Introduction to toric geometry

Transcription

1 Introduction to toric geometry Ugo Bruzzo Scuola Internazionale Superiore di Studi Avanzati and Istituto Nazionale di Fisica Nucleare Trieste

2 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.

3 Contents 1 Toric varieties Fans and toric varieties Convex polyhedral cones Affine toric varieties Constructing toric varieties from fans Torus action Limiting points The orbit-cone correspondence More properties of toric varieties Singularities Completeness Resolution of singularities Divisors and line bundles Base-point free, ample and nef line bundles on normal varieties Base point free line bundles and divisors Ample and numerically effective divisors Nef and Mori cones Polytopes Convex polytopes Canonical presentations Divisors in toric varieties The class group of a toric variety The Picard group of a toric variety v

4 vi CONTENTS Describing Cartier divisors Divisors versus polytopes Global sections of sheaves associated to toric divisors Base point free divisors in toric varieties Support functions Ample divisors in toric varieties The nef and Mori cones in toric varieties Cohomology of coherent sheaves Reflexive sheaves and Weil divisors Differential forms, canonical sheaf and Serre duality Zariski forms Euler sequences Serre duality Cohomology of toric divisors References 63

5 Chapter 1 Toric varieties Let T n be the n-dimensional algebraic torus T n = 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 P 2. Let [z 0, z 1, z 2 ] be homogeneous coordinates for it, and consider the open subset obtained by removing the three lines {z 0 = 0}, {z 1 = 0}, {z 2 = 0}. This set can be thought of as the affine 2-plane A 2 with two lines missing, so it is C C. 1 To get P 2 back, one must add three projective lines with two points removed, e.g., three copies of C, and three points (p 1, p 2, p 3 ). So one can write P 2 = (C C ) (C C C ) (p 1, p 2, p 3 ) Thus P 2 can be stratified in smaller and smaller tori. This will be a general feature of toric varieties. 1.1 Fans and toric varieties 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. 1 Note that A 2 minus two lines is isomorphic to the cartesian product of two copies of A 1 minus a point.

6 2 CHAPTER 1. TORIC VARIETIES We recall that a lattice is a free, finitely generated abelian group equipped with a nondegenerate quadratic Z-valued form. The dimension of a lattice N is dim N = dim R (N Z R). The standard example is Z n with the restriction of the euclidean product. We also introduce the dual lattice M = Hom(N, Z). Since M R = NR, we may think of M as a subset of NR. Let us fix a lattice N. Definition 1.1. A convex polyhedral cone (cpc for short) is a subset of N R = N Z R of the form { s σ = i=1 r i v i r i R 0 } N R where v 1,..., v s are fixed vectors in N R, called the generators of σ. The dimension of σ is the dimension of the vector space generated by v 1,..., v s. σ is said to be rational if it is generated by vectors in N. A 1-dimensional cpc is called a ray. 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}. Any nonzero u N R defines a normal hyperplane H u, and a closed half-space H + u = {v N R u(v) 0}. Given a polyhedral cone σ lying in H + u, we say that H u 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.

7 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 S σ = σ M. The latter has a natural semigroup structure. Proposition 1.5 (Gordon s lemma). If σ is a rational cpc, then S σ is a finitely generated semigroup. Proof. Since σ is rational, we can choose generators u 1,..., u s M for σ. The set { s } K = i=1 t i u i 0 t i 1 is a compact subset of σ, and since M is discrete, K M is finite. If u = s i=1 r i u i S σ, split r i = m i + t i with m i integers, and t i [0, 1). Then u = s m i u i + i=1 s t i u i i=1 and since u and s i=1 m iu i are in M, also s i=1 t iu i is, hence s i=1 t iu i M K. This says that S σ is generated by M K (note indeed that also u i M K). An immediate consequence is that the semigroup algebra C[S σ ] (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 S σ. Proposition 1.6. σ u = τ is an rcpc, and all faces of σ are of this type. Moreover, S τ = S σ + Z 0 ( u). (1.1)

8 4 CHAPTER 1. TORIC VARIETIES Proof. Only equation (1.1) needs a proof. Figure 1.1 shows this result in the example were N = Z 2 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 N R containing σ. The relative interior Relint(σ) of σ is the interior of σ in its span. Note that if σ generates N R, 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 H u = σ H u. Corollary 1.9. If σ and σ are rcpc that intersect along a common face τ, then S τ = S σ + S σ. Proof. Since τ lies in both σ and σ, then S τ contains both S σ and S σ, hence S τ S σ + S σ. 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 since u S σ. S τ = S σ + Z 0 ( u) S σ + S σ Example Let N = Z 2 and σ = (1, 2), (2, 1). Then σ = ( 1, 2), (2, 1), and S σ is the set of integral points of σ. If u = (2, 1), then τ = σ u is the face generated by (1, 2). Moreover, S τ is the right half plane delimited by the line generated by u, and S τ = S σ + Z 0 ( u). Definition A cpc σ is strongly convex if σ ( σ) = {0}.

