the face generated by (1, 2). Moreover, Sτ is the right half plane delimited by the line generated by u, and Sτ = Sσ + Z≥0 · (−u). 4

Deﬁnition 1.11. A cpc σ is strongly convex if σ ∩ (−σ) = {0}. 4 1.1. FANS AND TORIC VARIETIES 5

Proposition 1.12. For a cpc σ the following conditions are equivalent.

1. σ is strongly convex. 2. σ contains no nontrivial linear subspace. 3. There is u ∈ σ∨ such that σ ∩ u⊥ = {0}. 4. σ∨ generates N ∗. R

Exercise 1.13. Let σ ⊂ NR be a rcpc.

1. Prove that if σ is strongly convex, then Sσ is saturated in M, i.e., if u ∈ M and

pu ∈ Sσ for some positive integer p, then u in is Sσ.

∨ 2. If σ is strongly convex, then σ spans MR = M ⊗ R.

3. If σ is strongly convex, Sσ generates M as a group, that is, M = Sσ + (−Sσ). 4. Conversely, any ﬁnitely generated sub-semigroup of M which is saturated and gen-

erates M is of form Sσ for a unique strongly convex rational polyhedral cone σ.

1.1.2 Aﬃne toric varieties

Any additive semigroup S determines a group ring C[S], which is a commutative C-algebra. Any element u ∈ S provides an element χu of a basis of C[S]; multiplication is given by the addition in S: χu · χu0 = χu+u0 .

0 The identity 0 ∈ S corresponds to the unit in C[S], i.e., χ = 1. If S has generators {ui}, then {χui } are generators of C[S] as a C-algebra.

We shall associate affine (toric) varieties to the semigroup algebras C[Sσ], where σ is an rcpc. We recall that a commutative C-algebra determines a scheme Spec A over C. In our case, A will be finitely generated. The closed points of Spec A, corresponding to prime ideals that are maximal, form a complex affine variety X = Specm A, which can be regarded as the zero locus in An of a set of polynomials in the following way: given generators (x1, ..., xn) of A, then A = C[x1, ..., xn]/I, where I is the ideal generated by the relations among the generators. Then X is the locus of common zeroes of the polynomials in I. In the following, unless otherwise stated, we shall usually consider only closed points. 6 CHAPTER 1. TORIC VARIETIES

If f ∈ A is nonzero, one can localize A at f,2 and then

Xf = Spec Af ⊂ X = Spec A is an open subset, called the principal open subset associated with f. If A = C[x1, . . . , xn], n so that Spec A = A , and f = xi, then Spec Af is the open subset xi 6= 0. Since maximal ideals in a commutative C-algebra correspond to nontrivial homomor- phisms to C, one has an isomorphism

Specm A ' HomC-alg(A, C).

In the toric case there is also another identiﬁcation. If A = C[S] for some semigroup in the dual lattice M, one also has the identiﬁcation

Specm C[S] ' Homsemigroup(S, C) where C is considered as a semigroup under multiplication. Let us show how this correspondence can be established. A point p ∈ X = Specm C[S] deﬁnes a map S → C by sending u ∈ S to χu(p) (remember that χu is a regular function on X). This is a semigroup morphism. To establish the opposite correspondence, let ψ : S → C be a semigroup morphism. By sending χu ∈ C[S], with u ∈ S, to ψ(u) we obtain a surjective C-algebra morphism C[S] → C, whose kernel is a maximal ideal and therefore corresponds to a point of X. The two constructions are clearly one the inverse of the other.

Definition 1.14. Let σ be an rcpc. The affine scheme Uσ = Spec C[Sσ] is the affine toric variety associated to the rcpc σ. 4

Spec C[Sσ] is in integral domain and a finitely generated C-algebra, so it (or more precisely, the set of its closed points) is indeed a variety, thus justifying the terminology. In particular, it is a noetherian scheme. Before giving some examples, we recall that C∗ can be regarded as the affine variety in the affine 2-space cut by the equation xy = 1, so that [x, y] ∗ ' Spec C = Spec [x, x−1]. C (1 − xy) C

2We recall that, given a commutative ring R with unit and subset S which is closed under multiplication (and is such that 0 6= S and 1 ∈ S), the localization of R with respect to S, denoted S−1R, is the quotient of the product R × S under the equivalence relation

(a, s) ∼ (b, t) if there is u ∈ S such that u(at − bs) = 0.

The equivalence class [(a, s)] is usually denoted a/s. There is a natural morphism R → S−1R, a 7→ a/1. n If S = {s , n ∈ N} for some element s ∈ A, then the localization is denoted As. Intuitively this is the ring obtained by inverting the powers of s. 1.1. FANS AND TORIC VARIETIES 7

In the next two examples {e1, . . . , en} will be a basis for N. ∗ ∗ Example 1.15. Let σ = {0}; then Sσ = M. A semigroup basis for it is given by {±e1,..., ±en}. ∗ ∗ ei −1 −ei Deﬁne xi = χ , so that xi = χ . Then

−1 −1 C[Sσ] = C[M] = C[x1, x1 , . . . , xn, xn ] so that ∗ ∗ n Spec C[Sσ] ' C × · · · × C = T . Thus the trivial rcpc σ = {0} corresponds to the algebraic n-torus Tn. 4 Example 1.16. Fix a number k with 1 ≤ k ≤ n − 1 and consider the rcpc generated by

{e1, . . . , ek}. Then k n ∨ X ∗ X ∗ σ = R≥0 ei + R ei i=1 i=k+1 so that k n X ∗ X ∗ Sσ = Z≥0 ei + Z ei . i=1 i=k+1 Then −1 −1 C[Sσ] = C[x1, . . . , xk, xk+1, xk+1, . . . , xn, xn ] k n−k and Uσ = A × T . 4 Note that in both examples the aﬃne toric varieties are smooth. 3 Example 1.17. Let N = Z , and let σ be generated by v1, . . . , v4 ∈ N with the relation v1 + v3 = v2 + v4. Without loss of generality, we can assume v1 = (1, 0, 0), v2 = (0, 1, 0), ∗ ∗ ∗ ∗ ∗ ∗ v3 = (0, 0, 1), v4 = (1, −1, 1). Now Sσ is generated by e1, e2, e1 + e2, e2 + e3, so that

C[x1, x2, x3, x4] C[Sσ] = C[x1, x2, x1x2, x2x3] = . (x1x4 − x2x3)

4 The resulting variety is the hypersurface x1x4 = x2x3 in A (which is singular at the origin). 4

A morphism S → S0 of semigroups yields a morphism of algebras C[S] → C[S0], and therefore also a morphism of varieties Spec C[S0] → Spec C[S]. If τ is a subcone of a cpc σ, ∨ ∨ then σ ⊂ τ , so Sσ is a sub-semigroup of Sτ , and there is morphism of varieties Uτ → Uσ. n In particular, the torus U{0} = T maps to all the aﬃne toric varieties Uσ, whichever the

cone σ in NR is.

Proposition 1.18. If τ is a face of a strongly convex rational polyhedral cone σ, then Uτ

is a principal open subset of Uσ. 8 CHAPTER 1. TORIC VARIETIES

⊥ Proof. By Proposition 1.18 there is u ∈ Sσ such that τ = σ ∩u , and Sτ = Sσ +Z≥0 ·(−u). w−pu Basis elements in C[Sτ ] have the form χ , with w ∈ Sσ and p ∈ Z≥0, so that C[Sτ ] = C[Sσ]χu , i.e., Uτ is a principal open subset of Uσ.

In a slightly greater generality, we may consider a morphism of lattices φ: N 0 → N, and cones σ0 ⊂ N 0 , σ ⊂ N such that φ(σ0) ⊂ σ. Then φ induces a morphism between the R R 0 semigroups associated with σ and σ , and in turn a morphism of varieties φ: Uσ0 → Uσ.

1.1.3 Constructing toric varieties from fans

Deﬁnition 1.19. A fan Σ in NR is a ﬁnite collection of strongly convex rational polyhedral cones such that

1. all faces of cones in Σ are in Σ; 2. cones in Σ only intersect along faces.

A fan, being a collection of cones, provides a collection of affine toric varieties, to be thought of as an affine open cover of a toric variety XΣ associated to the fan, which will not be necessarily affine. The finiteness Σ corresponds to the fact that, as a scheme, the toric variety will be of finite type.3 Since all fans contain the origin, all toric varieties will contain an algebraic torus as an open dense subset. We give some examples just to gather some ideas before proceeding to the formal construction of the toric variety associated with a fan.

Example 1.20. Let N = Z and σ = R≥0. Then Σ = {σ, −σ, {0}} is a fan. From Example 1.16 we see that Sσ = Z≥0 · e1 and S−σ = Z≥0 · (−e1) so that Uσ ' U−σ ' C. Moreover, ∗ U{0} ' C , and by Proposition 1.18, this embeds into Uσ and U−σ as a principal open subset. Thus Xσ is formed by two copies of C glued along a common subset isomorphic to ∗ 1 C , hence XΣ ' P . 4 2 Example 1.21. Let N = Z and let σ1, σ2 be the cones generated by (1, 0) and (0, 1), ∗ ∗ respectively. Then Σ = {σ1, σ2, {0}} is a fan. We have Uσ1 ' C × C , Uσ2 ' C × C, and 2 2 U{0} ' T . The associated toric variety is C − {0} (see Figure). 4 Note that both varieties in the previous Examples are not aﬃne.

3 We recall that a scheme over a field k is of finite type if it has a finite affine cover {Ui = Spec Ai}, where the Ai are finitely generated k-algebras. 1.1. FANS AND TORIC VARIETIES 9

2 Exercise 1.22. Let N = Z and let τi, i = 1, 2, 3 be the rays generated by (1, 0), (−1, −1), (0, 1), respectively, and let σi be the closed sectors of the plane delimited by any two of the τ’s. Then −1 ∗ C[Sτi ] ' C[x, x , y] so that Uτi ' C × C 2 2 2 while Uσ ' C and U{0} ' T . Show that the toric variety XΣ is the projective plane P . 2 Exercise 1.23. Let N = Z and let τi, i = 1, 2, 3, 4 be the rays generated by the four unit versors of the axes, respectively, and let σi be the closed sectors of the plane delimited by any two consecutive τ’s (i.e., the closed quadrants). Note that the fan Σ can be regarded as the cartesian product of two copies of the fan of P1 given in Example 1.20. Show that 1 1 XΣ is the variety P × P .

Note that the fans in Example 1.20 and Exercises 1.22 and 1.23 generate NR, and the corresponding toric varieties are complete (in these cases, even projective). As we shall see this is a general fact. So, a fan provides us with a collection of affine toric varieties, with definite inclusion morphisms when two cones in the fan are one a subcone of the other. But the fan also prescribes how to glue these affine varieties. Starting from Σ, we construct the disjoint union of all the affine toric varieties corresponding to the cones in Σ; then, if two cones σ and τ intersect, then the intersection σ ∩ τ is a face of both cones, and Uσ∩τ is a principal open subset (hence an open subvariety) of both Uσ and Uτ ; these two varieties are then glued along Uσ∩τ . It is not difficult to check that this operation is consistent (a cocycle condition is satisfied on triple intersections, if any). A result in point set topology is that a topological space X if Hausdorff if and only if the diagonal in X ×X is a closed subset. This fact motivates the definition of the analogous property for schemes: a scheme X is separated if the diagonal morphism ∆: X → X × X is a closed immersion (i.e., its image is closed, and the morphism of sheaves of algebras ∗ ∆ : OX×X → ∆∗OX is surjective).

The following result implies that a toric variety XΣ is separated.

Lemma 1.24. If σ and τ are rcpc’s that intersect along a face, the diagonal map ∆: Uσ∩τ →

Uσ × Uτ is a closed embedding.

Proof. By Corollary 1.9, Sσ∩τ = Sσ + Sτ ; then the generators of C[Sσ] tensored with the ∗ generators of C[Sτ ] generate C[Sσ∩τ ], namely, the natural map ∆ : C[Sσ]⊗C[Sτ ] → C[Sσ∩τ ] is surjective. This implies the claim; note indeed that C[S ] ⊗ [S ] [S ] ' σ C τ , C σ∩τ ker ∆∗ ∗ i.e., ∆(Uσ∩τ ) is the locus cut in Uσ × Uτ by the ideal ker ∆ . 10 CHAPTER 1. TORIC VARIETIES

XΣ is separated as a consequence of the following fact.

Exercise 1.25. Let a variety X be covered by aﬃne sets {Uα}, and denote by

∆αβ : Uα ∩ Uβ → Uα × Uβ

the morphisms deﬁned by the product of the obvious inclusions. Then X is separated

if each ∆αβ is a closed embedding (note that if you think of X as being obtained by

glueing the {Uα} along their intersections, the image of ∆αβ is isomorphic to the diagonal

in Uα × Uα). 4 Note that the question of the separatedness of a toric variety arises because it might not be aﬃne (aﬃne varieties are always separated by the same argument as the one in the proof of Lemma 1.24).

Now we address the normality of the variety XΣ. We recall that a scheme X is said to be normal if all of its local rings are integrally closed domains. An integral domain is a nonzero commutative ring such that the product of two nonzero elements is never zero. An integrally closed domain is an integral domain R which coincides with its integral closure in its ring of fractions Rˆ, i.e., which has the following property: every element in Rˆ which is a root of a monic polynomial in R[x] is in R, cf. [1].4 An important feature of a normal scheme X is that it is smooth in codimension 1, i.e., the singularities of X form a subscheme of codimension at least 2.5 Thus, for instance, a normal surface is singular only at isolated points. Smooth varieties are normal: every local ring is isomorphic to the local ring of Cn at 0, which a unique factorization domain, hence is an integrally closed domain.6 We shall need the algebraic results expressed in the following exercise.

