Introduction to toric geometrypeople.sissa.it/~bruzzo/notes/toric.pdf · Introduction to toric geometry ... 1.1 Fans and toric varieties 1.1.1 Convex polyhedral cones ... At this

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

P2 = (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

dimN = 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,Z). Since M ⊗ R = N∗R, we may think of M as a subset of

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

σ =

s∑i=1

ri vi | ri ∈ R≥0

⊂ NR

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 N∗R defined as

σ∨ = u ∈ N∗R |u(v) ≥ 0 for all v ∈ σ.

4

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

H+u = v ∈ NR |u(v) ≥ 0.

Given a polyhedral cone σ lying in H+u , 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

K =

s∑i=1

ti ui | 0 ≤ ti ≤ 1

is a compact subset of σ∨, and since M is discrete, K ∩M is finite. If u =∑s

i=1 ri ui ∈ Sσ,

split ri = mi + ti with mi integers, and ti ∈ [0, 1). Then

u =s∑i=1

miui +s∑i=1

tiui

and since u and∑s

i=1miui are in M , also∑s

i=1 tiui is, hence∑s

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)


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 σ′ are cpc’s having a common face τ , there is

for any u ∈ Relint(σ∨ ∩ (−σ′∨)

τ = σ ∩Hu = σ′ ∩Hu.

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

Sτ = Sσ + Sσ′ .

Proof. Since τ lies in both σ and σ′, then Sτ contains both Sσ and Sσ′ , hence Sτ ⊃Sσ + Sσ′ . To show the opposite inclusion we note that by Lemma 1.8 we can take a

u ∈ σ∨ ∩ (−σ′)∨ ∩M such that τ = σ ∩ u⊥ = σ′ ∩ u⊥. By Proposition 1.6 we have

Sτ = Sσ + Z≥0 · (−u) ⊂ Sσ + Sσ′

since −u ∈ Sσ′ .

Example 1.10. Let N = Z2 and σ = 〈(1, 2), (2, 1)〉. Then σ∨ = 〈(−1, 2), (2,−1)〉, and Sσis 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


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 · χu′ = χu+u′ .

The identity 0 ∈ S corresponds to the unit in C[S], i.e., χ0 = 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 SpecA over C.

In our case, A will be finitely generated. The closed points of SpecA, corresponding to

prime ideals that are maximal, form a complex affine variety X = SpecmA, 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.


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

Xf = SpecAf ⊂ X = SpecA

is an open subset, called the principal open subset associated with f . If A = C[x1, . . . , xn],

so that SpecA = An, and f = xi, then SpecAf 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

SpecmA ' 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

SpecmC[S] ' Homsemigroup(S,C)

where C is considered as a semigroup under multiplication. Let us show how this corre-

spondence can be established. A point p ∈ X = SpecmC[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 semi-

group 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 = SpecC[Sσ] is the affine toric

variety associated to the rcpc σ. 4

SpecC[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

C∗ ' SpecC[x, y]

(1− xy)= SpecC[x, x−1].

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.

If S = sn, 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.


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 ±e∗1, . . . ,±e∗n.Define xi = χe

∗i , so that x−1

i = χ−e∗i . Then

C[Sσ] = C[M ] = C[x1, x−11 , . . . , xn, x

−1n ]

so that

SpecC[Sσ] ' C∗ × · · · × C∗ = Tn.

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

σ∨ =k∑i=1

R≥0 e∗i +

n∑i=k+1

R e∗i

so that

Sσ =k∑i=1

Z≥0 e∗i +

n∑i=k+1

Z e∗i .

Then

C[Sσ] = C[x1, . . . , xk, xk+1, x−1k+1, . . . , xn, x

−1n ]

and Uσ = Ak × Tn−k. 4

Note that in both examples the affine toric varieties are smooth.

Example 1.17. Let N = Z3, 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 e∗1, e∗2, e∗1 + e∗2, e∗2 + e∗3, so that

C[Sσ] = C[x1, x2, x1x2, x2x3] =C[x1, x2, x3, x4]

(x1x4 − x2x3).

The resulting variety is the hypersurface x1x4 = x2x3 in A4 (which is singular at the origin).

4

A morphism S → S ′ of semigroups yields a morphism of algebras C[S] → C[S ′], and

therefore also a morphism of varieties SpecC[S ′]→ SpecC[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.

In particular, the torus U0 = Tn 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 Uτis a principal open subset of Uσ.


Proof. By Proposition 1.18 there is u ∈ Sσ such that τ = σ∩u⊥, and Sτ = Sσ +Z≥0 · (−u).