9 1.1. FANS AND TORIC VARIETIES 5 Proposition 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 NR. Exercise Let σ N R be a rcpc. 1. Prove that if σ is strongly convex, then S σ is saturated in M, i.e., if u M and pu S σ for some positive integer p, then u in is S σ. 2. If σ is strongly convex, then σ spans M R = M R. 3. If σ is strongly convex, S σ generates M as a group, that is, M = S σ + ( S σ ). 4. Conversely, any finitely generated sub-semigroup of M which is saturated and generates M is of form S σ for a unique strongly convex rational polyhedral cone σ 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 χ u of a basis of C[S]; multiplication is given by the addition in S: χ u χ u = χ u+u. The identity 0 S corresponds to the unit in C[S], i.e., χ 0 = 1. If S has generators {u i }, then {χ u i } are generators of C[S] as a C-algebra. We shall associate affine (toric) varieties to the semigroup algebras C[S σ ], where σ is an rcpc. We recall that a commutative C-algebra determines a scheme Spec A over C. In our case, A will be finitely generated. The closed points of Spec A, corresponding to prime ideals that are maximal, form a complex affine variety X = Specm A, which can be regarded as the zero locus in A n of a set of polynomials in the following way: given generators (x 1,..., x n ) of A, then A = C[x 1,..., x n ]/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.

10 6 CHAPTER 1. TORIC VARIETIES If f A is nonzero, one can localize A at f, 2 and then X f = Spec A f X = Spec A is an open subset, called the principal open subset associated with f. If A = C[x 1,..., x n ], so that Spec A = A n, and f = x i, then Spec A f is the open subset x i 0. Since maximal ideals in a commutative C-algebra correspond to nontrivial homomorphisms to C, one has an isomorphism Specm A Hom C-alg (A, C). In the toric case there is also another identification. If A = C[S] for some semigroup in the dual lattice M, one also has the identification Specm C[S] Hom semigroup (S, C) where C is considered as a semigroup under multiplication. Let us show how this correspondence can be established. A point p X = Specm C[S] defines a map S C by sending u S to χ u (p) (remember that χ u is a regular function on X). This is a semigroup morphism. To establish the opposite correspondence, let ψ : S C be a semigroup morphism. By sending χ u 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 Let σ be an rcpc. The affine scheme U σ = Spec C[S σ ] is the affine toric variety associated to the rcpc σ. Spec C[S σ ] 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 Spec C[x, y] (1 xy) = Spec C[x, x 1 ]. 2 We recall that, given a commutative ring R with unit and subset S which is closed under multiplication (and is such that 0 S and 1 S), the localization of R with respect to S, denoted S 1 R, 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 1 R, a a/1. If S = {s n, n N} for some element s A, then the localization is denoted A s. Intuitively this is the ring obtained by inverting the powers of s.

11 1.1. FANS AND TORIC VARIETIES 7 In the next two examples {e 1,..., e n } will be a basis for N. Example Let σ = {0}; then S σ = M. A semigroup basis for it is given by {±e 1,..., ±e n}. Define x i = χ e i, so that x 1 i = χ e i. Then so that C[S σ ] = C[M] = C[x 1, x 1 1,..., x n, x 1 n ] Spec C[S σ ] C C = T n. Thus the trivial rcpc σ = {0} corresponds to the algebraic n-torus T n. Example Fix a number k with 1 k n 1 and consider the rcpc generated by {e 1,..., e k }. Then k n σ = R 0 e i + R e i so that Then and U σ = A k T n k. S σ = i=1 k Z 0 e i + i=1 i=k+1 n i=k+1 Z e i. C[S σ ] = C[x 1,..., x k, x k+1, x 1 k+1,..., x n, x 1 n ] Note that in both examples the affine toric varieties are smooth. Example Let N = Z 3, and let σ be generated by v 1,..., v 4 N with the relation v 1 + v 3 = v 2 + v 4. Without loss of generality, we can assume v 1 = (1, 0, 0), v 2 = (0, 1, 0), v 3 = (0, 0, 1), v 4 = (1, 1, 1). Now S σ is generated by e 1, e 2, e 1 + e 2, e 2 + e 3, so that C[S σ ] = C[x 1, x 2, x 1 x 2, x 2 x 3 ] = C[x 1, x 2, x 3, x 4 ] (x 1 x 4 x 2 x 3 ). The resulting variety is the hypersurface x 1 x 4 = x 2 x 3 in A 4 (which is singular at the origin). A morphism S S of semigroups yields a morphism of algebras C[S] C[S ], and therefore also a morphism of varieties Spec C[S ] Spec C[S]. If τ is a subcone of a cpc σ, then σ τ, so S σ is a sub-semigroup of S τ, and there is morphism of varieties U τ U σ. In particular, the torus U {0} = T n maps to all the affine toric varieties U σ, whichever the cone σ in N R is. Proposition If τ is a face of a strongly convex rational polyhedral cone σ, then U τ is a principal open subset of U σ.