4We recall that given an integral domain R, its integral closure is the ring

R¯ = {a ∈ Rˆ | a is a root of a monic polynomial in R[x]} where Rˆ is the field of fractions of R. Thus R is integrally closed if R = R¯. 5 The formal definition is that a scheme X is smooth in codimension 1 if every local ring OX,p of dimension one is regular. For such rings to be integrally closed or to be regular is equivalent [1, Prop. 9.2]. We recall that the dimension of the ring is the supremum of the heights of all prime ideals in A. The height of a prime ideal p is the supremum of the lengths of chains of prime ideals p0 ⊂ · · · ⊂ pn = p.A 2 local ring A with maximal ideal m and residue field k = A/m is regular if dimk m/m = dim A. 6We recall that an element in a commutative ring A is said to be a unit if it is multiplicatively invertible. An element x is said to be irreducible if x = yz implies that one between y and z is a unit. A ring A is said to be a unique factorization domain if every element can be written as x = x1 ··· xs, where each xi is irreducible, in a unique way up to reordering and multiplication by units. Example of unique factorization domains are the polynomial rings k[x1, . . . , xn], the rings of germs of holomorphic functions at a point of n a complex manifold, and the local rings of regular functions on affine space Ak . 1.1. FANS AND TORIC VARIETIES 11

Exercise 1.26. Prove that following statements.

1. If A is an integrally closed domain, and S ⊂ A a multiplicative subset, the localization

AS is A is an integrally closed domain.

2. If {Ai} are integrally closed domains whose ﬁeld of fractions are all isomorphic, then

∩iAi is an integrally closed domain (intersection inside the ﬁeld of fractions).

We note that a toric variety XΣ is irreducible for a very simple reason.

Proposition 1.27. If Σ is fan, the toric variety XΣ is irreducible.

n Proof. If Σ = {σ}, the irreducible aﬃne variety U{0} ' T is dense in every Uσ, so it is also dense in XΣ = ∪σ∈ΣUσ, and therefore XΣ is irreducible as well. Lemma 1.28. An irreducible aﬃne variety X = Spec A is normal if and only if the ring A is an integrally closed domain.

Proof. Let C(X) be the ring of rational functions on X. A rational function f in C(X) is contained in the local ring OX,p, with p ∈ X, exactly when f is regular at p. So if U ⊂ X is open, \ OX (U) = OX,p p∈U (intersection in C(X)). Then \ OX,p = OX (X) = A. p∈X

Now, if X is normal, all local rings OX,p are integrally closed domains with the same ﬁeld of fractions; by Exercise 1.26 A is an integrally closed domain. Conversely, let us assume that A is an integrally closed domain. Assume g ∈ C(X) satisﬁes k k−1 g + ai g + ... + ak Q where ai ∈ OX,p. We can write ai = hi/fi with hi, fi ∈ A and fi(p) 6= 0. Then f = i fi is such that f(p) 6= 0. By Exercise 1.26 the localization Af is an integrally closed domain, so that ai ∈ Af . Moreover, Af ⊂ OX,p as f(p) 6= 0. Since ai ∈ Af we have g ∈ Af , then g ∈ OX,p, which proves the claim.

For the sake of completeness we give the following Proposition in its full form, but we shall actually need only the implication 3 ⇒ 1. 12 CHAPTER 1. TORIC VARIETIES

Proposition 1.29. Let X be an aﬃne variety X = Spec C[S], where S is a ﬁnitely generated semigroup S ⊂ M. Assume X contains a torus Tn. The following conditions are equivalent.

1. X is normal.

2. S is saturated.

3. X = Spec C[Sσ] for a strongly convex rational polyhedral cone σ.

Proof. 1 ⇒ 2. If X is normal, C[S] is integrally closed in its ring of fractions C(X). Let m ∈ M and km ∈ S for some positive integer k. Then χm is a polynomial function on Tn and therefore a rational function on X. Also, χkm ∈ C[S] since km ∈ S. Thus χm is a root of the monic polynomial xk − χkm with coeﬃcients in C[S]. Since X is normal C[S] is an integrally closed domain, so that χm ∈ C[S] and m ∈ S, i.e., S is saturated. 2 ⇒ 3. Let K be a ﬁnite set of generators for S; these generate a rational polyhedral ∨ cone in MR containing S. Let σ ⊂ NR be its dual. Then S ⊂ σ ∩ M, with equality if S is saturated. So S = Sσ.

3 ⇒ 1. If σ ⊂ NR is a strongly convex rational polyhedral cone we need to show that C[Sσ] is an integrally closed domain. Now σ is generated by its rays ρi, and

∨ ∨ σ = ∩iρi .

Then Sσ = ∩iSρ , and i \ C[Sσ] = C[Sρi ] i

(intersection inside C[M]). By Exercise 1.26 it is enough to prove that C[Sρ] is normal

when ρ is a rational ray in NR. Let v be the (primitive) generator of the ray. We can complete v = e1 to a basis {e1, . . . , en} of N, and

−1 −1 C[Sρ] = C[x1, x2, x2 , . . . , xn, xn ].

This is a localization of C[x1, . . . , xn] (at the element x2 ····· xn), which is an integrally closed domain, so it is an integrally closed domain as well.7

Corollary 1.30. Toric varieties are normal.

7Put in a diﬀerent way,

−1 −1 ∗ ∗ Spec C[x1, x2, x2 , . . . , xn, xn ] ' C × C × · · · × C

−1 −1 is a smooth variety, hence it is normal, so that C[x1, x2, x2 , . . . , xn, xn ] is integrally closed. 1.1. FANS AND TORIC VARIETIES 13

Proof. Normality is a local issue so we can assume that the variety is an aﬃne toric variety

Uσ for some strongly convex polyhedral cone σ. Then the claim follows from Proposition 1.29 (in particular from the implication 3 ⇒ 1).

1.1.4 Torus action

n Finally, we check that the torus T acts on the variety XΣ (in particular, it acts on every aﬃne toric variety Uσ), and study this action. In general, given an algebraic group G and a variety X, a (left) action of G on X is a morphism ζ : G × X → X satisfying the condition ζ(e, p) = p for all p ∈ X, and ζ(g, ζ(h, p)) = ζ(gh, p) for all g, h ∈ G and p ∈ X. The two conditions are tantamount to the commutativity of the diagrams

id×ζ e×id G × G × X / G × X X = {∗} × X / G × X (1.2)

m×id ζ ζ id ζ G × X / X ' X

where m: G × G → G is the multiplication morphism and e: {∗} → G maps the point to the identity of G. The arrows in these diagrams are morphisms of varieties. If X is aﬃne, it is enough to specify the morphism ζ at the level of the rings of functions, by assigning ∗ 8 a morphism ζ : AX → AG ⊗ AX satisfying the diagrams dual to the ones in (1.2):

id⊗ζ∗ e∗⊗id AG ⊗ AG ⊗ AX o AG ⊗ AX AX o AG ⊗ AX (1.3) O O e O m∗⊗id ζ∗ ζ∗ id ζ∗ AG ⊗ AX o AX AX

n The action T × Uσ → Uσ we want to deﬁne is then given by the morphism

u u u φ: C[Sσ] → C[M] ⊗ C[Sσ], χ 7→ χ ⊗ χ .

To check that this extends the action of Tn on itself means checking that φ, regarded as a map C[M] → C[M] ⊗ C[M], coincides the morphism induced by the multiplication map m: Tn × Tn → Tn, namely,

∗ ∗ m C[M] → C[M] ⊗ C[M], m (f)(s, t) = f(st).

8 Here AX and AG denote the rings of regulars functions on X and G respectiverly, so that X = Spec AX , n G = Spec AG. In the case at hand G = T so that AG = M. 14 CHAPTER 1. TORIC VARIETIES

Note that by identifying C[M] with the coordinate ring of Tn = N ⊗ C∗ we have χu(t) = u(t), and (χu ⊗ χu)(s, t) = χu(s) χu(t) = u(s) u(t) = u(st) = χu(st)

as required.9 We want to study the ﬁxed points of this action. These can be detected in the following

way. For every cone σ in a lattice N, let xσ be the point in Uσ given (in view of the correspondence between (closed) points in Uσ and semigroup morphisms Sσ → C) by the morphism10 ( 1 if u ∈ σ⊥ u 7→ (1.4) 0 otherwise

(recall that C is regarded as a semigroup with the multiplicative structure, and note that the sum of two elements in σ∨ is in σ⊥ if and only if both are in σ⊥ since the latter is a face of σ∨, so that the morphism is well deﬁned).

Proposition 1.31. If a strongly convex rational polyhedral cone σ spans NR, then xσ is n the unique ﬁxed point of the action of T on Uσ. If σ does not span NR, the action has no ﬁxed points.

To prove this result we need some preliminary material.

Deﬁnition 1.32. A semigroup S ⊂ M is pointed if S ∩ (−S) = {0}. 4

So a semigroup is pointed if and only if its only invertible element is 0.

9 ∗ n Let us give some detail about the equality u(s) u(t) = u(st). If v ⊗ x ∈ N ⊗Z C = T , and u ∈ M, then u(v ⊗ x) = xu(v). Therefore, if s = v ⊗ x and t = w ⊗ y,

u(s) u(t) = xu(v) yu(w).

P P P ∗i On the other hand, if {ei} is a basis of N, and v = i viei, w = i wiei, u = uie , one has

X vi wi st = ei ⊗ x y i

and Y u(st) = (xvi ywi )ui = xu(v) yu(w). i

10 ⊥ The symbol σ denotes the subset of NR given by

⊥ 0 0 σ = {v ∈ NR | v · v = 0 for all v ∈ σ}. 1.1. FANS AND TORIC VARIETIES 15

If Z = {u1, . . . , us} is a finite subset of a semigroup S ⊂ M, let SZ be the sub-semigroup it generates, and let UZ = Spec C[SZ ] be the corresponding affine variety. This variety can be embedded into the projective space Ps−1 as follows (so that it is quasi-projective). n s Consider the map ΦZ : T → C defined as

u1 us ΦZ (t) = (χ (t), . . . , χ (t)) (1.5)

where χu : Tn → C∗ is the character of the torus defined by u ∈ M, χu(t) = t(u) identifying Tn = Hom(M, C∗). Note that the value of this map can never be zero,11 so that the image s−1 s−1 of ΦZ can be projected to P . Let XZ be the (Zariski) closure of the image in P . It is a projective variety by definition, so that UZ , which can be identified with a dense subset

of XZ , is quasi-projective. For future use we note the following fact.

s−1 Lemma 1.33. Let Ui be the open set in P where the i-th homogeneous coordinate does not vanish. Then

XZ ∩ Ui ' Spec C[N(Z − ui)].

Lemma 1.34. Let U = Spec C[S] be an aﬃne toric variety, equipped with the action of Tn deﬁned as before.

1. The torus action on U has a fixed point if and only if S is pointed. If that is the case, the fixed point corresponds to the semigroup morphism ψ : S → C defined as ( 1 if u = 0 ψ(u) = (1.6) 0 if u 6= 0

s 2. After writing X = UZ ⊂ C for Z ⊂ S − {0}, the torus action has a ﬁxed point if and only if 0 ∈ UZ , and then the ﬁxed point is 0.

Proof. If a point p ∈ U is represented by a semigroup morphism ψ : S → C, we have χu(p) = ψ(u) for u ∈ S. On other hand, denoting by tp the action of t ∈ Tn on p, we have χu(tp) = χu(t) χu(p). If p is ﬁxed, we have χu(t) χu(p) = χu(p) which translates into

χu(t) ψ(u) = ψ(u).

11 P Indeed if t = i vi ⊗ xi, then u Y u(vi) χ (t) = xi i which is never zero. 16 CHAPTER 1. TORIC VARIETIES

This condition is satisﬁed by u = 0, while if u 6= 0, then ψ = 0. So the (unique) ﬁxed point is given by the prescription (1.6). On the other hand, (1.6) is a semigroup homomorphism if and only if S is pointed. Indeed, if S is pointed, it is clear that (1.6) is a semigroup homomorphism. Conversely, S is not pointed, let u 6= 0 be in S ∩ (−S). Then if (1.6) were a semigroup homomorphism, 1 = ψ(0) = ψ(u − u) = ψ(u)ψ(−u) = 0.

To get Proposition 1.31 we only need the following result.

Lemma 1.35. Let σ be a strongly convex rational polyhedral cone. The corresponding semigroup Sσ is pointed if and only if σ generates NR.

∨ Proof. If σ generates NR, the dual cone σ is a strongly convex polyhedral cone which generates MR. Then Sσ must be pointed: if Sσ ∩ (−Sσ) had any point diﬀerent from 0, ∨ σ would not be strongly convex. On the other hand, if σ does not generate NR, then ∨ σ contains a vector subspace of MR which intersects M nontrivially, so that Sσ is not pointed.

1.1.5 Limiting points

n ∗ ∗ After identifying the n-dimensional torus T = N ⊗Z C as Hom(M, C ), and since Hom(C∗, C∗) ' Z, the group Hom(C∗, Tn) of co-characters of Tn (one-parameter subgroups) can be identiﬁed with N; thus any co-character of Tn is given by an element ∗ n 12 v ∈ N. We shall denote by λv : C → T this co-character. It satisﬁes

λv(xy) = λv(x)λv(y). If v ∈ N and u ∈ M, the composition

u ∗ λv n χ C −→ T −→ C maps z to zu(v), so that u u(v) u(λv(z)) = χ (λv(z)) = z if z ∈ C∗ and u ∈ M.

Given a fan Σ, we denote by |Σ| its support, i.e., the union in NR of its cones. 12 P Note indeed that λv(x) = v ⊗ x, and if we write v = i vi ei on a basis of N, we have ! ! X vi X vi X vi λv(x) λv(y) = ei ⊗ x ei ⊗ y = ei ⊗ (xy) = λv(xy). i i i 1.1. FANS AND TORIC VARIETIES 17

Proposition 1.36. 1. If v ∈ N, and σ is a strongly convex polyhedral cone in NR, then v ∈ σ if and only if limz→0 λv(z) exists in Uσ.

2. If v is in the relative interior of σ, then limz→0 λv(z) = xσ.

Proof. 1. We note that the following conditions are equivalent:

1. limz→0 λv(z) exists in Uσ;

u(v) 2. limz→0 u(λv(z)) = limz→0 z exists in C for all u ∈ Sσ;

3. u(v) ≥ 0 for all u ∈ σ∨ ∩ M;

4. v ∈ (σ∨)∨ = σ.

To prove the second claim, assume that v ∈ σ∩N, so that limz→0 λv(z) exists as a point in Uσ. Denote that point by xv. The corresponding semigroup homomorphism ψv : Sσ → C is u u(v) ψv(u) = χ (xv) = lim u(λv(z)) = lim z . z→0 z→0 ⊥ ⊥ Now, since v ∈ Relint(σ), one has u(v) > 0 if u ∈ Sσ \ σ , and u(v) = 0 if u ∈ Sσ ∩ σ , so that ψv(u) = 0 in the ﬁrst case, and ψv(u) = 1 in the second. By comparing with (1.4) we see that xv is the distinguished point xσ.