Basis elements in C[Sτ ] have the form χw−pu, 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 ′ → N ,

and cones σ′ ⊂ N ′R, σ ⊂ NR such that φ(σ′) ⊂ σ. Then φ induces a morphism between the

semigroups associated with σ and σ′, and in turn a morphism of varieties φ : Uσ′ → 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.

4

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,

U0 ' 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

C∗, hence XΣ ' P1. 4

Example 1.21. Let N = Z2 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

U0 ' T2. The associated toric variety is C2 − 0 (see Figure). 4

Note that both varieties in the previous Examples are not affine.

3We recall that a scheme over a field k is of finite type if it has a finite affine cover Ui = SpecAi,where the Ai are finitely generated k-algebras.


Exercise 1.22. Let N = Z2 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

C[Sτi ] ' C[x, x−1, y] so that Uτi ' C× C∗

while Uσ ' C2 and U0 ' T2. Show that the toric variety XΣ is the projective plane P2.

Exercise 1.23. Let N = Z2 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

XΣ is the variety P1 × P1.

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 ×Xis 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[Sσ∩τ ] 'C[Sσ]⊗ C[Sτ ]

ker ∆∗,

i.e., ∆(Uσ∩τ ) is the locus cut in Uσ × Uτ by the ideal ker ∆∗.


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 R, i.e., which has the following property: every element in R 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 ∈ R | a is a root of a monic polynomial in R[x]

where R is the field of fractions of R. Thus R is integrally closed if R = R.5The 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

local ring A with maximal ideal m and residue field k = A/m is regular if dimk m/m2 = dimA.

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

a complex manifold, and the local rings of regular functions on affine space Ank .


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).

4

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.

Proof. If Σ = σ, the irreducible affine variety U0 ' Tn 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 = SpecA 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) =⋂p∈U

OX,p

(intersection in C(X)). Then ⋂p∈X

OX,p = OX(X) = A.

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

gk + ai gk−1 + . . . + ak

where ai ∈ OX,p. We can write ai = hi/fi with hi, fi ∈ A and fi(p) 6= 0. Then f =∏

i fiis 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.


Proposition 1.29. Let X be an affine variety X = SpecC[S], where S is a finitely gen-

erated semigroup S ⊂ M . Assume X contains a torus Tn. The following conditions are

equivalent.

1. X is normal.

2. S is saturated.

3. X = SpecC[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∨ = ∩iρ∨i .

Then Sσ = ∩iSρi , and

C[Sσ] =⋂i

C[Sρi ]

(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

C[Sρ] = C[x1, x2, x−12 , . . . , xn, x

−1n ].

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,

SpecC[x1, x2, x−12 , . . . , xn, x

−1n ] ' C× C∗ × · · · × C∗

is a smooth variety, hence it is normal, so that C[x1, x2, x−12 , . . . , xn, x

−1n ] is integrally closed.


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

Finally, we check that the torus Tn 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

G×G×Xm×id

id×ζ // G×Xζ

G×X ζ // X

X = ∗ ×X

id''

e×id // G×XζX

(1.2)

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

a morphism ζ∗ : AX → AG ⊗ AX satisfying the diagrams dual to the ones in (1.2):8

AG ⊗ AG ⊗ AX AG ⊗ AXid⊗ζ∗oo

AG ⊗ AX

m∗⊗id

OO

AXζ∗oo

ζ∗

OO AX AG ⊗ AXe∗⊗idoo

AX

ζ∗

OO

id

ee (1.3)

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

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

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).

8Here AX and AG denote the rings of regulars functions on X and G respectiverly, so that X = SpecAX ,

G = SpecAG. In the case at hand G = Tn so that AG = M .


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

u 7→

1 if u ∈ σ⊥

0 otherwise(1.4)

(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

the unique fixed point of the action of Tn 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.

9Let us give some detail about the equality u(s)u(t) = u(st). If v ⊗ x ∈ N ⊗Z C∗ = Tn, 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).

On the other hand, if ei is a basis of N , and v =∑i viei, w =

∑i wiei, u =

∑uie∗i, one has

st =∑i

ei ⊗ xvi ywi

and

u(st) =∏i

(xvi ywi)ui = xu(v) yu(w).

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

σ⊥ = v′ ∈ NR | v′ · v = 0 for all v ∈ σ.


If Z = u1, . . . , us is a finite subset of a semigroup S ⊂M , let SZ be the sub-semigroup

it generates, and let UZ = SpecC[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).