12 8 CHAPTER 1. TORIC VARIETIES Proof. By Proposition 1.18 there is u S σ such that τ = σ u, and S τ = S σ +Z 0 ( u). Basis elements in C[S τ ] have the form χ w pu, with w S σ and p Z 0, so that C[S τ ] = C[S σ ] χ u, i.e., U τ is a principal open subset of U σ. In a slightly greater generality, we may consider a morphism of lattices φ: N N, and cones σ N R, σ N R 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: U σ U σ Constructing toric varieties from fans Definition A fan Σ in N R is a finite collection of strongly convex rational polyhedral cones such that 1. all faces of cones in Σ are in Σ; 2. cones in Σ only intersect along faces. A fan, being a collection of cones, provides a collection of affine toric varieties, to be thought of as an affine open cover of a toric variety X Σ 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 Let N = Z and σ = R 0. Then Σ = {σ, σ, {0}} is a fan. From Example 1.16 we see that S σ = Z 0 e 1 and S σ = Z 0 ( e 1 ) so that U σ U σ C. Moreover, U {0} C, and by Proposition 1.18, this embeds into U σ and U σ as a principal open subset. Thus X σ is formed by two copies of C glued along a common subset isomorphic to C, hence X Σ P 1. Example Let N = Z 2 and let σ 1, σ 2 be the cones generated by (1, 0) and (0, 1), respectively. Then Σ = {σ 1, σ 2, {0}} is a fan. We have U σ1 C C, U σ2 C C, and U {0} T 2. The associated toric variety is C 2 {0} (see Figure). Note that both varieties in the previous Examples are not affine. 3 We recall that a scheme over a field k is of finite type if it has a finite affine cover {U i = Spec A i }, where the A i are finitely generated k-algebras.

13 1.1. FANS AND TORIC VARIETIES 9 Exercise Let N = Z 2 and let τ i, i = 1, 2, 3 be the rays generated by (1, 0), ( 1, 1), (0, 1), respectively, and let σ i be the closed sectors of the plane delimited by any two of the τ s. Then C[S τi ] C[x, x 1, y] so that U τi C C while U σ C 2 and U {0} T 2. Show that the toric variety X Σ is the projective plane P 2. Exercise Let N = Z 2 and let τ i, i = 1, 2, 3, 4 be the rays generated by the four unit versors of the axes, respectively, and let σ i 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 P 1 given in Example Show that X Σ is the variety P 1 P 1. Note that the fans in Example 1.20 and Exercises 1.22 and 1.23 generate N R, 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 U σ τ is a principal open subset (hence an open subvariety) of both U σ and U τ ; these two varieties are then glued along U σ τ. It is not difficult to check that this operation is consistent (a cocycle condition is satisfied on triple intersections, if any). A result in point set topology is that a topological space X if Hausdorff if and only if the diagonal in X X is a closed subset. This fact motivates the definition of the analogous property for schemes: a scheme X is separated if the diagonal morphism : X X X is a closed immersion (i.e., its image is closed, and the morphism of sheaves of algebras : O X X O X is surjective). The following result implies that a toric variety X Σ is separated. Lemma If σ and τ are rcpc s that intersect along a face, the diagonal map : U σ τ U σ U τ is a closed embedding. Proof. By Corollary 1.9, S σ τ = S σ + S τ ; then the generators of C[S σ ] tensored with the generators of C[S τ ] generate C[S σ τ ], namely, the natural map : C[S σ ] C[S τ ] C[S σ τ ] is surjective. This implies the claim; note indeed that C[S σ τ ] C[S σ] C[S τ ] ker, i.e., (U σ τ ) is the locus cut in U σ U τ by the ideal ker.