Given a fan Σ, we denote by |Σ| its support, i.e., the union in NR of its cones. The previous result shows that if v is not in |Σ|, then then limz→0 λv(z) does not exist in XΣ. If it is in |Σ|, then either it is in {0}, which has no relative interior, so that the limiting point is not a fixed point (coherently with the fact that Tn has no fixed point), or it is in the relative interior of some cone (as the fan includes all faces), so that the limiting point is a fixed point of the torus action.

1.1.6 The orbit-cone correspondence

The previous analysis of the limiting points of the torus action on a toric variety allows one to establish a one-to-one correspondence between orbits of the torus action, and cones in a fan. Example 1.37. Example 3.2.1 pag. 115 of Cox-Little-Schenck. 4

Given a cone σ in a lattice N, let Nσ be the sublattice generated by σ, i.e.,

Nσ = (σ ∩ N) ∪ (−σ ∩ N). 18 CHAPTER 1. TORIC VARIETIES

Since σ is saturated, so is Nσ, so that the quotient N(σ) = N/Nσ is torsion-free, and is therefore a lattice. The exact sequence

0 → Nσ → N → N(σ) → 0 (1.7)

∗ splits, as N(σ) is free over Z. Tensoring the exact sequence (1.7) by C and letting TN(σ) = ∗ n N(σ) ⊗Z C we obtain a surjective group homomorphim T → TN(σ) (here n = dim N as n usual), so that T acts transitively on TN(σ). Lemma 1.38. Let σ be a strongly convex rational polyhedral cone in N.

1. The pairing M × N → Z induces a nondegenerate pairing (σ⊥ ∩ M) × N(σ) → Z;

⊥ ∗ 2. this pairing induces an isomorphism TN(σ) ' Hom(σ ∩ M, C ).

Proof. Both claims follows from the fact that σ⊥ ∩ M is the dual lattice to N(σ).

For every cone σ in a fan Σ, we denote by O(σ) the orbit of the distinguished point n xσ ∈ Uσ ⊂ XΣ under the action of T . Next lemma shows that all orbits O(σ) are algebraic tori.

Lemma 1.39. Let σ be a strongly convex rational polyhedral cone in N. Then

⊥ O(σ) = {ψ : Sσ → C | ψ(u) 6= 0 iﬀ u ∈ σ ∩ M} ⊥ ∗ ' Hom(σ ∩ M, C ) ' TN(σ).

Proof. We deﬁne

0 ⊥ O = {ψ : Sσ → C | ψ(u) 6= 0 iﬀ u ∈ σ ∩ M}.

This space is invariant under the action of Tn

t · ψ : u 7→ χu(t) ψ(u) and contains ψσ, i.e., the semigroup homomorphism corresponding to the distinguished point xσ. ⊥ ∨ ⊥ Note now that σ is the largest vector subspace of MR contained in σ , so that σ ∩ M ∨ 0 ⊥ is a subgroup of Sσ = σ ∩ M. The restriction of an element γ ∈ O to σ ∩ M yields a a group homomorphism σ⊥ ∩ M → C∗, so that there is an (injective) map

0 ⊥ ∗ O → Hom(σ ∩ M, C ). 1.1. FANS AND TORIC VARIETIES 19

On the other hand, itγ ˆ : σ⊥ ∩M → C∗ is an element in Hom(σ⊥ ∩M, C∗), we deﬁne γ ∈ O0 by letting ( γˆ(u) if u ∈ σ⊥ ∩ M γ(u) = 0 otherwise

so that O0 ' Hom(σ⊥ ∩ M, C∗). So we obtain bijections

⊥ ∗ 0 TN(σ) ' Hom(σ ∩ M, C ) ' O

n that are compatible with the T -action. Since this group acts transitively on TN(σ), it also 0 0 0 acts transitively on O . But O contains ψσ, so that O = O(σ).

We shall write τ ≤ σ if τ is a face of σ.

Theorem 1.40 (Orbit-cone correspondence). 1. There is a bijective correspondence be- n tween cones in Σ and orbits of T in XΣ given by

⊥ ∗ σ ←→ O(σ) ' Hom(σ ∩ M, C ).

2. dim O(σ) = n − dim σ.

3. [ Uσ = O(τ) τ≤σ

4. τ ≤ σ if and only if O(σ) ⊂ O(τ), and [ O(τ) = O(σ). (1.8) τ≤σ

Proof. 1. Let O be an orbit. Since the open aﬃne toric varieties Uσ are torus-invariant, there is a cone σ ∈ Σ such that O ⊂ Uσ. We may assume this cone to be minimal, and then it is unique (indeed if σ1 and σ2 are two such cones, then O ⊂ Uσ1 ∩ Uσ2 = Uσ1∩σ2 so that by minimality σ1 = σ2). ∨ Let ψ ∈ O and assume that u ∈ Sσ satisﬁes ψ(u) 6= 0; then u lies on a face of σ , which can be written as σ∨ ∩ τ ⊥ for some face τ of σ, i.e.,

∨ ⊥ {u ∈ Sσ | ψ(u) 6= 0} = σ ∩ τ ∩ M.

This implies ψ ∈ Uτ and then τ = σ since σ is minimal. Then

∨ {u ∈ Sσ | ψ(u) 6= 0} = σ ∩ M. 20 CHAPTER 1. TORIC VARIETIES

In view of Lemma 1.39 we have ψ ∈ O(σ) and then O = O(σ).

2. Follows from Lemma 1.39 since dim TN(σ) = n − dim σ.

3. We need to put together three things: any Uσ is a union of orbits; if τ is a face of

σ, then O(τ) ⊂ Uτ ⊂ Uσ; any orbit contained in Uσ is of the type O(τ) for a face τ of σ. The ﬁrst two claims are clear, the third follows from the proof of point 1. 4. At ﬁrst we think of O(τ) as the closure of O(τ) in the analytic topology, and we shall later prove that it coincides with the Zariski closure. We take as a known fact that the closure of an orbit is a union of orbits. Let O(σ) ⊂ O(τ) (where we use the fact that any orbit is associated with a cone). Then O(τ) ⊂ Uσ, indeed if O(τ) ∩ Uσ = ∅ then

O(τ) ∩ Uσ = ∅ since Uσ is open (also in the analytic topology), and this is impossible since

O(τ) contains O(σ). By part 3, O(τ) ⊂ Uσ implies that τ is a face of σ. To prove the converse, if τ is a face of σ we show that O(τ) ∩ O(σ) 6= ∅ (then O(σ) ⊂ n O(τ) necessarily). We can do this using one-parameter subgroups of T . Let ψτ be the semigroup homomorphism corresponding to the distinguished point of Uτ , and let v ∈ ∗ Relint(σ). For every z ∈ C we deﬁne a semigroup homomorphism ψz = λv(z) · ψτ , whose action is

u u(v) ψz(u) = χ (λv(z)) ψτ(u) = z ψτ (u).

Since O(τ) is the orbit of ψτ , ψz is in O(τ) for all z. Now we note that u(v) > 0 of u ∈ σ∨ \ σ⊥ and u(v) = 0 if u ∈ σ. Then

ψ0 = lim ψz z→0

exists in Uσ, and is a point in O(σ), so that O(τ) ∩ O(σ) 6= ∅. Equation (1.8) now follows. Then we show that the analytic closure O(τ) coincides with the Zariski closure. We intersect O(τ) with an aﬃne open toric subvariety Uσ0 ; then

[ O(τ) ∩ Uσ0 = O(σ). τ≤σ0≤σ

One can show that this is the an aﬃne subvariety of Uσ0 given by the ideal

u ⊥ 0 ∨ 0 ∨ I = < χ | u ∈ τ ∩ (σ ) ∩ M > ⊂ C[(σ ) ∩ M] = Sσ0 .

Hence O(τ) is Zariski closed in XΣ, and therefore it is also the Zariski closure of O(τ). 1.2. MORE PROPERTIES OF TORIC VARIETIES 21 1.2 More properties of toric varieties

1.2.1 Singularities

We know that whenever a cone is generated by a basis of N, the corresponding affine toric variety is a copy of affine space Cn, and is therefore smooth. We also know that when the k generators of the cone can be completed to a basis of N, the corresponding affine variety is isomorphic to Ck × Tn−k, hence it is smooth as well. These are actually the only instances of affine toric varieties obtained from cones in a lattice that are smooth. ⊥ Indeed, let us consider at first the case when σ generates NR, so that σ = {0}. If Uσ is nonsingular, in particular it is nonsingular at the distinguished point xσ. Let m be the ∨ 2 maximal ideal of Aσ = C[σ ∩M] corresponding to that point. Then m/m is the cotangent 2 u space at xσ, so that dimC m/m = n. Now, m is generated by {χ } for all nonzero u ∈ Sσ. Then m2 is generated by the elements u that are sum of two other elements. So m/m2 is generated by the images of the elements χu with u indecomposable, for instance, the generators of the edge of the cone. This implies that σ cannot have more than n edges, and the minimal generators of these edges span M. So σ is generated by a basis of N, and n Uσ ' C .

Now let σ have a non-maximal dimension k, and let still assume that Uσ is nonsingular.

Let us introduce the sublattice Nσ generated by σ, and the quotient lattice N(σ) as before. After splitting the exact sequence (1.7) we set

σ = σ0 ⊕ {0}.

If M = Mσ ⊕ M(σ) is the dual splitting, we have

∨ Sσ = (σ0 ∩ Mσ) ⊕ M(σ) and n−k Uσ ' Uσ0 × T .

The meaning of the symbols Uσ and Uσ0 is that these are the toric varieties associated with the same cone σ, the ﬁrst regarding σ as a cone in NR, and the second by regarding it as a cone in NσR. Now, if Uσ is nonsingular, then Uσ0 is nonsingular as well, so that the 0 k previous analysis applies; σ is generated by a basis of Nσ, and Uσ0 ' C . So we have proved:

Proposition 1.41. An aﬃne toric variety Uσ is smooth if and and only if σ is generated k n−k by k elements of a basis of N, and then Uσ ' C × T . 22 CHAPTER 1. TORIC VARIETIES

A cone with this property will be called smooth. Example 1.42. We study now an example which generalizes Example 1.17. Let N = Z2 and consider the cone σ generated by e2 and me1 − e2; for m = 2 we recover indeed Example 0 0 0 1.17. Let N be the lattice generated by e2 and e1 = me1 − e2, and let σ be σ regarded as a cone in N (note that N 0 = N ). Now σ0 is generated by the generators of N 0, so that R R 2 0 0 0 Uσ0 = C . Moreover we have inclusions N ⊂ N and M ⊂ M . Actually M is generated 1 ∗ ∗ 1 ∗ 1 ∗ ∗ 0 by m e1 and e2, while Sσ is generated by m e1 and m e1 + e2. We associate variables 1 1 e∗ s, e∗ y, e∗ + e∗ t, m 1 2 m 1 2 so that sy = t. 0 Now, the inclusion N,→ N yields a map σ → σ, deﬁned by a map Aσ → Aσ0 . One has Aσ0 = C[s, t], while to identify Aσ we note that Sσ is generated by ∗ ∗ ∗ ∗ ∗ ∗ ∗ e1, e1 + e2, e1 + 2e2, . . . , e1 + me2 m m so that Aσ = C[x, xy, . . . , xy ]. By setting x = s , y = t/s we obtain m m−1 m Aσ = C[s , s t, . . . , t ].

2 Now note that µm ' Zm, the group of m-th roots of unity, acts on C in the standard µm way ζ(s, t) = (ζs, ζt). The ring Aσ may be described as C[s, t] , the ring of µm-invariant 2 polynomials in 2 variables, and therefore Uσ is the quotient C /µm, a variety which is singular at the origin, where it has a quotient singularity (the variety is a cone of a normal curve of degree m, i.e., it is the “aﬃne part” of the cone over a curve in Pm+1; “normal” here is an old terminology and does not refer to normality). 4

This example is easily generalized if we consider a cone generated by e2 and me1 − ke2, 2 k with 0 < k < m. In this case µm acts on C by ζ(s, t) = (ζs, ζ t). To formalise in a way which is valid in all dimensions, we consider the case when σ is an n-symplex in N, where as usual n = dim N, i.e., σ is generated by n independent vectors (which, however,

need not be a basis for N). The previous analysis goes through, and Uσ turns out to be n 0 n a quotient C /G, where G is the ﬁnite abelian group G = N/N , Uσ0 = C , and G has an action on Uσ0 obtained as before. Note that Aσ can be obtained by intersecting Aσ0 with C[M 0]G = C[M]. A fan Σ will be said simplicial if all its cones are simplicial.

Deﬁnition 1.43. An orbifold of dimension n is a variety which is locally of the form Cn/G, where G is a ﬁnite group. 4

The previous discussion proves the following theorem. Theorem 1.44. If the fan Σ is simplicial, the associated toric variety is an orbifold. 1.2. MORE PROPERTIES OF TORIC VARIETIES 23

1.2.2 Completeness

A reminder about morphisms of schemes.

1. A morphism of schemes f : X → Y is separated if the diagonal morphism ∆f : X →

X ×Y X is a closed immersion.

2. f is of finite type if there is an affine open cover {Ui = Spec Bi} of Y and for every −1 i, an affine open cover {Uij = Spec Aij} of f (Vi), where each Aij is a finitely generated

Bi-algebra. 3. f is universally closed if for every morphism Y 0 → Y , the corresponding morphism 0 0 0 0 0 f : X → Y is closed, where X is the ﬁber product X ×Y Y :

f 0 X0 / Y 0

f X / Y

(in particular, f is closed). 4. f is proper if it is separated, of ﬁnite type, and universally closed. 5. A scheme X over a ﬁeld k is complete if the structural morphism X → Spec k is proper. One proves that a variety over C is proper if and only if is compact in the analytic topology.

Theorem 1.45. A toric variety XΣ is complete if and only if |Σ| = NR.

It is easy to prove the “only if”. Indeed, if |Σ| is not the whole of NR, let v ∈ N be a lattice point which is not in |Σ|. Then by Proposition 1.36 limz→0 λv(z) does not exist,

which contradicts the completeness of XΣ. To prove the “if” party, we actually generalize Theorem 1.45, and prove the generalization using the valuative criterion. Let φ: N 0 → N be a morphism of lattice that maps a fan Σ0 to a fan Σ, and denote as usual with the same

symbol the corresponding morphism φ: XΣ0 → XΣ.