Consider the map ΦZ : Tn → Cs defined as

ΦZ(t) = (χu1(t), . . . , χus(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

of ΦZ can be projected to Ps−1. Let XZ be the (Zariski) closure of the image in Ps−1. 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.

Lemma 1.33. Let Ui be the open set in Ps−1 where the i-th homogeneous coordinate does

not vanish. Then

XZ ∩ Ui ' SpecC[N(Z − ui)].

Lemma 1.34. Let U = SpecC[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

ψ(u) =

1 if u = 0

0 if u 6= 0(1.6)

2. After writing X = UZ ⊂ Cs 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).

11Indeed if t =∑i vi ⊗ xi, then

χu(t) =∏i

xu(vi)i

which is never zero.


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

After identifying the n-dimensional torus Tn = 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 sub-

groups) can be identified with N ; thus any co-character of Tn is given by an element

v ∈ N . We shall denote by λv : C∗ → Tn this co-character. It satisfies12

λv(xy) = λv(x)λv(y).

If v ∈ N and u ∈M , the composition

C∗ λv−→ Tn χu−→ C

maps z to zu(v), so that

u(λv(z)) = χu(λv(z)) = zu(v)

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.

12Note indeed that λv(x) = v ⊗ x, and if we write v =∑i vi ei on a basis of N , we have

λv(x)λv(y) =

(∑i

ei ⊗ xvi)(∑

i

ei ⊗ yvi)

=∑i

ei ⊗ (xy)vi = λv(xy).


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;

2. limz→0 u(λv(z)) = limz→0 zu(v) 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 → Cis

ψv(u) = χu(xv) = limz→0

u(λv(z)) = limz→0

zu(v).

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).


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(σ) ⊗Z C∗ we obtain a surjective group homomorphim Tn → TN(σ) (here n = dimN as

usual), so that Tn 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

xσ ∈ Uσ ⊂ XΣ under the action of Tn. 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

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⊥∩Mis 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

O′ → Hom(σ⊥ ∩M,C∗).


On the other hand, it γ : σ⊥∩M → C∗ is an element in Hom(σ⊥∩M,C∗), we define γ ∈ O′

by letting

γ(u) =

γ(u) if u ∈ σ⊥ ∩M

0 otherwise

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

TN(σ) ' Hom(σ⊥ ∩M,C∗) ' O′

that are compatible with the Tn-action. Since this group acts transitively on TN(σ), it also

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-

tween cones in Σ and orbits of Tn in XΣ given by

σ ←→ O(σ) ' Hom(σ⊥ ∩M,C∗).

2. dimO(σ) = n− dimσ.

3.

Uσ =⋃τ≤σ

O(τ)

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

O(τ) =⋃τ≤σ

O(σ). (1.8)

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.


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

2. Follows from Lemma 1.39 since dimTN(σ) = n− dimσ.

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) ⊂O(τ) necessarily). We can do this using one-parameter subgroups of Tn. 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

ψz(u) = χu(λv(z))ψτ(u) = zu(v) 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 = limz→0

ψz

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 Uσ′ ; then

O(τ) ∩ Uσ′ =⋃

τ≤σ′≤σ

O(σ).

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

I = < χu |u ∈ τ⊥ ∩ (σ′)∨ ∩M > ⊂ C[(σ′)∨ ∩M ] = Sσ′ .

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 Uσis nonsingular, in particular it is nonsingular at the distinguished point xσ. Let m be the

maximal ideal of Aσ = C[σ∨∩M ] corresponding to that point. Then m/m2 is the cotangent

space at xσ, so that dimC m/m2 = n. Now, m is generated by χu 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

Uσ ' Cn.

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 = σ′ ⊕ 0.

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

Sσ = (σ′∨ ∩Mσ)⊕M(σ)

and

Uσ ' Uσ′ × Tn−k.

The meaning of the symbols Uσ and Uσ′ 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 Uσ′ is nonsingular as well, so that the

previous analysis applies; σ′ is generated by a basis of Nσ, and Uσ′ ' Ck.

So we have proved:

Proposition 1.41. An affine toric variety Uσ is smooth if and and only if σ is generated

by k elements of a basis of N , and then Uσ ' Ck × Tn−k.


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

1.17. Let N ′ be the lattice generated by e2 and e′1 = me1− e2, and let σ′ be σ regarded as

a cone in N (note that N ′R = NR). Now σ′ is generated by the generators of N ′, so that

Uσ′ = C2. Moreover we have inclusions N ′ ⊂ N and M ⊂ M ′. Actually M ′ is generated

by 1me∗1 and e∗2, while Sσ′ is generated by 1

me∗1 and 1

me∗1 + e∗2. We associate variables

1

me∗1 s, e∗2 y,

1

me∗1 + e∗2 t,

so that sy = t.