14 10 CHAPTER 1. TORIC VARIETIES X Σ is separated as a consequence of the following fact. Exercise Let a variety X be covered by affine sets {U α }, and denote by αβ : U α U β U α U β 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 {U α } along their intersections, the image of αβ is isomorphic to the diagonal in U α U α ). 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 X Σ. 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 ˆR, i.e., which has the following property: every element in ˆR 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 C n 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. 4 We recall that given an integral domain R, its integral closure is the ring R = {a ˆR a is a root of a monic polynomial in R[x]} where ˆR is the field of fractions of R. Thus R is integrally closed if R = R. 5 The formal definition is that a scheme X is smooth in codimension 1 if every local ring O X,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 p 0 p n = p. A local ring A with maximal ideal m and residue field k = A/m is regular if dim k m/m 2 = dim A. 6 We 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 = x 1 x s, where each x i is irreducible, in a unique way up to reordering and multiplication by units. Example of unique factorization domains are the polynomial rings k[x 1,..., x n ], the rings of germs of holomorphic functions at a point of a complex manifold, and the local rings of regular functions on affine space A n k.

15 1.1. FANS AND TORIC VARIETIES 11 Exercise Prove that following statements. 1. If A is an integrally closed domain, and S A a multiplicative subset, the localization A S is A is an integrally closed domain. 2. If {A i } are integrally closed domains whose field of fractions are all isomorphic, then i A i is an integrally closed domain (intersection inside the field of fractions). We note that a toric variety X Σ is irreducible for a very simple reason. Proposition If Σ is fan, the toric variety X Σ is irreducible. Proof. If Σ = {σ}, the irreducible affine variety U {0} T n is dense in every U σ, so it is also dense in X Σ = σ Σ U σ, and therefore X Σ is irreducible as well. Lemma An irreducible affine variety X = Spec A is normal if and only if the ring A is an integrally closed domain. Proof. Let C(X) be the ring of rational functions on X. A rational function f in C(X) is contained in the local ring O X,p, with p X, exactly when f is regular at p. So if U X is open, O X (U) = p U O X,p (intersection in C(X)). Then O X,p = O X (X) = A. p X Now, if X is normal, all local rings O X,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 g k + a i g k a k where a i O X,p. We can write a i = h i /f i with h i, f i A and f i (p) 0. Then f = i f i is such that f(p) 0. By Exercise 1.26 the localization A f is an integrally closed domain, so that a i A f. Moreover, A f O X,p as f(p) 0. Since a i A f we have g A f, then g O X,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.

16 12 CHAPTER 1. TORIC VARIETIES Proposition Let X be an affine variety X = Spec C[S], where S is a finitely generated semigroup S M. Assume X contains a torus T n. The following conditions are equivalent. 1. X is normal. 2. S is saturated. 3. X = Spec C[S σ ] for a strongly convex rational polyhedral cone σ. Proof 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 χ m is a polynomial function on T n and therefore a rational function on X. Also, χ km C[S] since km S. Thus χ m is a root of the monic polynomial x k χ km with coefficients in C[S]. Since X is normal C[S] is an integrally closed domain, so that χ m C[S] and m S, i.e., S is saturated Let K be a finite set of generators for S; these generate a rational polyhedral cone in M R containing S. Let σ N R be its dual. Then S σ M, with equality if S is saturated. So S = S σ If σ N R is a strongly convex rational polyhedral cone we need to show that C[S σ ] is an integrally closed domain. Now σ is generated by its rays ρ i, and σ = i ρ i. Then S σ = i S ρi, and C[S σ ] = i C[S ρi ] (intersection inside C[M]). By Exercise 1.26 it is enough to prove that C[S ρ ] is normal when ρ is a rational ray in N R. Let v be the (primitive) generator of the ray. We can complete v = e 1 to a basis {e 1,..., e n } of N, and C[S ρ ] = C[x 1, x 2, x 1 2,..., x n, x 1 n ]. This is a localization of C[x 1,..., x n ] (at the element x 2 x n ), which is an integrally closed domain, so it is an integrally closed domain as well. 7 Corollary Toric varieties are normal. 7 Put in a different way, Spec C[x 1, x 2, x2 1,..., x n, x 1 n ] C C C is a smooth variety, hence it is normal, so that C[x 1, x 2, x 1 2,..., x n, x 1 n ] is integrally closed.