Proposition 1.46. The morphism φ is proper if and only if φ−1(Σ) = Σ0.

This reduces to Theorem 1.45 if we take N = {0}. To prove the “if” part of this proposition, we use the valuative criteria for properness [?]. We recall that a valuation ring is a integral domain R such that, given an element x in its ring of fractions K, at least one between x and x−1 belong to R. Moreover there is a totally ordered abelian group Γ, called the value group, and a surjective morphism ord: K∗ → Γ such that

R = {x ∈ K∗ | ord(x) ≥ 0} ∪ {0}. 24 CHAPTER 1. TORIC VARIETIES

The valuation ring is discrete if Γ is the group of integers.13 Discrete valuation rings may also be deﬁned as principal ideal domains having a unique maximal ideal.

Theorem 1.47. [?, Thm. II.4.7] Let f : → Y be a scheme morphism of ﬁnite type, with X noetherian. f is proper if and only if for every discrete valuation ring R and every commutative diagram Spec K / X (1.9) ; f Spec R / Y there is a morphism Spec R → X making the completed diagram commutative.

Note that if X = Spec(B) and Y = Spec(A) are affine, the diagram (1.9) is equivalent to Spec K / X ; f Spec R / Y Remark 1.48. If X is irreducible, one can assume that Spec K maps into a fixed nonempty open subset. Moreover, properness is local on the target, so that one can assume Y to be affine. 4

Proof of Theorem 1.47. We want to apply this to X = XΣ0 , Y = XΣ, f = φ. In view of the previous remark, we may assume that Y is aﬃne, Y = Uσ, and assume that Spec K n maps into an open subset U ⊂ XΣ which is isomorphic to the n-dimensional torus T , with n = dim N 0. The morphism Spec K → U corresponds to a homomorphism of algebras C[M 0] → K given by a group homorphism α: M 0 → K∗. We need to ﬁnd a cone σ0 ⊂ Σ0 such that the diagram α 0 _ K o C[M ] o ? C[Sσ0 ] O O : ∗ φ∗ φ ? u R o C[Sσ]

commutes. Now, Spec R maps to Uσ, so that the composition

∗ φ 0 α ord C[Sσ] −→ C[M ] −→ K −−→ Z (1.10)

13An example of a discrete valuation ring is the set R of pairs (f, g) of elements in k[x], where k is a field, and g(0) 6= 0. If we write such a pair as f/g, the ring structure is given by (f/g)·(f 0/g0) = (ff 0/gg0). K is the field of fractions, i.e., the field of rational functions k(x), and the valuation is the order of f at 0. The maximal ideal is the ideal generated by x/1; the projection R → k is the evaluation at 0. 1.2. MORE PROPERTIES OF TORIC VARIETIES 25

has nonnegative values. By hypothesis, there is a cone σ0 in N 0 such that φ(σ0) ⊂ σ, so that the composition (1.2.2) can be written as

0 α ord C[Sσ] → C[Sσ0 ] → C[M ] −→ K −−→ Z which implies that we can complete the diagram with the arrow C[Sσ0 ] → R. Example 1.49. Theorem 1.47 gives a very easy way to check if a toric map is proper. As an example, let us describe the blowup of a toric variety at a ﬁxed point of the torus action.

Let σ be a cone in the fan Σ associated to a toric variety generated by a basis {v1, . . . , vn}

of N. Set v0 = v1 + ··· + vn, and replace σ by the collection of cones generated by subsets

of {v0, v1, . . . , vn} that do not contain {v1, . . . , vn}. This change only affects the variety n Uσ, so we can assume that Σ consists only of σ and its faces, so that XΣ = Uσ ' C . We n may also assume N = Z and vi = ei. The new toric variety XΣ0 is covered by affine open ∨ toric varieties Uσi , where σi is the cone generated by {e0, e1,..., eî, . . . , en}. Then σi is ∗ ∗ ∗ ∗ ∗ generated by {ei , e1 − ei , . . . , en − ei }, and the coordinate rings of the varieties Uσi are

−1 −1 Aσi = C[xi, x1xi , . . . , xnxi ].

n Thus Uσi ' C . Note that xj/tj = xi/ti when ti, tj 6= 0. On the other hand, the blowup Cˆ n of Cn at the origin is the subvariety of Cn × Pn−1 n deﬁned by the equations xitj = xjti, with {x1, . . . , xn} aﬃne coordinates in C ), and n−1 ˆ n {t1, . . . , tn} homogeneous co-ordinates in P . The sets Ui ⊂ C given by ti 6= 0 are n copies of C , and glue with the transition functions xj = xi(ti/tj) with i 6= j. Thus n XΣ0 ' C are obtained by glueing the same open sets with the same transition functions, hence they are isomorphic. Actually Σ0 is obtained by subdividing σ into subcones, so that φ−1(Σ) = Σ0, and the

morphism Xσ0 → XΣ is proper. 4 Example 1.50. Let N = Z2 and let Σ the standard fan for P2. If we add a further ray generated by (−1, m), with m a positive integer, we get a complete, smooth toric surface 2 Fm, called the m-th Hirzebruch surface. The map Fm → P given by the subdivision of Σ is a blowup at a ﬁxed point of the torus action. The inspection of the fan Σ0 provides a 1 1 lot of information about Fm (for instance, that it is a P ﬁbration over P . 4

Remark 1.51. If Y is a smooth projective surface, the same construction that one does to blowup C2 at the origin allows one to blow up Y at a point p, constructing a new smooth projective surface X and a proper birational morphism π : X → Y , which is an isomorphism away from the (smooth rational) curve E = π−1(p). Moreover, E2 = −1. The curve E is called the exceptional divisor. Castelnuovo’s criterion in a sense tells that 26 CHAPTER 1. TORIC VARIETIES

this is the general situation: if X is a smooth projective surface, and C a rational curve in it such that C2 = −1, then the surface Y that one obtains by shrinking C to a point is smooth, and X is the blowup of Y at the resulting point [?]. 4

The complete (smooth) toric surfaces we have so far seen in our examples are the projective plane and the Hirzebruch surfaces. Applying the procedure of Exercise 1.50 we can construct new complete smooth toric surfaces. Some fan combinatorics shows that these are actually all complete smooth toric surfaces [?].

Theorem 1.52. Every complete smooth toric surface is obtained by a sequence of blowups at ﬁxed points of the toric action starting either from the projective plane or from a Hirze- bruch surface.

Corollary 1.53. All complete toric surfaces are projective.

1.3 Resolution of singularities

The previous discussion showed a technique to blow up smooth toric surfaces at a fixed point of the torus action. The same tool can be used to resolve the singularities of a 2 singular toric surface. Let σ be the cone in Z generated by 2e1 − e2 and e2, and let Σ be the fan consisting in σ and its faces. Note that the matrix that expresses these generators in terms of e1 and e2 is not unimodular, so the affine variety XΣ is singular. If we add the 0 ray spanned by e1, the cones in the new fan Σ are all smooth. We have a proper birational morphism π : XΣ0 → XΣ; adapting what we saw in Example 1.50, this is shown to be an isomorphism away from the inverse image of the singular locus of XΣ, i.e., the morphism π is a resolution of singularities. 2 This procedure can be iterated. If σ is the cone in Z generated by 3e1 − 2e2 and e2, and Σ is the fan consisting in σ and its faces, again XΣ is singular. If we add the ray generated by e1, now the affine toric variety associated with the cone generated by e1 and e2 is smooth, but the one given by the cone generated by e1 and 3e1 − 2e2 is still singular. 0 But if we add the ray generated by 2e1 − e2, the cones in the new fan Σ are all smooth.

We have a proper birational morphism π : XΣ0 → XΣ; adapting what we saw in Example 1.50, this is shown to be an isomorphism away from the inverse image of the singular locus of XΣ, i.e., the morphism π is a resolution of singularities. More generally, given a cone σ which is not smooth, as we already saw we can choose 2 0 (after identifying N with Z ) generators v2 and v = me1 − ke2, with m ≥ 2, 0 < k < m, and m and k coprime. We add the ray generated by e1. The variety associated to new cone 1.3. RESOLUTION OF SINGULARITIES 27

2 generated by e1 and e2 is smooth, isomorphic to C , while in general the variety associated 0 to the cone generated by e1 and v is still singular. The point however is this variety is “less singular” than the original one. To see this we ﬁrst rotate the cone by 90◦ counteclockwise,

obtaining the vectors e2 and ke1 + me2, then apply the base change given by the matrix ! 1 0 c 1

00 with c chosen in a way that the new vector v = m1e1 −k1e2 satisﬁes m1 = k, 0 ≤ k1 < m1,

k1 = a1 − m for some integer a1 ≥ 2. If k1 = 0 the new cone is smooth, otherwise the procedure can be iterated. This boils down to writing the faction m/k in the form of the Hirzebruch-Jung continued fraction m 1 = a − k 1 1 a − 2 1 ··· + ar

where ai ≥ 2. This gives rise to a sequence of r blowups, the last of which produces a smooth surface. At each blowup a new exceptional divisor Ei is created, and one can check 2 that each Ei = −ai (note that, as suggested by Castelnuovo’s criterion, the self-intersection must be smaller than -1 otherwise by blowing down one would produce a surface with the same singularity). Moreover these exceptional divisor form a chain, i.e., Ei · Ej = 1 for 1 ≤ i < j ≤ r. This procedure is fully local, and applies to any non-smooth cone in a fan. Moreover it can be adapted to work in any dimension. Doing this for all non-smooth cones in a fan we obtain:

0 Theorem 1.54. For every toric variety XΣ there is a reﬁnement Σ of the fan Σ such that

the natural morphism XΣ0 → XΣ is a resolution of singularities.

Remark 1.55. 1. This blowup procedure is equivariant with respect to the torus action. 2. The torus corresponding to the origin is smooth, and all blowups leave it unchanged. 4 28 CHAPTER 1. TORIC VARIETIES Chapter 2

Divisors and line bundles in toric varieties

2.1 Base-point free, ample and nef line bundles on normal varieties

2.1.1 Base point free line bundles and divisors

Let X be variety, and L a line bundle on X.

Deﬁnition 2.1. Let W be a subspace of Γ(X,L). A point p ∈ X is a base point of W if s(p) = 0 for all s ∈ W . The subspace W is base point free if it has no base point. L is said to be base point free if Γ(X,L) is base point free. A Cartier divisor D is base point free if

Γ(X, OX (D)) is base point free. 4

A global section s of L gives rise to a sheaf morphism OX → L by mapping f ∈ OX (U)

to f · s|U . So given a subspace of W of Γ(X,L) there is morphism f : W ⊗C OX → L.

Lemma 2.2. W is base point free if and only if f : W ⊗C OX → L is surjective.

Proof. If p is a base point, then the ﬁbre fp of f at p is zero, so that f is not surjective.

On the other hand, if f is not surjective at a point p, since L has rank 1, then fp = 0, i.e., all sections in W vanish at p.

Suppose that X is normal, and let D be a Cartier divisor, with local data {(Ui, fi)}.

Let L = OX be the associated line bundle. Assume that L has a global section s. This can −1 be regarded as a rational function f ∈ C(X) by recalling that L is locally generated by fi 30 CHAPTER 2. DIVISORS AND LINE BUNDLES

−1 and writing s|Ui = si · fi with si ∈ OX (Ui). By its very definition, f satisfies D + (f) ≥ 0. We write (s) = D + (f). This (the divisor of zeroes of s) is an effective divisor linearly equivalent to D.

Deﬁnition 2.3. A line bundle L is generated by global sections if the evaluation morphism

Γ(X,L) ⊗ OX → L is surjective. 4

Proposition 2.4. Let D be a Cartier divisor. The following conditions are equivalent.

1. OX (D) is generated by global sections.

2. D is base point free.

3. For every p ∈ X there is a section s of OX (D) such that p∈ / supp(s) (i.e., there is a section s that does not vanish at p).

Proof. 2 and 3 are clearly equivalent. 1 and 2 are equivalent by Lemma 2.2.

Lemma 2.5. Let L be a line bundle on a scheme X. If f : Y → X is a surjective morphism of schemes, and L is generated by global sections, then f ∗L is generated by global sections.

∗ Proof. By applying the left-exact functor f to the surjective morphism Γ(X,L)⊗OX → L ∗ we get a surjective morphism Γ(X,L) ⊗ OY → f L. Since f is surjective, Γ(X,L) is a subspace of Γ(Y, f ∗L) , so that we obtain the claim.

2.1.2 Ample and numerically eﬀective divisors

We need a preliminary. Let L be a line bundle on X, and let V (L) be its total space, with projection π : V (L) → X. Let f : Y → X be a morphism, where Y is another variety.

Then the ﬁbre product Y ×X V (L) is a geometric rank one bundle on Y . Its sheaf of sections is an invertible sheaf on Y , that we denote f ∗L, and call the pullback of L to Y . ∗ −1 −1 This coincides with the usual pullback of sheaves of modules, f L = f L ⊗f OX OY . Let L be a line bundle on X, and W a base point free subspace of Γ(X,L). We construct ∨ a morphism φW : X → PW as follows. Given a point p ∈ X, pick any νp ∈ Lp (the ﬁbre of L at p) and for any s ∈ W let λs be the complex number such that s(p) = λs · νp. The map

`p : W → C, `p(s) = λs

∨ is linear and nonzero since W is base point free. Thus `p ∈ W − {0}. This depends on ∨ the choice of νp but its class in PW does not. So we set φW (p) = [`p]. 2.1. BASE-POINT FREE, AMPLE AND NEF LINE BUNDLES ON NORMAL VARIETIES31

A basis-dependent description of this map shows that it is a morphism of varieties.

Choose a basis (s0, . . . , sN ) of W . For a suitable open set U, let ψU : L|U → OU be a trivialisation (regarded as an isormphism of sheaves of OU -modules). We deﬁne a morphism N+1 φU : U → C by letting

φU (p) = (ψU ◦ s0(p), . . . , ψU ◦ sN (p)).

Since two overlapping trivialisations diﬀer by multiplication by a nonzero regular function, N this deﬁnes a morphism φ: X → P , which actually coincides with φW by identifying W N+1 with C via the chosen basis. This shows that φL is a morphism.