Now, the inclusion N → N yields a map σ′ → σ, defined by a map Aσ → Aσ′ . One

has Aσ′ = C[s, t], while to identify Aσ we note that Sσ is generated by

e∗1, e∗1 + e∗2, e

∗1 + 2e∗2, . . . , e

∗1 +me∗2

so that Aσ = C[x, xy, . . . , xym]. By setting x = sm, y = t/s we obtain

Aσ = C[sm, sm−1t, . . . , tm].

Now note that µm ' Zm, the group of m-th roots of unity, acts on C2 in the standard

way ζ(s, t) = (ζs, ζt). The ring Aσ may be described as C[s, t]µm , the ring of µm-invariant

polynomials in 2 variables, and therefore Uσ is the quotient C2/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,

with 0 < k < m. In this case µm acts on C2 by ζ(s, t) = (ζs, ζkt). 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 = dimN , 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

a quotient Cn/G, where G is the finite abelian group G = N/N ′, Uσ′ = Cn, and G has an

action on Uσ′ obtained as before. Note that Aσ can be obtained by intersecting Aσ′ with

C[M ′]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.


1.2.2 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 = SpecBi of Y and for every

i, an affine open cover Uij = SpecAij of f−1(Vi), where each Aij is a finitely generated

Bi-algebra.

3. f is universally closed if for every morphism Y ′ → Y , the corresponding morphism

f ′ : X ′ → Y ′ is closed, where X ′ is the fiber product X ×Y Y ′:

X ′

f ′ // Y ′

X

f // 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 ′ → N

be a morphism of lattice that maps a fan Σ′ to a fan Σ, and denote as usual with the same

symbol the corresponding morphism φ : XΣ′ → XΣ.

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

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.


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

SpecK

// X

f

SpecR //

;;

Y

(1.9)

there is a morphism SpecR→ 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

SpecK

// X

f

SpecR //

;;

Y

Remark 1.48. If X is irreducible, one can assume that SpecK 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 = XΣ′ , Y = XΣ, f = φ. In view of

the previous remark, we may assume that Y is affine, Y = Uσ, and assume that SpecK

maps into an open subset U ⊂ XΣ which is isomorphic to the n-dimensional torus Tn, with

n = dimN ′. The morphism SpecK → U corresponds to a homomorphism of algebras

C[M ′] → K given by a group homorphism α : M ′ → K∗. We need to find a cone σ′ ⊂ Σ′

such that the diagram

K C[M ′]αoo C[Sσ′ ]? _oo

uuR?

OO

C[Sσ]oo

φ∗

OO

φ

∗::

commutes. Now, SpecR maps to Uσ, so that the composition

C[Sσ]φ∗−→ C[M ′]

α−→ Kord−−→ 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 ′/g′) = (ff ′/gg′).

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.


has nonnegative values. By hypothesis, there is a cone σ′ in N ′ such that φ(σ′) ⊂ σ, so

that the composition (1.2.2) can be written as

C[Sσ]→ C[Sσ′ ]→ C[M ′]α−→ K

ord−−→ Z

which implies that we can complete the diagram with the arrow C[Sσ′ ]→ 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, . . . , vnof 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

Uσ, so we can assume that Σ consists only of σ and its faces, so that XΣ = Uσ ' Cn. We

may also assume N = Zn and vi = ei. The new toric variety XΣ′ is covered by affine open

toric varieties Uσi , where σi is the cone generated by e0, e1, . . . , ei, . . . , en. Then σ∨i is

generated by e∗i , e∗1 − e∗i , . . . , e∗n − e∗i , and the coordinate rings of the varieties Uσi are

Aσi = C[xi, x1x−1i , . . . , xnx

−1i ].

Thus Uσi ' Cn. Note that xj/tj = xi/ti when ti, tj 6= 0.

On the other hand, the blowup Cn of Cn at the origin is the subvariety of Cn × Pn−1

defined by the equations xitj = xjti, with x1, . . . , xn affine coordinates in Cn), and

t1, . . . , tn homogeneous co-ordinates in Pn−1. The sets Ui ⊂ Cn given by ti 6= 0 are

copies of Cn, and glue with the transition functions xj = xi(ti/tj) with i 6= j. Thus

XΣ′ ' Cn are obtained by glueing the same open sets with the same transition functions,

hence they are isomorphic.