17 1.1. FANS AND TORIC VARIETIES 13 Proof. Normality is a local issue so we can assume that the variety is an affine toric variety U σ for some strongly convex polyhedral cone σ. Then the claim follows from Proposition 1.29 (in particular from the implication 3 1) Torus action Finally, we check that the torus T n acts on the variety X Σ (in particular, it acts on every affine toric variety U σ ), 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 X id ζ G X X = { } X e id G X (1.2) m id G X ζ X ζ id ζ X where m: G G G is the multiplication morphism and e: { } G maps the point to the identity of G. The arrows in these diagrams are morphisms of varieties. If X is affine, it is enough to specify the morphism ζ at the level of the rings of functions, by assigning a morphism ζ : A X A G A X satisfying the diagrams dual to the ones in (1.2): 8 A G A G A X m id A G A X id ζ e A G A X A id X A G A X (1.3) ζ ζ A X id ζ A X The action T n U σ U σ we want to define is then given by the morphism φ: C[S σ ] C[M] C[S σ ], χ u χ u χ u. To check that this extends the action of T n 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: T n T n T n, namely, m C[M] C[M] C[M], m (f)(s, t) = f(st). 8 Here A X and A G denote the rings of regulars functions on X and G respectiverly, so that X = Spec A X, G = Spec A G. In the case at hand G = T n so that A G = M.

18 14 CHAPTER 1. TORIC VARIETIES Note that by identifying C[M] with the coordinate ring of T n = N C we have χ u (t) = u(t), and as required. 9 (χ u χ u )(s, t) = χ u (s) χ u (t) = u(s) u(t) = u(st) = χ u (st) 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 x σ be the point in U σ given (in view of the correspondence between (closed) points in U σ and semigroup morphisms S σ C) by the morphism 10 u { 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 If a strongly convex rational polyhedral cone σ spans N R, then x σ is the unique fixed point of the action of T n on U σ. If σ does not span N R, the action has no fixed points. To prove this result we need some preliminary material. Definition A semigroup S M is pointed if S ( S) = {0}. So a semigroup is pointed if and only if its only invertible element is 0. 9 Let us give some detail about the equality u(s) u(t) = u(st). If v x N Z C = T n, and u M, then u(v x) = x u(v). Therefore, if s = v x and t = w y, u(s) u(t) = x u(v) y u(w). On the other hand, if {e i } is a basis of N, and v = i v ie i, w = i w ie i, u = u i e i, one has st = i e i x vi y wi and u(st) = i (x vi y wi ) ui = x u(v) y u(w). 10 The symbol σ denotes the subset of N R given by σ = {v N R v v = 0 for all v σ}.

19 1.1. FANS AND TORIC VARIETIES 15 If Z = {u 1,..., u s } is a finite subset of a semigroup S M, let S Z be the sub-semigroup it generates, and let U Z = Spec C[S Z ] be the corresponding affine variety. This variety can be embedded into the projective space P s 1 as follows (so that it is quasi-projective). Consider the map Φ Z : T n C s defined as Φ Z (t) = (χ u 1 (t),..., χ us (t)) (1.5) where χ u : T n C is the character of the torus defined by u M, χ u (t) = t(u) identifying T n = Hom(M, C ). Note that the value of this map can never be zero, 11 so that the image of Φ Z can be projected to P s 1. Let X Z be the (Zariski) closure of the image in P s 1. It is a projective variety by definition, so that U Z, which can be identified with a dense subset of X Z, is quasi-projective. For future use we note the following fact. Lemma Let U i be the open set in P s 1 where the i-th homogeneous coordinate does not vanish. Then X Z U i Spec C[N(Z u i )]. Lemma Let U = Spec C[S] be an affine toric variety, equipped with the action of T n 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 0 (1.6) 2. After writing X = U Z C s for Z S {0}, the torus action has a fixed point if and only if 0 U Z, and then the fixed point is 0. Proof. If a point p U is represented by a semigroup morphism ψ : S C, we have χ u (p) = ψ(u) for u S. On other hand, denoting by tp the action of t T n on p, we have χ u (tp) = χ u (t) χ u (p). If p is fixed, we have χ u (t) χ u (p) = χ u (p) which translates into χ u (t) ψ(u) = ψ(u). 11 Indeed if t = i v i x i, then which is never zero. χ u (t) = i x u(vi) i

20 16 CHAPTER 1. TORIC VARIETIES This condition is satisfied by u = 0, while if u 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 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 Let σ be a strongly convex rational polyhedral cone. semigroup S σ is pointed if and only if σ generates N R. The corresponding Proof. If σ generates N R, the dual cone σ is a strongly convex polyhedral cone which generates M R. Then S σ must be pointed: if S σ ( S σ ) had any point different from 0, σ would not be strongly convex. On the other hand, if σ does not generate N R, then σ contains a vector subspace of M R which intersects M nontrivially, so that S σ is not pointed Limiting points After identifying the n-dimensional torus T n = N Z C as Hom(M, C ), and since Hom(C, C ) Z, the group Hom(C, T n ) of co-characters of T n (one-parameter subgroups) can be identified with N; thus any co-character of T n is given by an element v N. We shall denote by λ v : C T n this co-character. It satisfies 12 If v N and u M, the composition maps z to z u(v), so that if z C and u M. λ v (xy) = λ v (x)λ v (y). C λ v T n χu C u(λ v (z)) = χ u (λ v (z)) = z u(v) Given a fan Σ, we denote by Σ its support, i.e., the union in N R of its cones. 12 Note indeed that λ v (x) = v x, and if we write v = i v i e i on a basis of N, we have ( ) ( λ v (x) λ v (y) = e i x vi i i e i y vi ) = i e i (xy) vi = λ v (xy).