If L is base point free, we shall denote by φL = φΓ(X,L).

Deﬁnition 2.6. The line bundle L is said to be very ample if it is base point free, and the ∨ m morphism φL : X → PΓ(X,L) is a closed embedding. L is said to be ample if L is very ample for some m > 0. A Cartier divisor D is said to be ample if OX (D) is ample. 4

Exercise 2.7. Let L be a very ample line bundle on X, and V = Γ(X,L)∨. Prove that ∗ φLOPV (1) ' L. 4 We want to deﬁne a kind of “degeneration” of the notion of ample divisor. The basic idea (which at the moment may appear to be quite unrelated) is that we want to “pair” a Cartier divisor D and a curve C in such a way that, when D and C intersect “nicely”, the pairing counts their intersection points. We shall call this pairing the intersection product of D and C and will denote it D · C. We want the following properties to be satisﬁed. (E is another Cartier divisor).

• (D + E) · C = D · C + E · C.

• D · C = E · C when D ∼ E.

• If D is a prime divisor and D ∩ C is ﬁnite, assume that p ∈ D ∩ C is smooth in X, D and C and that the tangent spaces to p in C and D are transversal (their sum in

TpX is TpX). Then D · C = #(D ∩ C).

The key notion to deﬁne the intersection product between a Cartier divisor and a curve is that of degree of a divisor, or a line bundle, in/over a smooth complete irreducible curve C. P Deﬁnition 2.8. Let C be a smooth complete irreducible curve, and D = i aipi a divisor P in C. The degree of D is the integer number deg D = i ai. 4 32 CHAPTER 2. DIVISORS AND LINE BUNDLES

We have the following properties.

Proposition 2.9. 1. If D, E are divisors on C, then deg(D + E) = deg D + deg E.

2. If D is a principal divisor, then deg D = 0.

As a consequence, deg D = deg E whenever D ∼ E. One knows that every smooth complete irreducible curve is projective [?]. As a conse-

quence, any line bundle L on C can be written as L = OC (D) for some divisor D in C. By the previous Proposition, we can set deg L = deg D. We have the following properties.

• deg L = deg L0 if L ' L0.

• deg(L ⊗ L0) = deg L + deg L0.

Recalling that an aﬃne scheme X is normal if A = C[X] is an integrally closed ring, one can deﬁne the normalisation of a scheme X = Spec A as X¯ = Spec A¯, where A¯ is the integral closure of A. The inclusion of A into A0 yields a morphism X¯ → X. This construction is local, hence it extends to any scheme X. In particular, we have the following fact:

Proposition 2.10. If C is an irreducible curve, the normalisation C¯ of C is smooth. If C is complete, C¯ is complete as well.

The ﬁrst claim follows from the fact that a normal variety is smooth in codimension one. For the second see [?].

Deﬁnition 2.11. Let D be a Cartier divisor in a normal variety X, and let C be an irreducible complete curve in X. Denote by φ: C¯ → C the normalisation of C. The intersection product D · C is deﬁned as

∗ D · C = deg φ OX (D).

One checks that this intersection product has all the sought-for properties (the ﬁrst two are easily consequences of the properties of the operation of taking the degree, the third is less trivial; we refer the reader to [?].

Deﬁnition 2.12. A Cartier divisor D in a normal variety X is numerically eﬀective (nef) is D · C ≥ 0 for all complete irreducible curves C in X. 4 2.1. BASE-POINT FREE, AMPLE AND NEF LINE BUNDLES ON NORMAL VARIETIES33

Proposition 2.13. Every base point free divisor is nef.

Proof. If D is a base point free divisor, and C a complete irreducible curve C in X, let ¯ ∗ φ: C → C be the normalisation of C. Then L = φ OX (D) is generated by global sections by Lemma 2.1.1. Chosen a section s of L, let D0 be the (base point free) divisor (s); then 0 L 'OC¯(D ). Thus, D · C = deg L = deg(s) ≥ 0.

The notion of nefness is in a sense a limit of the notion of ampleness. To see this, we recall the Nakai-Moishezon criterion for ampleness [?]. This requires a more extended notion of intersection product with respect to the one we rigorously introduced. We shall use the fact that given a Cartier divisor D in an n-dimensional complete variety X, that we assume to be smooth for simplicity, and given a closed subvariety V of X of dimension k, one can give sense to the intersection product Dk · V . We refer to [?] and [?] for deﬁnitions and properties. The Nakai-Moishezon criterion states that D is ample if and only if Dk · V > 0 for all k, 0 < k ≤ n, and all closed subvarieties V of X of dimension k. For k = n the condition means Dn > 0. The “limit” condition Dk · V ≥ 0 can be shown to be equivalent to the nefness of D [?] (i.e., the condition holds true for all irreducible subvarieties if and only if it holds for curves).

Deﬁnition 2.14. A Weil divisor D in a normal variety X is said to be Q-Cartier if mD is Cartier for some positive integer m. A normal variety X is said to be Q-factorial if all Weil divisors are Q-Cartier. 4

The intersection product can extended to Q-Cartier divisors (on which it is Q-valued) by letting 1 D · C = (mD) · C m and so also for these divisors the notion of numerical eﬀectiveness makes sense.

2.1.3 Nef and Mori cones

Deﬁnition 2.15. Let X be a normal variety. Two Cartier divisors D and E in X are numerically equivalent, written D ≡ E, if D · C = E · E for all irreducible complete curves C in X. A Cartier divisor is numerically equivalent to 0 if D · C = 0. Analogously, a complete curve C is numerically equivalent to 0 if D · C = 0 for all Cartier divisors D in X, and two complete curves C and C0 are numerically equivalent if C − C0 ≡ 0. 4 34 CHAPTER 2. DIVISORS AND LINE BUNDLES

Deﬁnition 2.16. Let X be a normal variety. We associate with it the following objects.

1 • The real vector space N (X) = (Cdiv(X)/ ≡) ⊗Z R. • The free abelian group Z1(X) generated over Z be the irreducible complete curves C in X. Its elements are called 1-cycles.

• The real vector space N1(X) = (Z1(X)/ ≡) ⊗Z R. • The cone Nef(X) in N 1(X) generated by the nef Cartier divisors, called the nef cone.

• The cone NE(X) in N1(X) generated by the irreducible complete curves.

• The closure NE(X) of NE(X) in N1(X), called the Mori cone.

1 The intersection product deﬁnes a pairing N (X) × N1(X) → R, which is easily shown to be nondegenerate. A less trivial fact is that these spaces are ﬁnite-dimensional [].

Lemma 2.17. 1. Nef(X) and NE(X) are closed convex cones, dual to each other.

2. NE(X) has full dimension in N1(X). 3. Nef(X) is strongly convex in N 1(X).

Proof. It is obvious that the three cones we have introduced are convex. Moreover, Nef(X) is closed and Nef(X) = NE(X)∨ by the deﬁnition of nefness. This also means Nef(X) = NE(X)∨. Since the closure of a convex cone is its double dual, we have

NE(X) = NE(X)∨∨ = Nef(X)∨.

Point 2 is obvious. Then also NE(X) has full dimension in N1(X). This implies that Nef(X) is strongly convex, as its dual has full dimension.

2.2 Polytopes

Divisors in toric varieties are conveniently studied by means of another sort of combinatorial objects, called polytopes.

2.2.1 Convex polytopes

A convex polytope ∆ in a ﬁnite-dimensional real vector space V is the convex hull of a ﬁnite set of points. A proper face F on ∆ is the intersection with a supporting hyperplane

F = {v ∈ ∆ | u(v) = r} 2.2. POLYTOPES 35

where u ∈ V ∗ is such that u(r) ≥ r for all v ∈ ∆. The latter is considered to be an improper face of itself. We shall make the standard assumptions that dim ∆ = dim V (say n) and that ∆ contains 0 in its interior. Our previous results about cones can be transferred to analogous results on polytopes by introducing the cones σ over the set ∆ × {1} ⊂ V × R. The faces of σ are the cones over the faces of ∆, with the cone {0} corresponding to the empty face of ∆.

Deﬁnition 2.18. The polar set of a convex polytope ∆ is the set

∆◦ = {u ∈ V ∗ | u(v) ≥ −1 for all v ∈ ∆}.

Proposition 2.19. The polar set ∆◦ of a convex polytope ∆ is a convex polytope, and ∆ is the polar set of ∆◦. If F is a face of ∆, then

F ◦ = {u ∈ ∆◦ | u(v) = −1 for all v ∈ F } is a face of ∆◦. This establishes a one-to-one correspondence between the faces of ∆ and ∆◦, with dim F + dim F ◦ = n − 1. If ∆ is rational, meaning that its vertexes lie in some lattice of V , then ∆◦ is rational as well, and its vertexes lie in the dual lattice.

Proof. If σ is the cone over ∆ × {1} ⊂ V × R, the dual cone is the set of points (u, r) in V ∗ × R such that u(r) + r ≥ 0, and coincides with the cone over ∆◦ × {1} in V ∗ × R. The claimed results then follow from the analogous results about cones. In particular, (∆◦)◦ = ∆ follows from (σ∨)∨ = σ, and the duality between faces follows from the fact that if τ is the cone over F × {1} for a face F of ∆, then τ ∨ = σ∨ ∩ τ ⊥ is the cone over F ◦ × {1}.

One applies this to toric varieties starting with a lattice N of rank n and considering

n-dimensional rational polytopes ∆ in MR. We shall relax the requirement that ∆ contains

the origin. The we construct a fan Σ in NR by associating cones σF to the faces F of ∆:

0 0 σF = {v ∈ NR | u(v) ≤ u (v) for all u ∈ F, u ∈ ∆}.

Proposition 2.20. The set {σF | F is a face of ∆} is a fan Σ in NR. If ∆ contains the origin in its interior, then Σ is made of the cones over the faces of the dual polytope ∆◦.

The following are easy but instructive examples: 36 CHAPTER 2. DIVISORS AND LINE BUNDLES

• If ∆ is a standard simplex in Rn, the associated fan deﬁnes the projective n-space Pn.

3 ∗ ∗ ∗ • If ∆ is the cube in R with vertices in the points ±e1 ± e2 ± e3, then Σ is the fan made by the cones over the faces of the octahedron with vertexes ±ei, and the corresponding toric variety is the product P1 × P1 × P1.

2.2.2 Canonical presentations

We say that a polytope ∆ is full dimensional if dim ∆ = dim MR. In this case every facet F of ∆ has a unique supporting aﬃne hyperplane. Let us deﬁne

+ HF = {u ∈ MR | u(vF ) = −aF } and HF = {u ∈ MR | u(vF ) ≥ −aF } for some set of pairs (vF , aF ) ∈ NR ×R+, that are unique up to multiplication by a positive real number. Then vF is an inward-pointing normal vector to the facet F . It follows that ∆ can be represented as

\ + ∆ = HF = {u ∈ MR | u(vF ) ≥ −aF for all facets F < ∆}. (2.1) F We say that ∆ is rational of all its vertexes are in M. In this case the normals to the facets lie on a rational ray, and vF can be taking as a generator of that ray. Then the numbers aF are integers, since u(vF ) = −aF when u is a vertex.

2.3 Divisors in toric varieties

2.3.1 The class group of a toric variety

Let XΣ be an n-dimensional toric variety associated with a fan Σ. We want to describe the class group of XΣ. If ρ is a ray in Σ, by the orbit-cone correspondence the closure of its n orbit under the torus action is a T -invariant prime Weil divisor Dρ in XΣ. Let us denote by νρ the corresponding valuation morphism.

Lemma 2.21. If vρ is the minimal generator of ρ, and u ∈ M, then

u νρ(χ ) = u(vρ).

n Proof. Extend vρ to a basis {ei} of N, with e1 = vρ. Thus N ' Z and ρ is the cone generated by e1. The corresponding toric aﬃne variety is

± ± ∗ n−1 Uρ = Spec[x1, x2 , . . . , xn ] ' C × (C ) , 2.3. DIVISORS IN TORIC VARIETIES 37

and Dρ ∩ Uρ has equation x1 = 0. Thus the associated discrete valuation ring is

± OXΣ,Dρ = C[x1, . . . , xn]x1 = C[x1 , x2, . . . , xn].

∗ Then a rational function f ∈ C(x1, . . . , xn) has valuation ` if g f = x` with g, h ∈ [x , . . . , x ]. 1 h C 2 n

Note now that the xi are the characters of the dual basis to {ei}, so that

u Y u(ei) χ = xi . i

u So νρ(χ ) = u(e1) = u(vρ).

This essentially also proves:

u Proposition 2.22. Given u ∈ M, the character χ is a rational function on XΣ, whose corresponding divisor is u X (χ ) = u(vρ) Dρ. ρ∈Σ(1)

Proof. We only need to prove that (χu) is a linear combination of torus-invariant divisors; then the coeﬃcients are as in the formula in view of the previous Lemma.

We need the following results (see e.g. [?] for related results). Note that if X is an integral scheme and U an open subset in it, there is a natural morphism Cl(X) → Cl(U) which is actually surjective. The morphism is given by intersecting a prime divisor in X with U, and given a prime divisor in U, by taking its closure in X we obtain a counterimage in Cl(X).

Theorem 2.23. Let X be a normal variety, U a nontempty open subset, and let D1,...,Ds be the components of X − U of codimension 1. The sequence

s M ZDj → Cl(X) → Cl(U) → 0 j=1 is exact.

s Proof. We already hinted that Cl(XΣ) → Cl(U) is surjective. That the image ⊕j=1ZDj goes to 0 in Cl(U) is obvious since Dj ∩ U = ∅. On the other hand, if the class of the Weil ∗ divisor D restricts to 0 in Cl(U), we have D|U = (f) for some f ∈ C(U) . However f can be regarded as a rational function on X; then D − (f) is supported on X − U, so that it

a linear combination of the Dj. 38 CHAPTER 2. DIVISORS AND LINE BUNDLES

We denote by DivT (XΣ) the subgroup of Div XΣ generated by the torus invariant prime P divisors. One has DivT (XΣ) = ρ∈Σ(1) ZDρ.