Actually Σ′ is obtained by subdividing σ into subcones, so that φ−1(Σ) = Σ′, and the

morphism Xσ′ → 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

Fm, called the m-th Hirzebruch surface. The map Fm → P2 given by the subdivision of Σ

is a blowup at a fixed point of the torus action. The inspection of the fan Σ′ provides a

lot of information about Fm (for instance, that it is a P1 fibration over P1. 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


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

singular toric surface. Let σ be the cone in Z2 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

ray spanned by e1, the cones in the new fan Σ′ are all smooth. We have a proper birational

morphism π : XΣ′ → 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.

This procedure can be iterated. If σ is the cone in Z2 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.

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 : XΣ′ → 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

(after identifying N with Z2) 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

generated by e1 and e2 is smooth, isomorphic to C2, while in general the variety associated

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

)

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

k= a1 −

1

a2 −1

· · ·+ 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

that each E2i = −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:

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

the natural morphism XΣ′ → 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


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

be regarded as a rational function f ∈ C(X) by recalling that L is locally generated by f−1i

30 CHAPTER 2. DIVISORS AND LINE BUNDLES

and writing s|Ui = si ·f−1i 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 .

This coincides with the usual pullback of sheaves of modules, f ∗L = f−1L⊗f−1OX 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 ANDNEF 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 ofOU -modules). We define a morphism

φU : U → CN+1 by letting

φU(p) = (ψU s0(p), . . . , ψU sN(p)).

Since two overlapping trivialisations differ by multiplication by a nonzero regular function,

this defines a morphism φ : X → PN , which actually coincides with φW by identifying W

with CN+1 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

morphism φL : X → PΓ(X,L)∨ is a closed embedding. L is said to be ample if Lm 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

φ∗LOPV (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.

Definition 2.8. Let C be a smooth complete irreducible curve, and D =∑

i aipi a divisor

in C. The degree of D is the integer number degD =∑

i ai. 4


We have the following properties.

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

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

As a consequence, degD = degE 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 degL = degD. We have the following properties.

• degL = degL′ if L ' L′.

• deg(L⊗ L′) = degL+ degL′.

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 = SpecA as X = Spec A, where A is

the integral closure of A. The inclusion of A into A′ 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).

4

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 ANDNEF 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 D′ be the (base point free) divisor (s); then

L ' OC(D′). Thus,

D · C = degL = 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

D · C =1

m(mD) · C

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 C ′ are numerically equivalent if C − C ′ ≡ 0. 4


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

• The real vector space N1(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 N1(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.

4

The intersection product defines a pairing N1(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 N1(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 ∆ = dimV (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 ∈ ∆.

4

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 dimF + dimF = 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 ∆:

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:


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

Pn.

• If ∆ is the cube in R3 with vertices in the points ±e∗1 ± e∗2 ± e∗3, 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 ∆ = dimMR. In this case every facet

F of ∆ has a unique supporting affine hyperplane. Let us define

HF = u ∈MR |u(vF ) = −aF and H+F = 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

∆ =⋂F

H+F = u ∈MR |u(vF ) ≥ −aF for all facets F < ∆. (2.1)

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

orbit under the torus action is a Tn-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

νρ(χu) = u(vρ).

Proof. Extend vρ to a basis ei of N , with e1 = vρ. Thus N ' Zn and ρ is the cone

generated by e1. The corresponding toric affine variety is

Uρ = Spec[x1, x±2 , . . . , x

±n ] ' C× (C∗)n−1,

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[x±1 , x2, . . . , xn].

Then a rational function f ∈ C(x1, . . . , xn)∗ has valuation ` if

f = x`1g

hwith g, h ∈ C[x2, . . . , xn].

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

χu =∏i

xu(ei)i .

So νρ(χu) = u(e1) = u(vρ).

This essentially also proves:

Proposition 2.22. Given u ∈ M , the character χu is a rational function on XΣ, whose

corresponding divisor is

(χu) =∑ρ∈Σ(1)

u(vρ)Dρ.

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⊕j=1

ZDj → Cl(X)→ Cl(U)→ 0

is exact.

Proof. We already hinted that Cl(XΣ) → Cl(U) is surjective. That the image ⊕sj=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.


We denote by DivT (XΣ) the subgroup of DivXΣ generated by the torus invariant prime

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

if NR is generated by the elements vρ, with ρ a ray.1

Proof. Since the irreducible components of the complement of Tn in XΣ are exactly the

torus-invariant divisors Dρ, from Theorem 2.23 we have an exact sequence

DivT (XΣ)→ Cl(XΣ)→ Cl(Tn)→ 0

(as usual we regard Tn 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

f ∈ C(XΣ). The support of D does not meet Tn, hence (f) restricts to zero in Tn, 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.