21 1.1. FANS AND TORIC VARIETIES 17 Proposition If v N, and σ is a strongly convex polyhedral cone in N R, then v σ if and only if lim z 0 λ v (z) exists in U σ. 2. If v is in the relative interior of σ, then lim z 0 λ v (z) = x σ. Proof. 1. We note that the following conditions are equivalent: 1. lim z 0 λ v (z) exists in U σ ; 2. lim z 0 u(λ v (z)) = lim z 0 z u(v) exists in C for all u S σ ; 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 lim z 0 λ v (z) exists as a point in U σ. Denote that point by x v. The corresponding semigroup homomorphism ψ v : S σ C is ψ v (u) = χ u (x v ) = lim u(λ v (z)) = lim z u(v). z 0 z 0 Now, since v Relint(σ), one has u(v) > 0 if u S σ \ σ, and u(v) = 0 if u S σ σ, so that ψ v (u) = 0 in the first case, and ψ v (u) = 1 in the second. By comparing with (1.4) we see that x v is the distinguished point x σ. Given a fan Σ, we denote by Σ its support, i.e., the union in N R of its cones. The previous result shows that if v is not in Σ, then then lim z 0 λ v (z) does not exist in X Σ. 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 T n 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 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 Example pag. 115 of Cox-Little-Schenck. Given a cone σ in a lattice N, let N σ be the sublattice generated by σ, i.e., N σ = (σ N) ( σ N).

22 18 CHAPTER 1. TORIC VARIETIES Since σ is saturated, so is N σ, so that the quotient N(σ) = N/N σ is torsion-free, and is therefore a lattice. The exact sequence 0 N σ N N(σ) 0 (1.7) splits, as N(σ) is free over Z. Tensoring the exact sequence (1.7) by C and letting T N(σ) = N(σ) Z C we obtain a surjective group homomorphim T n T N(σ) (here n = dim N as usual), so that T n acts transitively on T N(σ). Lemma 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 T N(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 x σ U σ X Σ under the action of T n. Next lemma shows that all orbits O(σ) are algebraic tori. Lemma Let σ be a strongly convex rational polyhedral cone in N. Then O(σ) = {ψ : S σ C ψ(u) 0 iff u σ M} Hom(σ M, C ) T N(σ). Proof. We define O = {ψ : S σ C ψ(u) 0 iff u σ M}. This space is invariant under the action of T n t ψ : u χ u (t) ψ(u) and contains ψ σ, i.e., the semigroup homomorphism corresponding to the distinguished point x σ. Note now that σ is the largest vector subspace of M R contained in σ, so that σ M is a subgroup of S σ = 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 ).

23 1.1. FANS AND TORIC VARIETIES 19 On the other hand, it ˆγ : σ M C is an element in Hom(σ M, C ), we define γ O by letting { ˆγ(u) if u σ M γ(u) = 0 otherwise so that O Hom(σ M, C ). So we obtain bijections T N(σ) Hom(σ M, C ) O that are compatible with the T n -action. Since this group acts transitively on T N(σ), it also acts transitively on O. But O contains ψ σ, 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 between cones in Σ and orbits of T n in X Σ given by σ O(σ) Hom(σ M, C ). 2. dim O(σ) = n dim σ. 3. U σ = 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 U σ are torus-invariant, there is a cone σ Σ such that O U σ. We may assume this cone to be minimal, and then it is unique (indeed if σ 1 and σ 2 are two such cones, then O U σ1 U σ2 = U σ1 σ 2 so that by minimality σ 1 = σ 2 ). Let ψ O and assume that u S σ satisfies ψ(u) 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 S σ ψ(u) 0} = σ τ M. This implies ψ U τ and then τ = σ since σ is minimal. Then {u S σ ψ(u) 0} = σ M.