Theorem 2.24. There is an exact sequence

M → DivT (XΣ) → Cl(XΣ) → 0 where the ﬁrst map sends u ∈ M to (χu). The sequence is exact also on the left if and only 1 if NR is generated by the elements vρ, with ρ a ray.

n Proof. Since the irreducible components of the complement of T in XΣ are exactly the torus-invariant divisors Dρ, from Theorem 2.23 we have an exact sequence

n DivT (XΣ) → Cl(XΣ) → Cl(T ) → 0

n (as usual we regard T as the open aﬃne toric subvariety of XΣ corresponding to the cone {0}). Since Tn is the spectrum of the UFD C[M], its class group vanishes [?], hence DivT (XΣ) → Cl(XΣ) is surjective. Moreover, the composition M → DivT (XΣ) → Cl(XΣ) is obviously zero since elements in M are mapped to principal divisors.

Suppose now that D ∈ DivT (XΣ) maps to zero in Cl(XΣ), i.e., D = (f) for some n n f ∈ C(XΣ). The support of D does not meet T , hence (f) restricts to zero in T , and regarded as an element in C(Tn), the rational function f yields a zero divisor in Tn, so that we can regard f as an element in C[M]∗ (what we are saying is that on Tn, f is a morphism, not only a rational function). Hence, f = cχu for some c ∈ C∗ and u ∈ M. Thus D = (f) = (χu) which proves exactness in the middle. u P Concerning the second claim, if u ∈ M and (χ ) = ρ∈Σ(1) u(vρ) Dρ = 0, then u(vρ) for

all rays ρ; if the rays span NR, then u = 0, hence the full sequence is exact. On the other

hand, assume that the rays do not generate NR, i.e., we can eliminate one ray ρ without

changing the subspace of NR generated by the rays. Up to an automorphism of the lattice, we can ﬁnd u ∈ M such that u(ρ) = 1 and u is zero on the complement of ρ. Then u is

mapped to 0 in DivT (XΣ), so that M → DivT (XΣ) is not injective.

If we choose a basis in N, and represent torus-invariant divisors by the their coeﬃcients

on the divisors Dρ, the morphism M → DivT (XΣ) is represented by a matrix with integer

coeﬃcients, and the class group of XΣ is isomorphic to the cokernel of the associated linear map.

1 When this is true we say that XΣ has no torus factors; indeed, if the rays do not generate Σ, the n0 0 0 toric variety XΣ is of the type XΣ = XΣ × T for some n , with Σ a fan of lower dimension. 2.3. DIVISORS IN TORIC VARIETIES 39

2 Example 2.25. We know that the 2-dimensional cone generated in Z by v1 = de1 − e2 and v2 = e2 is associated for d ≥ 2 with a singular toric surface, a singular cone Cbd. 2 2 M → DivT (XΣ) becomes a map Z → Z given by the matrix A ! d −1 A = . 0 1

Then Cl(Cbd) ' Zd. 4

2.3.2 The Picard group of a toric variety

We start now a similar analysis of Cartier divisors in normal toric varieties. We know that any Cartier divisor D deﬁnes a Weil divisor, so, we a slight abuse of language, we can write P D = i aiDi for a Cartier divisor D. We denote by CDivT (XΣ) the subgroup of DivT (XΣ) consisting of Cartier divisor. Since by deﬁnition (χu) is Cartier for all u ∈ M, Theorem 2.24 has an immediate corollary.

Corollary 2.26. There is an exact sequence

M → CDivT (XΣ) → Pic(XΣ) → 0 (2.2) where the ﬁrst map sends u ∈ M to (χu). The sequence is exact also on the left if and only if NR is generated by the elements vρ, with ρ a ray.

This, together with the analogous result for the class group, allows one the write a commutative diagram

0 / M / CDivT (XΣ) / Pic(XΣ) / 0 _ _

0 / M / DivT (XΣ) / Cl(XΣ) / 0

We want to study the structure of the group CDivT (XΣ). We need the following (very plausible) Lemma, for a proof see [?], Lemma 1.1.16. Recall that Tn = Spec C[M], and therefore Tn has an action on Spec C[M].

Lemma 2.27. If A ⊂ C[M] is a Tn-invariant subspace,

M u A = C · χ . χu∈A 40 CHAPTER 2. DIVISORS AND LINE BUNDLES

Proposition 2.28. Let σ be a strongly convex polyhedral cone in NR.

n 1. Every T -invariant Cartier divisor in Uσ is the divisor of a character.

2. Pic(Uσ) = 0.

∨ Proof. Let Aσ = C[σ ∩ M], i.e., Uσ = Spec Aσ. Assume at ﬁrst that D is eﬀective. One has

Γ(Uσ, OUσ (−D)) = {f ∈ C(Uσ) | f = 0, or f 6= 0 and ( f) ≥ D}. n This is a T -invariant ideal I in Aσ, so that, be the previous Lemma,

M u M u I = C · χ = C · χ χu∈A ( χu)≥D By the orbit-cone correspondence we have \ O(σ) ⊂ Dρ. ρ∈Σ(1) After ﬁxing a point p ∈ O(σ), since every Cartier divisor is locally principal, there is a

neighborhood U of p in which D is principal, and we may assume that that U = Spec Ug, ∗ where g ∈ Aσ is such that g(p) 6= 0. Then D|U = (f)|U for some f ∈ C(Uσ) . Since D is eﬀective, f|U is a regular function, and as h in invertible on U, we have f ∈ Aσ. So X X X (f) = νDρ (f) Dρ + νE(f) E ≥ νDρ (f) Dρ = D.

ρ E6=Dρ ρ

P ui ∗ ui The inequality (f) ≥ D implies f ∈ I, so that f = i ai χ with ai ∈ C and (χ ) ≥ D. ui ui Restricting to U, we have (χ )|U ≥ (f)|U , so that χ /f is morphism U → C. Thus

P ui ui ai χ X χ 1 = i = a f i f i

so that (χui /f)(p) 6= 0 for at least one i; for this value of i, χui /f does not vanish in neighbourhood V of p, contained in U. This implies

ui (χ )|V = (f)|V = D|V .

ui Moreover, p ∈ V ∩ Dρ for all ρ, hence all Dρ intersect V , and therefore (χ ) = D. So the first claim is proved when D is effective. In general, let us note that since σ is ∨ u strongly convex, we can find u ∈ σ ∩ M such that u(vρ) > 1 for all ρ ∈ Σ(1). Then (χ ) 0 ku is a linear combination of the Dρ with positive coefficients, and D = D + (χ ) ≥ 0 for k large enough, hence D0 is the divisor of a character; but then the same is true for D. The second claim is equivalent to the first. 2.3. DIVISORS IN TORIC VARIETIES 41

Example 2.29. This example shows that the Picard group of a smooth toric variety may 2 have torsion. Consider the fan Σ formed by the rays in Z generated by de1 − e2, with d ≥ 2, e2, and the origin. The corresponding toric variety is Cbd minus the ﬁxed point of the torus action (which corresponds to the 2-dimensional cone in the fan of Cbd). Now note that all cones in Σ are smooth (indeed, the vector de1 − e2 can be completed to a basis of N by adding e1). So the toric variety Xσ is smooth. Now, Xσ and Cbd have the same rays, so that they have the same class group. So

Pic(XΣ) = Cl(XΣ) = Cl(Cbd) = Zd.

Note that Cbd is normal. 4

Proposition 2.30. If the fan Σ contains a cone of dimension n = dim N, then Pic(XΣ) is a free abelian group.

Proof. The exact sequence (2.2) shows that Pic(XΣ) is ﬁnitely generated, hence it is enough to show that is it torsion-free. Again by that sequence, that amounts to showing that if kD is the divisor of a character for some k > 0, then the same is true for D. So let P u D = ρ aρ Dρ and kD = (χ ) for u ∈ M. Let σ be a cone of dimension n. We have

X D|Uσ = aρ Dρ ρ∈σ(1)

w By Proposition 2.28, D|Uσ = (χ ) for some w ∈ M, hence aρ = w(vρ) for all ρ ∈ σ(1). On u the other hand, kD = (χ ) implies kaρ = u(vρ), so that

kw(vρ) = kaρ = u(vρ).

kw w Since dim σ = n, the vρ span NR, so that kw = u. Hence, kD = (χ ), and D = (χ ).

We can now compare Cartier and Weil divisor on a toric variety.

Proposition 2.31. The following conditions are equivalent.

1. Every Weil divisor in XΣ is Cartier.

2. The Picard and the class groups of XΣ are isomorphic.

3. XΣ is smooth. 42 CHAPTER 2. DIVISORS AND LINE BUNDLES

Proof. 1 and 2 are clearly equivalent, and 3 implies 2 by general theory. We only need to

show that 1 and 2 imply 3. Pick a cone σ. The morphism Cl(XΣ) → Cl(Uσ) is surjective, so that every divisor in Uσ is Cartier. On the other hand, Pic(Uσ) = 0, so that the morphism X M → DivT (Uσ) = Z Dρ ρ∈σ(1)

is surjective. We may write this as a morphism M → Zr, where r is the number of rays in σ, by letting

u 7→ (u(vρ1 ), . . . , u(vρr ). This is actually dual to the map

r r X φ: Z → N, Φ(a1, . . . , ar) = ai vρi . i=1

It is an easy algebraic fact that Φ∗ is surjective if and only if Φ is injective and Φ(Zr) is r primitive in N (i.e., N/Φ(Z ) is torsion-free), and also, if and only if (vρ1 , . . . , vρr ) can be ∗ extended to a basis of N. In our case, Φ is surjective, so that (vρ1 , . . . , vρr ) can be extended to a basis of N, i.e., σ is smooth. Since this is true for all cones, XΣ is smooth.

Proposition 2.32. The following conditions are equivalent.

1. Every Weil divisor in XΣ has a positive integer multiple which is Cartier (i.e., every Weil divisor is Q-Cartier).

2. Pic(XΣ) has ﬁnite index in Cl(XΣ).

3. The natural morphism Pic(XΣ → Cl(XΣ) induces an isomorphism Pic(XΣ) ⊗ Q ' Cl(XΣ) ⊗ Q. 4. Σ is simplicial, i.e., all its cones are simplexes.

Note that if Σ has a cone of dimension n, condition 2 is equivalent to rk Pic(XΣ) = r−n, where r is the number of rays in Σ.

2.3.3 Describing Cartier divisors

We say that a cone is Σ is maximal if it is not properly contained in a cone of higher dimension. P Proposition 2.33. Let D = ρ aρ Dρ be Weil divisor. The following conditions are equivalent. 2.4. DIVISORS VERSUS POLYTOPES 43

1. E is Cartier.

2. D restricted to Uσ is principal for all σ ∈ Σ.

3. For each σ ∈ Σ there is uσ ∈ M such that uσ(vρ) = −aρ for all ρ ∈ σ(1).

4. For each maximal cone σ in Σ, if any, there is uσ ∈ M such that vρ(uσ) = −aρ for all ρ ∈ σ(1).

Proof. The equivalence between 1, 2 and 3 follows from Proposition 2.28. 3 obviously implies 4; moreover, every cone is the face of a maximal cone, and if uσ does the job for a cone, it also does for all faces of σ (the rays of a cone are the rays of its faces), so 4 implies 3. P Corollary 2.34. If D is Cartier and D = ρ uσ(vρ) Dρ, then

⊥ 1. each uσ is unique modulo M(σ) = σ ∩ M;

2. if τ ≤ σ then uσ = uτ mod M(τ).

Proof. The following three conditions are equivalent

0 1. uσ(vρ) − uσ(vρ) = 0 for all ρ ∈ σ(1); 0 2. uσ(v) − uσ(v) = 0 for all v ∈ σ; 0 ⊥ 3. uσ − uσ ∈ σ ∩ M = M(σ).

Thus uσ is unique modulo M(σ). Since uσ works for all faces of σ, claim 2 follows.

In this sense, the set uσ, σ ∈ Σ speciﬁes the divisor D. These are called the Cartier data of D.

2.4 Divisors versus polytopes

If ∆ is a full-dimensional rational polytope, we can construct the associated the associated fan Σ ; the resulting toric variety will be denoted X∆. We can represent the polytope as in (2.1), with aF ∈ Z, and vF identiﬁed with the minimal generator of a ray ρF . The cones in Σ are indexed by the faces Φ of ∆, i.e.,

σΦ = Cone(uF | F contains Φ).

The fan Σ is complete (since ∆ is full-dimensional), the vertexes of ∆ correspond to the the maximal cones, and the facets to the rays. The latter in turn yield prime, torus-invariant

divisors DF . 44 CHAPTER 2. DIVISORS AND LINE BUNDLES

Should we try to reconstruct the polytope from the fan, we could get the fact normal uF , P but there is no way to get the number aF . To get them, we need to divisor D∆ = F aF DF . P Proposition 2.35. The divisor D∆ = F aF DF is Cartier.

Proof. A vertex u ∈ M of ∆ corresponds to a maximal cone σu, and a ray ρF is in σu if

and only if u ∈ F . But this implies u(vF ) = −aF . This happens for all rays in σu, so that D is Cartier by Proposition 2.33.

2.4.1 Global sections of sheaves associated to toric divisors

Proposition 2.36. Let D be a torus-invariant Weil divisor in XΣ.Then

M m Γ(XΣ, OXΣ ) = C · χ . (χm)+D≥0

Proof. Recall that sections of OXΣ can be regarded as rational functions on D. Since m D|Tn = 0, the condition (χ ) + D ≥ 0 gives (f)|Tn ≥ 0, so that f ∈ C[M]. Then n Γ(XΣ, OXΣ ) ⊂ C[M]. This subspace is T invariant, and then by Lemma 2.27

M m Γ(XΣ, OXΣ ) = C · χ . m χ ∈Γ(XΣ,OXΣ ) m m Now χ ∈ Γ(XΣ, OXΣ ) if and only if (χ )+D ≥ 0, so that we get the required formula.

We can produce another representation of this space of global sections by associating a P m polyhedron to a Cartier divisor. If we write D = ρ aρ Dρ, and u ∈ M, then (χ )+D ≥ 0 is equivalent to the condition

u(vρ) + aρ ≥ 0 for all ρ ∈ Σ(1).