Concerning the second claim, if u ∈M and (χu) =∑

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.

1When this is true we say that XΣ has no torus factors; indeed, if the rays do not generate Σ, the

toric variety XΣ is of the type XΣ = XΣ × Tn′for some n′, with Σ′ a fan of lower dimension.


Example 2.25. We know that the 2-dimensional cone generated in Z2 by v1 = de1 − e2

and v2 = e2 is associated for d ≥ 2 with a singular toric surface, a singular cone Cd.

M → DivT (XΣ) becomes a map Z2 → Z2 given by the matrix A

A =

(d −1

0 1

).

Then Cl(Cd) ' 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

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 = SpecC[M ], and

therefore Tn has an action on SpecC[M ].

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

A =⊕χu∈A

C · χu.


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

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

2. Pic(Uσ) = 0.

Proof. Let Aσ = C[σ∨ ∩M ], i.e., Uσ = SpecAσ. 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.

This is a Tn-invariant ideal I in Aσ, so that, be the previous Lemma,

I =⊕χu∈A

C · χu =⊕

(χu)≥D

C · χu

By the orbit-cone correspondence we have

O(σ) ⊂⋂

ρ∈Σ(1)

Dρ.

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 = SpecUg,

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

(f) =∑ρ

νDρ(f)Dρ +∑E 6=Dρ

νE(f)E ≥∑ρ

νDρ(f)Dρ = D.

The inequality (f) ≥ D implies f ∈ I, so that f =∑

i ai χui with ai ∈ C∗ and (χui) ≥ D.

Restricting to U , we have (χui)|U ≥ (f)|U , so that χui/f is morphism U → C. Thus

1 =

∑i ai χ

ui

f=∑i

aiχui

f

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

(χui)|V = (f)|V = D|V .

Moreover, p ∈ V ∩Dρ for all ρ, hence all Dρ intersect V , and therefore (χui) = D.

So the first claim is proved when D is effective. In general, let us note that since σ is

strongly convex, we can find u ∈ σ∨ ∩M such that u(vρ) > 1 for all ρ ∈ Σ(1). Then (χu)

is a linear combination of the Dρ with positive coefficients, and D′ = D + (χku) ≥ 0 for k

large enough, hence D′ is the divisor of a character; but then the same is true for D.

The second claim is equivalent to the first.


Example 2.29. This example shows that the Picard group of a smooth toric variety may

have torsion. Consider the fan Σ formed by the rays in Z2 generated by de1 − e2, with

d ≥ 2, e2, and the origin. The corresponding toric variety is Cd minus the fixed point of

the torus action (which corresponds to the 2-dimensional cone in the fan of Cd). 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 Cd have the same rays,

so that they have the same class group. So

Pic(XΣ) = Cl(XΣ) = Cl(Cd) = Zd.

Note that Cd is normal. 4

Proposition 2.30. If the fan Σ contains a cone of dimension n = dimN , 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

D =∑

ρ aρDρ and kD = (χu) for u ∈M .

Let σ be a cone of dimension n. We have

D|Uσ =∑ρ∈σ(1)

aρDρ

By Proposition 2.28, D|Uσ = (χw) for some w ∈M , hence aρ = w(vρ) for all ρ ∈ σ(1). On

the other hand, kD = (χu) implies kaρ = u(vρ), so that

kw(vρ) = kaρ = u(vρ).

Since dimσ = n, the vρ span NR, so that kw = u. Hence, kD = (χkw), and D = (χw).

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.


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

M → DivT (Uσ) =∑ρ∈σ(1)

ZDρ

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

φ : Zr → N, Φ(a1, . . . , ar) =r∑i=1

ai vρi .

It is an easy algebraic fact that Φ∗ is surjective if and only if Φ is injective and Φ(Zr) is

primitive in N (i.e., N/Φ(Zr) 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.

Proposition 2.33. Let D =∑

ρ aρDρ 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.

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

1. u′σ(vρ)− uσ(vρ) = 0 for all ρ ∈ σ(1);

2. u′σ(v)− uσ(v) = 0 for all v ∈ σ;

3. u′σ − 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 .


Should we try to reconstruct the polytope from the fan, we could get the fact normal uF ,

but there is no way to get the number aF . To get them, we need to divisor D∆ =∑

F aF DF .