24 20 CHAPTER 1. TORIC VARIETIES In view of Lemma 1.39 we have ψ O(σ) and then O = O(σ). 2. Follows from Lemma 1.39 since dim T N(σ) = n dim σ. 3. We need to put together three things: any U σ is a union of orbits; if τ is a face of σ, then O(τ) U τ U σ ; any orbit contained in U σ 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 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) U σ, indeed if O(τ) U σ = then O(τ) U σ = since U σ is open (also in the analytic topology), and this is impossible since O(τ) contains O(σ). By part 3, O(τ) U σ 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) (then O(σ) O(τ) necessarily). We can do this using one-parameter subgroups of T n. Let ψ τ be the semigroup homomorphism corresponding to the distinguished point of U τ, and let v Relint(σ). For every z C we define a semigroup homomorphism ψ z = λ v (z) ψ τ, whose action is ψ z (u) = χ u (λ v (z)) ψτ(u) = z u(v) ψ τ (u). Since O(τ) is the orbit of ψ τ, ψ z 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 ψ 0 = lim z 0 ψ z exists in U σ, and is a point in O(σ), so that O(τ) O(σ). 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 U σ ; then O(τ) U σ = τ σ σ O(σ). One can show that this is the an affine subvariety of U σ given by the ideal I = < χ u u τ (σ ) M > C[(σ ) M] = S σ. Hence O(τ) is Zariski closed in X Σ, and therefore it is also the Zariski closure of O(τ).

25 1.2. MORE PROPERTIES OF TORIC VARIETIES More properties of toric varieties 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 C n, 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 C k T n 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 N R, so that σ = {0}. If U σ is nonsingular, in particular it is nonsingular at the distinguished point x σ. Let m be the maximal ideal of A σ = C[σ M] corresponding to that point. Then m/m 2 is the cotangent space at x σ, so that dim C m/m 2 = n. Now, m is generated by {χ u } for all nonzero u S σ. Then m 2 is generated by the elements u that are sum of two other elements. So m/m 2 is generated by the images of the elements χ u 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 U σ C n. Now let σ have a non-maximal dimension k, and let still assume that U σ is nonsingular. Let us introduce the sublattice N σ 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 = M σ M(σ) is the dual splitting, we have S σ = (σ M σ ) M(σ) and U σ U σ T n k. The meaning of the symbols U σ and U σ is that these are the toric varieties associated with the same cone σ, the first regarding σ as a cone in N R, and the second by regarding it as a cone in N σr. Now, if U σ is nonsingular, then U σ is nonsingular as well, so that the previous analysis applies; σ is generated by a basis of N σ, and U σ C k. So we have proved: Proposition An affine toric variety U σ is smooth if and and only if σ is generated by k elements of a basis of N, and then U σ C k T n k.

26 22 CHAPTER 1. TORIC VARIETIES A cone with this property will be called smooth. Example We study now an example which generalizes Example Let N = Z 2 and consider the cone σ generated by e 2 and me 1 e 2 ; for m = 2 we recover indeed Example Let N be the lattice generated by e 2 and e 1 = me 1 e 2, and let σ be σ regarded as a cone in N (note that N R = N R). Now σ is generated by the generators of N, so that U σ = C 2. Moreover we have inclusions N N and M M. Actually M is generated by 1 m e 1 and e 2, while S σ is generated by 1 m e 1 and 1 m e 1 + e 2. We associate variables so that sy = t. 1 m e 1 s, e 2 y, 1 m e 1 + e 2 t, Now, the inclusion N N yields a map σ σ, defined by a map A σ A σ. One has A σ = C[s, t], while to identify A σ we note that S σ is generated by e 1, e 1 + e 2, e 1 + 2e 2,..., e 1 + me 2 so that A σ = C[x, xy,..., xy m ]. By setting x = s m, y = t/s we obtain A σ = C[s m, s m 1 t,..., t m ]. Now note that µ m Z m, the group of m-th roots of unity, acts on C 2 in the standard way ζ(s, t) = (ζs, ζt). The ring A σ may be described as C[s, t] µm, the ring of µ m -invariant polynomials in 2 variables, and therefore U σ is the quotient C 2 /µ m, 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 P m+1 ; normal here is an old terminology and does not refer to normality). This example is easily generalized if we consider a cone generated by e 2 and me 1 ke 2, with 0 < k < m. In this case µ m acts on C 2 by ζ(s, t) = (ζs, ζ k t). To formalise in a way which is valid in all dimensions, we consider the case when σ is an n-symplex in N, where as usual n = dim N, i.e., σ is generated by n independent vectors (which, however, need not be a basis for N). The previous analysis goes through, and U σ turns out to be a quotient C n /G, where G is the finite abelian group G = N/N, U σ = C n, and G has an action on U σ obtained as before. Note that A σ can be obtained by intersecting A σ with C[M ] G = C[M]. A fan Σ will be said simplicial if all its cones are simplicial. Definition An orbifold of dimension n is a variety which is locally of the form C n /G, where G is a finite group. The previous discussion proves the following theorem. Theorem If the fan Σ is simplicial, the associated toric variety is an orbifold.