We associate with D the polyhedron

PD = {u ∈ MR | u(vρ) + aρ ≥ 0 for all ρ ∈ Σ(1)}. As the intersection of a ﬁnite number of closed half spaces, this is a polyhedron, but need not be a polytope because it may not be bounded. By comparison with the previous Proposition we have

Proposition 2.37. Let D be a torus-invariant Cartier divisor in XΣ.Then

M m Γ(XΣ, OXΣ ) = C · χ . u∈PD∩M 2.4. DIVISORS VERSUS POLYTOPES 45

Given a polyhedron P ∈ MR we can deﬁne kP for k ∈ R+ as the polyhedron obtained by multiplying all points of P by k, and given another polyhedron Q, we deﬁne the Minkowski sum P + Q as the subset of MR given by the sums of all elements in P and Q. One easily checks that this is a polyhedron. Then the correspondence D 7→ PD has the following properties:

• PkD = kPD for all k ∈ N;

• P(χm)+D = PD − m;

• PD + PE ⊂ PD+E.

2.4.2 Base point free divisors in toric varieties P Let D = ρ aρ Dρ be a torus-invariant Cartier divisor. Proposition 2.38. Assume that the maximal cones in Σ have dimension n = dim N. Then the following conditions are equivalent

1. D is base point free, i.e., OXΣ is generated by global sections.

2. uσ ∈ PD for all σ ∈ Σ(n).

Proof. Assume D is globally generated and let σ ∈ Σ(n). Then the orbit of σ under Tn is a ﬁxed point x of the torus action, and by the orbit-cone correspondence, \ {x} = Dρ. ρσ(1) By point 3 of Proposition 2.4, there is a global section s such that x is not in the support of u (s). In view of Proposition 2.37 we can identify s with a character χ for some u ∈ PD ∩M. On the other hand, u X (s) = D + (χ ) = (aρ + vρ(u))Dρ. ρ So the point x, on the one hand is not in the support of (s), but on the other hand lies in

Dρ for all ρσ(1). The previous formula implies

aρ + vρ(u) = 0 for all ρ ∈ σ(1).

We consider this for uσ and use that σ is n-dimensional to conclude that uσ ∈ PD.

uσ To show the converse, pick up σ ∈ Σ(n). Since uσ ∈ PD, χ is a global section s whose zero divisor is uσ X (s) = D + (χ ) = (aρ + vρ(uσ))Dρ. ρ 46 CHAPTER 2. DIVISORS AND LINE BUNDLES

This shows that the support of (s) does not meet Uσ. Since the varieties Uσ cover XΣ, D is base point free.

Example 2.39. Show the example of OP2 (3), including the counting of sections. 4

2.4.3 Support functions

Let Σ be a fan in NR.

Deﬁnition 2.40. A support function for Σ is a function ϕ: |Σ| → R which is linear on each cone of Σ. A support function is integral if restricted to the rational points of Σ it takes integer values. 4

We denote by SF(Σ) the set of all support functions for Σ.

Theorem 2.41. 1. If D is a Cartier divisor with Cartier data uσ, σ ∈ Σ, the function

ϕD : |Σ| → R

v 7→ uσ(v) when v ∈ σ

is an integral support function. P 2. ϕD(vρ) = −aρ for all ρ ∈ Σ(1), so that D = − ρ∈Σ(1) ϕD(vρ) Dρ.

3. The association D 7→ ϕD deﬁnes an isomorphism CDivT (XΣ) ' SF(Σ).

⊥ Proof. By Proposition 2.33, each uσ is unique modulo σ ∩ M, and moreover, uσ = uσ0 0 ⊥ mod (σ ∩ σ ) ∩ M. This implies that ϕD is well deﬁned. φD is linear on each cone and integral just by deﬁnition. Point 2 is obvious. We need only to prove 3.

The map CDivT (XΣ) → SF(Σ) is a homomorphism because if D, E are toric Cartier divisor, one has ϕD+E = ϕD +ϕE. The map is injective because of 2. To prove surjectivity, let ϕ be an integral support function. For every cone σ, it deﬁnes an N-linear map σ∩N → Z, which extends to an N-linear map ϕσ : Nσ → Z. Since

HomZ(Nσ, Z) ' M/M(σ), there is v ∈ M such that ϕσ(v) = uσ(v) for v ∈ Σ. Then the divisor X D = − ϕ(vρ) Dρ ρ is a Cartier divisor that maps to ϕ. 2.4. DIVISORS VERSUS POLYTOPES 47

A full-dimensional rational polytope ∆ allows one to deﬁne a nice support function for the associated fan. In turn this support function corresponds to the divisor D∆ previously introduced.

Proposition 2.42. Let ∆ ⊂ MR be a full-dimensional rational polytope with associated fan Σ. Then, the function ϕ∆ : NR → R deﬁned as

ϕ∆(v) = min u(v) u∈∆

is an integral support function for Σ, and the corresponding Cartier divisor is D∆.

Proof. We give ∆ the canonical presentation as in (2.1) and deﬁne the associated Cartier P divisor D∆ = F ∈∆(n−1) aF DF . Theorem 2.41 shows that the corresponding support function evaluated on vF yields −aF .

We need to show that ϕ∆ is a support function for Σ. The maximal cones of Σ correspond to the vertices of ∆; ifu ¯ is a vertex, the corresponding maximal cone is

σu¯ = Cone(vF | u¯ ∈ F )

(i.e., the cone generated by the normals vF where F runs over all faces that containu ¯). Take X v = λF vF , u vertex of F with λF ≥ 0. If u ∈ ∆ then X X u(v) = λF u(vF ) ≥ − λF aF . (2.3) u vertex of F u vertex of F P Then ϕ∆(v) ≥ − u vertex of F λF aF . Equality occurs in (2.3) when u =u ¯, so that X ϕ∆(v) ≥ − λF aF =u ¯(v). u vertex of F

This expression deﬁnes a support function. Moreover, ifu ¯ is a vertex of F , then ϕ∆(vF ) =

u¯(vF ) = −aF as required.

Support functions happen to be convex, and this fact plays an important role.

Deﬁnition 2.43. Let S be a convex subset of NR. A function ϕ: S → R is convex if

ϕ(tv + (1 − t)w) ≥ tϕ(v) + (1 − t)ϕ(w) for all v, w ∈ S and t ∈ [0, 1]. 4 48 CHAPTER 2. DIVISORS AND LINE BUNDLES

One sees by inspection that support functions are convex. In relation to convexity, it is natural to consider fans such that dim |Σ| = n = dim M, and whose support |Σ| is convex. When these two conditions are satisﬁed we say that Σ has convex support of full dimension. These fans satisfy the characterisations [ Σ = cone(vρ | ρ ∈ Σ(1)) = . σ∈Σ(n) Note that these cones are not necessarily complete, however, it is still true for them that the maximal cones have dimension n = dim N. Deﬁnition 2.44. A cone σ ∈ Σ(n − 1) is a wall if it is the common face of a cone of dimension n. Note that if Σ is complete every τ ∈ Σ(n − 1) is a wall. 4 Lemma 2.45. Let Σ be a convex fan of full dimension, and let D be a Cartier divisor in

XΣ. The following conditions are equivalent:

1. the support function ϕD associated to D is convex;

2. ϕD(v) ≤ uρ(v) for all v ∈ |Σ| and σ ∈ Σ(n);

3. ϕD(v) = minσ∈Σ(n) uσ(v) for all v ∈ |Σ|; 0 0 4. for every wall τ = σ ∩ σ , there is v0 ∈ σ − σ such that ϕD(v0) ≤ uρ(v0).

The polyhedron of a Cartier divisor can be characterized in terms of the support function of the divisor.

Lemma 2.46. Let Σ be a fan, and D be a Cartier divisor in XΣ. Then

PD = {u ∈ MR | ϕD(v) ≤ u(v) for all v ∈ |Σ|}. Proposition 2.47. Let Σ be a convex fan of full dimension, and let D be a Cartier divisor in XΣ. The following conditions are equivalent:

1. D is base point free;

2. uσ ∈ PD for all σ ∈ Σ(n);

3. ϕD(v) = minσ∈Σ(n) uσ(v) for all v ∈ |Σ|;

4. the support function ϕD is convex. Proposition 2.48. If in addition Σ is complete, the conditions in the previous Proposition are also equivalent to each of the following conditions:

1. PD is the convex hull of the points uσ, for σ ∈ Σ(n);

2. {uσ | σ ∈ Σ(n)} is the set of vertexes of PD;

3. ϕD(v) = minσ∈Σ(n) uσ(v) for all v ∈ NR. 2.4. DIVISORS VERSUS POLYTOPES 49

2.4.4 Ample divisors in toric varieties

Support functions provide a simple characterization of ample divisors. Deﬁnition 2.49. A full dimensional rational polytope ∆ is very ample if for all vertices

ui of ∆ the semigroup N(∆ ∩ M − ui) is saturated. 4

We shall need the following fact (see [?], Corollary 2.2.19). Proposition 2.50. Let ∆ be a rational full-dimensional polytope of dimension n ≥ 2. Then k∆ is very ample for all k ≥ n − 1.

We consider the fan Σ associated to ∆, the corresponding toric variety XΣ, and the

Cartier divisor D∆ built from the facet presentation of ∆, as before. Then:

Theorem 2.51. 1. D = D∆ is ample and base point free;

2. if n ≥ 2, kD∆ is very ample for k ≥ n − 1;

3. D∆ is very ample if and only if ∆ is a very ample polytope.

Proof. Since ∆ coincides with the polyhedron of the divisor D, by Proposition 2.38 D is base point free, and moreover, the space W = Γ(XΣ, OXΣ ) is spanned by the characters u r−1 χ with u ∈ ∆ ∩ M = {u1, . . . , ur}. Then we can write the morphism φD : XΣ → P as

u1 ur φD(x) = [(χ (x), . . . , χ (x))].

If we go back to equation (1.5) and the discussion around it, we see that we can factor φD as ψD r−1 XΣ −−→ X∆∩M ,→ P . We want to show that the ﬁrst arrow is an isomorphism. Let I ⊂ {1, . . . , r} be the set of indices such that ui is a vertex of ∆; these also correspond to the maximal cones of the fan Σ.

ui Let si be the global section of OXΣ (D) corresponding to χ . The support of the ui associated divisor (si) = D + (χ ) consists exactly of the invariant divisors that do not intersect the aﬃne toric variety Ui = Uσi . Moreover, let Vi be the intersection of X∆∩M with the standard open subset of Pr−1 where the i-th homogeneous coordinate does not vanish. Since the Ui cover XΣ, and the Vi cover X∆∩M , after denoting ψi = ψD|Ui , it is enough to show that each

ψi : Ui → Vi is an isomorphism. Now, from Lemma 1.33

Vi ' Spec C[N(∆ ∩ M − ui)] 50 CHAPTER 2. DIVISORS AND LINE BUNDLES

∨ and since σi = Cone(∆ ∩ M − ui), one has an inclusion

∨ N(∆ ∩ M − ui) ⊂ σi ∩ M.

The only way the two semigroups may diﬀer is that the ﬁrst is not saturated. On the other ∨ hand, Ui = Spec C[σi ∩ M]. We have a chain of equivalences between the following facts:

a) D is very ample;

ψD b) XΣ −−→ X∆∩M is an isomorphism;

c) ψi : Ui → Vi is an isomorphism for all i ∈ I; ∨ d) C[N(∆ ∩ M − ui)] → C[σi ∩ M] is an isomorphism for all i ∈ I;

e) N(∆ ∩ M − ui) is saturated for all i ∈ I, i.e., ∆ is very ample.

This proves point 3. Point 2 follows from Proposition 2.50. Then D is ample, i.e., point 1 is proved as well.

Deﬁnition 2.52. Assume the fan Σ has full dimensional convex support. The support

function ϕD of a Cartier divisor D in XΣ is strictly convex if it is convex, and for all σ ∈ Σ(n) satisﬁes the condition

ϕD(v) = uσ(v) if and only if v ∈ σ.

Lemma 2.53. Let D be a Cartier divisor in a toric variety XΣ, where Σ has convex

support of full dimension. Then, the support function ϕD is strictly convex if and only if 0 0 it is convex, and uσ 6= uσ0 whenever σ and σ are in Σ(n) and σ ∩ σ is a wall.

Theorem 2.54. Let XΣ be complete, and let D be a Cartier divisor in XΣ.

1. D is ample if and only if ϕD is strictly convex. 2. If n ≥ 2 and D is ample, then kD is very ample for all k ≥ n − 1.

Proof. Point 1. We start by proving the “if” part, assuming at ﬁrst that D is very ample.

Since D is base point free, ϕD is convex. Assume that it is not strictly convex. By Lemma 0 2.53, there is wall τ = σ ∩ σ in Σ such that uσ = uσ0 . Let V (τ) be the corresponding close toric subvariety obtained from the orbit-cone correspondence, i.e., V (τ) = O(τ). Let ∆ be

polyhedron associated to D; it is a polytope since Σ is complete. Set ∆∩M = {u1, . . . , ur} r−1 and write φD : XΣ → P as

u1 ur φD(x) = [(χ (x), . . . , χ (x))]. 2.4. DIVISORS VERSUS POLYTOPES 51

0 0 Note there is a value of i0 ∈ {1, . . . , r} such that uσ = uσ = ui0 . We restrict φD to Uσ ∪Uσ .

We ﬁrst consider Uσ. Since D is base point free we have uσ ∈ ∆, and as we saw, the uσ section corresponding to χ does not vanish in Uσ; so we can express φD as

u1−uσ ur−uσ φD(x) = [(χ (x), . . . , χ (x))].

r−1 This takes values in the open set Ui0 of P where xi0 does not vanish. The same is true 0 0 0 for Uσ , so that φD maps Uσ ∪ Uσ to Ui0 . Now, σ and σ and the only n-dimensional cones containing τ, since the latter is a wall. So be the orbit-cone correspondence,

1 V (τ) ⊂ Uσ ∪ Uσ0 ' P .

1 P is complete and φD maps into into the aﬃne space Ui0 , hence φD maps V (τ) to a point, which is not possible since D is very ample. Hence ϕD is strictly convex.

If D is just ample, then kD is very ample for k big enough, so that ϕkD is strictly

convex, and ϕD is strictly convex as well.

Let us assume now that ϕD is strictly convex. In particular it is convex, and by

Proposition 2.48, the Cartier data {uσ}σ∈Σ(n) are the vertexes of ∆. So ∆ is a rational

polytope. We prove that it is full dimensional; if not, there is v 6= 0 in NR and k ∈ R such that uσ(v) = k for all σ ∈ Σ(n) (i.e., all vertexes lie on a hyperplane). By Proposition 2.48,

ϕD(v) = uσ(v) = k for all σ ∈ Σ(n). By strict convexity, v ∈ σ for all σ ∈ Σ(n); but then v = 0 since Σ is complete. This contradicts u 6= 0 and therefore ∆ is full dimensional.