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

Γ(XΣ,OXΣ) =

⊕(χm)+D≥0

C · χm.

Proof. Recall that sections of OXΣcan be regarded as rational functions on D. Since

D|Tn = 0, the condition (χm) + D ≥ 0 gives (f)|Tn ≥ 0, so that f ∈ C[M ]. Then

Γ(XΣ,OXΣ) ⊂ C[M ]. This subspace is Tn invariant, and then by Lemma 2.27

Γ(XΣ,OXΣ) =

⊕χm∈Γ(XΣ,OXΣ

)

C · χm.

Now χm ∈ Γ(XΣ,OXΣ) if and only if (χm)+D ≥ 0, so that we get the required formula.

We can produce another representation of this space of global sections by associating a

polyhedron to a Cartier divisor. If we write D =∑

ρ aρDρ, and u ∈M , then (χm)+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

Γ(XΣ,OXΣ) =

⊕u∈PD∩M

C · χm.


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

Let D =∑

ρ aρDρ be a torus-invariant Cartier divisor.

Proposition 2.38. Assume that the maximal cones in Σ have dimension n = dimN .

Then the following conditions are equivalent

1. D is base point free, i.e., OXΣis 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 =⋂ρσ(1)

Dρ.

By point 3 of Proposition 2.4, there is a global section s such that x is not in the support of

(s). In view of Proposition 2.37 we can identify s with a character χu for some u ∈ PD∩M .

On the other hand,

(s) = D + (χu) =∑ρ

(aρ + vρ(u))Dρ.

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.

To show the converse, pick up σ ∈ Σ(n). Since uσ ∈ PD, χuσ is a global section s whose

zero divisor is

(s) = D + (χuσ) =∑ρ

(aρ + vρ(uσ))Dρ.


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| → Rv 7→ uσ(v) when v ∈ σ

is an integral support function.

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 = uσ′

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

D = −∑ρ

ϕ(vρ)Dρ

is a Cartier divisor that maps to ϕ.


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) = minu∈∆

u(v)

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

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 corre-

spond to the vertices of ∆; if u 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 contain u).

Take

v =∑

u vertex of F

λF vF ,

with λF ≥ 0. If u ∈ ∆ then

u(v) =∑

u vertex of F

λF u(vF ) ≥ −∑

u vertex of F

λF aF . (2.3)

Then ϕ∆(v) ≥ −∑

u vertex of F λF aF . Equality occurs in (2.3) when u = u, so that

ϕ∆(v) ≥ −∑

u vertex of F

λF aF = u(v).

This expression defines a support function. Moreover, if u 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


One sees by inspection that support functions are convex.

In relation to convexity, it is natural to consider fans such that dim |Σ| = n = dimM ,

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 = dimN .

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|;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 func-

tion 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.


2.4.4 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

χu with u ∈ ∆ ∩M = u1, . . . , ur. Then we can write the morphism φD : XΣ → Pr−1 as

φD(x) = [(χu1(x), . . . , χur(x))].

If we go back to equation (1.5) and the discussion around it, we see that we can factor φDas

XΣψD−−→ X∆∩M → Pr−1.

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.

Let si be the global section of OXΣ(D) corresponding to χui . The support of the

associated divisor (si) = D + (χui) 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 ' SpecC[N(∆ ∩M − ui)]


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

N(∆ ∩M − ui) ⊂ σ∨i ∩M.

The only way the two semigroups may differ is that the first is not saturated. On the other

hand, Ui = SpecC[σ∨i ∩M ]. We have a chain of equivalences between the following facts:

a) D is very ample;

b) XΣψD−−→ X∆∩M is an isomorphism;

c) ψi : Ui → Vi is an isomorphism for all i ∈ I;

d) C[N(∆ ∩M − ui)]→ C[σ∨i ∩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.

4

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

it is convex, and uσ 6= uσ′ 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

2.53, there is wall τ = σ ∩ σ′ in Σ such that uσ = uσ′ . 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, . . . , urand write φD : XΣ → Pr−1 as

φD(x) = [(χu1(x), . . . , χur(x))].


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

section corresponding to χuσ does not vanish in Uσ; so we can express φD as

φD(x) = [(χu1−uσ(x), . . . , χur−uσ(x))].

This takes values in the open set Ui0 of Pr−1 where xi0 does not vanish. The same is true

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,

V (τ) ⊂ Uσ ∪ Uσ′ ' P1.

P1 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 uσσ∈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∈∆u(v)

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.


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.

Theorem 2.56. Any complete fan Σ has a refinement Σ′ such that XΣ′ is projective.

Proof. (Sketch) Define Σ′ by letting

Σ′ =⋃

τ∈Σ(n−1)

Span(τ).

We can choose uτ ∈M so that

Span(τ) = v ∈ NR |uτ (v)

and define ϕ : NR → R by letting

ϕ(v) = −∑

τ∈Σ(n−1)

|uτ (v)|.

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.

The divisor in XΣ′ defined as D′ = −∑

ρ′ ϕ(vρ′)Dρ′ is ample, hence XΣ′ 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


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.

If 2 holds, then L = OX(E) where E =∑

i aiEi is a Cartier divisor in U . Let Di be

the closure of Ei and D =∑

i aiDi. 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)|Uis 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 du-

ality

3.2.1 Zariski forms

Let X be a normal variety. In general the cotangent sheaf Ω1X is not locally free (it fails

to be so at the singular points of X), and in particular the sheaf

ΩnX = ∧nΩ1

X

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

ΩpX = (Ωp

X)∨∨ ' j∗(ΩpX |U)


where ΩpX = ∧pΩ1

X . The sections of ΩpX are called Zariski p-forms.

In particular the rank one reflexive sheaf ωX = ΩnX 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 Oρthe structure sheaf of the Weil divisor Dρ regarded as a (torsion) sheaf on XΣ. We can

form a sequence of morphisms

0→ Ω1XΣ

α−→M ⊗OXΣ

β−→ ⊕ρOρ → 0 (3.4)

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 Ω1XΣ

α−→M ⊗OXΣ.

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

0→ Ω1XΣ→M ⊗OXΣ

→ ⊕ρOρ (3.5)

is exact. If XΣ is simplicial, the sequence

0→ Ω1XΣ→M ⊗OXΣ

→ ⊕ρOρ → 0 (3.6)

is exact.

Theorem 3.8. If XΣ is a simplicial toric varieties with no torus factors there is an exact

sequence

0→ Ω1XΣ→ ⊕ρOXΣ

(−Dρ)→ Cl(XΣ)⊗OXΣ→ 0. (3.7)

If XΣ is smooth this can be written as

0→ Ω1XΣ→ ⊕ρOXΣ

(−Dρ)→ Pic(XΣ)⊗OXΣ→ 0. (3.8)

3.3. COHOMOLOGY OF TORIC DIVISORS 59

Proof.

0

0

0

0 // Ω1

XΣ//

M ⊗OXΣ//

⊕ρOρ //

0

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

H i(X,F)∨ ' Extn−1(F , ωX).

3.3 Cohomology of toric divisors

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 = Uσσ∈Σmax .

We label these cones as σi, with some given, arbitrary ordering. Let D =∑

ρ aρDρ be a

Cartier divisor. We know that

Γ(Uσ,OXΣ(D)) =

⊕u(vρ)≥−aρρ∈σ(1)

C · χu.


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 Cech complex, and since the Cech differential is compatible with the grading,

the Cech cohomology groups are graded as well.

Example 3.10. Example 9.1.1 Cox-Little-Schenck. 4

Definition 3.11. Let D =∑

ρ aρDρ be a divisor in XΣ. For u ∈ M we define the subset

|Σ|NR

VD,u =⋃σΣ

Conv(vρ | ρ ∈ σ(1), u(vρ) < −aρ).

If D is Q-Cartier, we also define1

VsuppD,u = v ∈ |Σ| | u(v) < ϕD(v).

4

Theorem 3.12. Let D =∑

ρ aρDρ be a Weil divisor in XΣ. For every u ∈M and k ≥ 0

one has2

Hk(XΣ,OXΣ(D))u ' Hk−1(VD,u,C).

If D is Q-Cartier, one also has

Hk(XΣ,OXΣ(D))u ' Hk−1(V

suppD,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.

Theorem 3.15. Let D =∑

ρ aρDρ be a Weil divisor in complete toric surface XΣ. The

dimension of Hk(XΣ,OXΣ(D))u is

H0(XΣ,OXΣ(D))u =

1 if signD(u) = + · · ·+, i.e., VD,u = ∅0 otherwise

1If D is a Q-Cartier divisor such that kD is Cartier, one defines ϕD = 1kϕkD. 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

H1(XΣ,OXΣ(D))u = max(0, ]connected components of VD,u)− 1

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

H2(XΣ,OXΣ(D))u =

1 if signD(u) = − · · · − i.e., VD,u is a cycle

0 otherwise


Bibliography

[1] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-

Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969.

63

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