Then ∆ produces a a toric variety X∆, with an ample divisor D∆, whose support function is

ϕD = min u(v) ∆ u∈∆

which on the other hand is the support function ϕD by Proposition 2.48, so that ϕD∆ is strictly convex both with respect to Σ and to the fan associated with ∆. Now, one

can check that the maximal cones of a fan are exactly the maximal subsets of NR where a strictly convex support function is linear. This tells that the an associated with ∆ is

Σ. Then, D∆ since the two divisors, now thought of in the same variety, have the same

support function. But we know that D∆ is ample, so that D is ample as well. The second claim follows from previous results.

Collecting some of the results in the last sections, one can prove the following important fact. 52 CHAPTER 2. DIVISORS AND LINE BUNDLES

Theorem 2.55. A Cartier divisor in a smooth complete variety is ample if and only it is very ample.

Finally, we apply the techniques in this section to show that, while not any complete toric variety is projective, it is always dominated by a projective toric variety. According to what we saw in Chapter 1, this will follow from the following fact.

0 Theorem 2.56. Any complete fan Σ has a reﬁnement Σ such that XΣ0 is projective.

Proof. (Sketch) Deﬁne Σ0 by letting [ Σ0 = Span(τ). τ∈Σ(n−1)

We can choose uτ ∈ M so that

Span(τ) = {v ∈ NR | uτ (v)} and deﬁne ϕ: NR → R by letting X ϕ(v) = − |uτ (v)|. τ∈Σ(n−1)

This function is convex by deﬁnition. One can show that it is linear on every cone (so that it is a support function) and that it is actually strictly convex. 0 P The divisor in XΣ0 deﬁned as D = − ρ0 ϕ(vρ0 ) Dρ0 is ample, hence XΣ0 is projective.

2.5 The nef and Mori cones in toric varieties

The use of support functions allows us to strengthen Proposition 2.13.

Proposition 2.57. Let D be a Cartier divisor in a toric variety XΣ, where Σ has convex support of full dimension. The following conditions are equivalent:

1. 2. D is base point free: 3. D is nef; 4. D · C ≥ 0 for all torus-invariant irreducible complete curves C in X. 2.5. THE NEF AND MORI CONES IN TORIC VARIETIES 53

Proof. We know that 1 implies 2 from Proposition 2.13, while 2 implies 3 by the deﬁnition

of nefness. We prove that 3 implies 1. Since the Picard group of XΣ is generated by torus-invariant divisors, we can assume that D is torus-invariant. Then it is enough to show that ϕD is convex.

Traccia: Teorema 6.3.12 Teorema 6.3.20 Corollary 6.3.21 Teorema 6.3.22 Proposizione 6.3.25 54 CHAPTER 2. DIVISORS AND LINE BUNDLES Chapter 3

Cohomology of coherent sheaves on toric varieties

3.1 Reﬂexive sheaves and Weil divisors

We recall that a coherent sheaf F on a scheme X is reﬂexive if the natural morphism F → F ∨∨ is an isomorphism. For the basics of the theory of reﬂexive sheaves see [?]. In the following an open set U is a scheme X is called big if codim(X − U) ≥ 2.

Proposition 3.1. Let X be a normal variety, F a coherent sheaf on X, and U a big subset of X. Denote by j : U,→ X the inclusion.

1. F ∨ is reﬂexive.

2. If F is reﬂexive then F' j∗(F|U ). ∨∨ 3. If F|U is locally free then F ' j∗(F|U ).

Proof. 1 and 2 are proved in [?]. We prove 3. Since taking dual is compatible with restriction to open subschemes, and using 1 and 2, we have

∨∨ ∨∨ ∨∨ F ' j∗((F )|U ) ' j∗((F|U ) ) ' j∗(F|U ).

Proposition 3.2. If X is a locally factorial variety and F has rank one, then F is reﬂexive if and only if it is an invertible sheaf. 56 CHAPTER 3. COHOMOLOGY OF COHERENT SHEAVES

Proof. Assume F is reﬂexive. Since X is normal and F is torsion-free, the latter is locally free on a big subset U. Note that Pic(X) ' Pic(U) since X is locally factorial. So there is

an invertible sheaf L on X that on U restricts to F|U . By Proposition 3.1 we have

∨∨ F'F ' j∗(F|U ) ' j∗(L|U ) 'L.

More generally, this argument shows that if two reﬂexive sheaves are isomorphic on a big subset, then they are isomorphic.

Theorem 3.3. Let X be a normal variety, and F a coherent sheaf on X. The following conditions are equivalent.

1. F is reﬂexive and has rank one; 2. F is the extension by zero of an invertible sheaf L on a big subset U of X;

3. F'OX (D) for some Weil divisor D.

Proof. If 1 holds, as X is normal its smooth locus U is a big subset of X, so that by

Proposition 3.2 L = F|U is invertible, hence by Proposition 3.1 F' j∗L, where j : U,→ X is the inclusion. So 2 holds. P If 2 holds, then L = OX (E) where E = i ai Ei is a Cartier divisor in U. Let Di be P the closure of Ei and D = i ai Di. By Proposition 3.1 we have

∨∨ F' j∗F'OX (D) .

Note now that if U is a big subset of X, and f is a rational function on X, then

(f) + D ≥ 0 if and only if ((f) + D)|U ≥ 0. (3.1)

since codim(X − U) ≥ 2, which implies that the restriction morphism OX (D) → OX (D)|U

is surjective (a section of OX (D)|U , regarded as rational function, always extends to a

section of OX (D)). The same is true by replacing X with any subset, so that OX (D) is normal. As it is torsion-free (it is the ideal sheaf of D), it is reﬂexive. Together with

equation (3.1) this implies F'OX (D), i.e., 3 holds.

If 3 holds, as seen in the previous step, OX (D) is reﬂexive, and obviously has rank 1.

If D and E are Weil divisor in a normal variety X, the multiplication of rational functions deﬁnes a morphism

OX (D) ⊗ OX (E) → OX (D + E) (3.2) 3.2. DIFFERENTIAL FORMS, CANONICAL SHEAF AND SERRE DUALITY 57

which may fail to be an isomorphism when neither divisor is Cartier. Things get better if we take double duals.

Proposition 3.4. The double dual of the morphism 3.2

∨∨ (OX (D) ⊗ OX (E)) → OX (D + E)

is an isomorphism.

Proof. Both sides are reﬂexive sheaves and they are locally free on the smooth locus of X, where they are isomorphic, so that they are isomorphic.

The morphism 3.2 in the case E = −D yields a morphism

∨ OX (−D) → OX (D) (3.3)

which turns out to be an isomorphism for the same reason as in the proof of the previous Proposition.

Proposition 3.5. Let D and E be Weil divisors in a normal variety X. Then OX (D) '

OX (E) if and only if D and E are linearly equivalent.

Proof. The “if” part is known by general theory. If OX (D) 'OX (E) we have

OX (D) ⊗ OX (−E) 'OX (E) ⊗ OX (−E).

Taking double duals we get OX (D −E) 'OX , so that D −E is lineary equivalent to 0.

3.2 Diﬀerential forms, canonical sheaf and Serre duality

3.2.1 Zariski forms

1 Let X be a normal variety. In general the cotangent sheaf ΩX is not locally free (it fails to be so at the singular points of X), and in particular the sheaf

n n 1 ΩX = ∧ ΩX may fail to be a line bundle. However we can at least define reflexive sheaves of “differential forms” by using the fact that the smooth locus U of X is a big subset. We set

ˆ p p ∨∨ p ΩX = (ΩX ) ' j∗(ΩX |U ) 58 CHAPTER 3. COHOMOLOGY OF COHERENT SHEAVES

p p 1 ˆ p where ΩX = ∧ ΩX . The sections of ΩX are called Zariski p-forms. ˆ n In particular the rank one reﬂexive sheaf ωX = ΩX is the canonical sheaf of X. By the previous theory, it is the sheaf associated with a Weil divisor; any such divisor is called as canonical divisor. In general a canonical divisor is not Cartier; when it is, X is said to be Gorenstein.

3.2.2 Euler sequences

Consider now the toric variety XΣ associated with a fan Σ. For every ray ρ denote by Oρ the structure sheaf of the Weil divisor Dρ regarded as a (torsion) sheaf on XΣ. We can form a sequence of morphisms

β 0 → Ω1 −→α M ⊗ O −→⊕ O → 0 (3.4) XΣ XΣ ρ ρ defined as follows. β is defined from the morphisms M → Z, u 7→ u(vρ), tonsuring by OXΣ , composing with the evaluation morphisn OXΣ → Oρ, and summing over ρ. The morphism β is defined as dχu 7→ u ⊗ χu in the affine patches Uσ and checking that these morphisms glue.

Theorem 3.6. The sequence (3.4) is exact when XΣ is smooth.

When XΣ is not smooth we can recover exactness by using Zariski forms. Note that the double dual of α deﬁnes a morphism Ωˆ 1 −→α M ⊗ O . XΣ XΣ

Theorem 3.7. Let XΣ be a normal toric variety. Then the sequence

0 → Ωˆ 1 → M ⊗ O → ⊕ O (3.5) XΣ XΣ ρ ρ is exact. If XΣ is simplicial, the sequence

0 → Ωˆ 1 → M ⊗ O → ⊕ O → 0 (3.6) XΣ XΣ ρ ρ is exact.

Theorem 3.8. If XΣ is a simplicial toric varieties with no torus factors there is an exact sequence 0 → Ωˆ 1 → ⊕ O (−D ) → Cl(X ) ⊗ O → 0. (3.7) XΣ ρ XΣ ρ Σ XΣ

If XΣ is smooth this can be written as

0 → Ω1 → ⊕ O (−D ) → Pic(X ) ⊗ O → 0. (3.8) XΣ ρ XΣ ρ Σ XΣ 3.3. COHOMOLOGY OF TORIC DIVISORS 59

Proof. 0 0 0

0 / Ωˆ 1 / M ⊗ O / ⊕ O / 0 XΣ XΣ ρ ρ

0 / ⊕ρOXΣ (−Dρ) / ⊕ρOXΣ / ⊕ρOρ / 0

0 / Cl(XΣ) ⊗ OXΣ / Cl(XΣ) ⊗ OXΣ / 0 / 0

0 0 0

3.2.3 Serre duality

We recall that a local ring A is said to be Cohen-Macauley (CM) if its depth equals its dimension. The depth is deﬁned as follows. A sequence {f1, . . . , fr} of elements of A is said to be regular if for every i, fi is not a zero divisor in A/(f1, . . . , fi−1). The depth of A is the supremum of the lengths of regular sequences. A scheme is CM if all its local rings are CM. Smooth Theorem 3.9. Let X be a complete normal Cohen-Macaulay variety of dimension n, and F a coherent sheaf of X. There are natural isomorphisms

i ∨ n−1 H (X, F) ' Ext (F, ωX ).

3.3 Cohomology of toric divisors

ˇ In computing the Cech cohomology of a torus-invariant sheaf on a toric variety XΣ there is an obvious candidate for the cover to choose, namely, that formed by the open toric aﬃne varieties associated with the maximal cones of the fan. So we consider the cover

U = {Uσ}σ∈Σmax . P We label these cones as σi, with some given, arbitrary ordering. Let D = ρ aρ Dρ be a Cartier divisor. We know that

M u Γ(Uσ, OXΣ (D)) = C · χ . u(vρ)≥−aρ ρ∈σ(1) 60 CHAPTER 3. COHOMOLOGY OF COHERENT SHEAVES

We can look at this as an M-grading for the group Γ(Uσ, OXΣ ), where the summand corresponding to a u ∈ is zero if the condition in the sum is not met. This gives a grading to the whole Cechˇ complex, and since the Cechˇ differential is compatible with the grading, the Cechˇ cohomology groups are graded as well. Example 3.10. Example 9.1.1 Cox-Little-Schenck. 4 P Definition 3.11. Let D = ρ aρ Dρ be a divisor in XΣ. For u ∈ M we define the subset |Σ|N R [ VD,u = Conv(vρ | ρ ∈ σ(1), u(vρ) < −aρ). σΣ If D is Q-Cartier, we also define1 supp VD,u = {v ∈ |Σ| | u(v) < ϕD(v)}. 4 P Theorem 3.12. Let D = ρ aρ Dρ be a Weil divisor in XΣ. For every u ∈M and k ≥ 0 one has2 k ˜ k−1 H (XΣ, OXΣ (D))u ' H (VD,u, C). If D is Q-Cartier, one also has k ˜ k−1 supp H (XΣ, OXΣ (D))u ' H (VD,u , C). Example 3.13. Example 9.1.4 Cox-Little-Schenck. 4 The computation is easy in the case of complete toric surfaces. In this case, arranging the minimal generators of the rays in a counterclockwise order around the origin, define the sign pattern signD of D as the string of length r (the number of rays) whose ith entry

is + if u(vi) ≥ −ai and − otherwise. Example 3.14. Examples 9.1.7 e 9.1.8 Cox-Little-Schenck. 4 Prop. 9.1.6 Cox-Little-Schenck. P Theorem 3.15. Let D = ρ aρ Dρ be a Weil divisor in complete toric surface XΣ. The k dimension of H (XΣ, OXΣ (D))u is ( 1 if sign (u) = + ··· +, i.e., V = ∅ H0(X , O (D)) = D D,u Σ XΣ u 0 otherwise

1 1 If D is a Q-Cartier divisor such that kD is Cartier, one deﬁnes ϕD = k ϕkD. It is easy to check that

D is Q-Cartier if and only if for every cone σ there is uσ ∈ MQ such that uσ(vρ) = −aρ for all ρ ∈ σ(1). When this is true one has ϕD(v) = uσ(v) for all v ∈ σ. 2For X a topological space, the group H˜ −1(Z, C) is deﬁned as 0 if X is empty, C otherwise. 3.3. COHOMOLOGY OF TORIC DIVISORS 61

1 H (XΣ, OXΣ (D))u = max(0,]connected components of VD,u) − 1

= max(0,]strings of consecutive −’s in signD(u)) − 1

( 1 if sign (u) = − · · · − i.e., V is a cycle H2(X , O (D)) = D D,u Σ XΣ u 0 otherwise 62 CHAPTER 3. COHOMOLOGY OF COHERENT SHEAVES Bibliography

[1] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison- Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.