27 1.2. MORE PROPERTIES OF TORIC VARIETIES 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 {U i = Spec B i } of Y and for every i, an affine open cover {U ij = Spec A ij } of f 1 (V i ), where each A ij is a finitely generated B i -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 (in particular, f is closed). X f Y 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 A toric variety X Σ is complete if and only if Σ = N R. It is easy to prove the only if. Indeed, if Σ is not the whole of N R, let v N be a lattice point which is not in Σ. Then by Proposition 1.36 lim z 0 λ v (z) does not exist, which contradicts the completeness of X Σ. 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: X Σ X Σ. Proposition 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}.

28 24 CHAPTER 1. TORIC VARIETIES The valuation ring is discrete if Γ is the group of integers. 13 Discrete valuation rings may also be defined as principal ideal domains having a unique maximal ideal. Theorem [?, Thm. II.4.7] Let f : Y be a scheme morphism of finite type, with X noetherian. f is proper if and only if for every discrete valuation ring R and every commutative diagram Spec K Spec R there is a morphism Spec R X making the completed diagram commutative. to X Y f (1.9) Note that if X = Spec(B) and Y = Spec(A) are affine, the diagram (1.9) is equivalent Spec K Spec R Remark If X is irreducible, one can assume that Spec K maps into a fixed nonempty open subset. Moreover, properness is local on the target, so that one can assume Y to be affine. Proof of Theorem We want to apply this to X = X Σ, Y = X Σ, f = φ. In view of the previous remark, we may assume that Y is affine, Y = U σ, and assume that Spec K maps into an open subset U X Σ which is isomorphic to the n-dimensional torus T n, with n = dim N. The morphism Spec K 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 α C[M ] C[S σ ] R φ C[S σ ] φ commutes. Now, Spec R maps to U σ, so that the composition X Y f C[S σ ] φ C[M ] α K ord Z (1.10) 13 An 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) 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.

29 1.2. MORE PROPERTIES OF TORIC VARIETIES 25 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[S σ ] C[S σ ] C[M ] α K ord Z which implies that we can complete the diagram with the arrow C[S σ ] R. Example 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 {v 1,..., v n } of N. Set v 0 = v v n, and replace σ by the collection of cones generated by subsets of {v 0, v 1,..., v n } that do not contain {v 1,..., v n }. This change only affects the variety U σ, so we can assume that Σ consists only of σ and its faces, so that X Σ = U σ C n. We may also assume N = Z n and v i = e i. The new toric variety X Σ is covered by affine open toric varieties U σi, where σ i is the cone generated by {e 0, e 1,..., ê i,..., e n }. Then σi is generated by {e i, e 1 e i,..., e n e i }, and the coordinate rings of the varieties U σi are A σi = C[x i, x 1 x 1 i,..., x n x 1 i ]. Thus U σi C n. Note that x j /t j = x i /t i when t i, t j 0. On the other hand, the blowup Ĉn of C n at the origin is the subvariety of C n P n 1 defined by the equations x i t j = x j t i, with {x 1,..., x n } affine coordinates in C n ), and {t 1,..., t n } homogeneous co-ordinates in P n 1. The sets U i Ĉn given by t i 0 are copies of C n, and glue with the transition functions x j = x i (t i /t j ) with i j. Thus X Σ C n 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 X σ X Σ is proper. Example Let N = Z 2 and let Σ the standard fan for P 2. If we add a further ray generated by ( 1, m), with m a positive integer, we get a complete, smooth toric surface F m, called the m-th Hirzebruch surface. The map F m P 2 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 F m (for instance, that it is a P 1 fibration over P 1. Remark If Y is a smooth projective surface, the same construction that one does to blowup C 2 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, E 2 = 1. The curve E is called the exceptional divisor. Castelnuovo s criterion in a sense tells that

30 26 CHAPTER 1. TORIC VARIETIES this is the general situation: if X is a smooth projective surface, and C a rational curve in it such that C 2 = 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 [?]. 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 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 Hirzebruch surface. Corollary 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 Z 2 generated by 2e 1 e 2 and e 2, and let Σ be the fan consisting in σ and its faces. Note that the matrix that expresses these generators in terms of e 1 and e 2 is not unimodular, so the affine variety X Σ is singular. If we add the ray spanned by e 1, the cones in the new fan Σ are all smooth. We have a proper birational morphism π : X Σ X Σ ; 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 X Σ, i.e., the morphism π is a resolution of singularities. This procedure can be iterated. If σ is the cone in Z 2 generated by 3e 1 2e 2 and e 2, and Σ is the fan consisting in σ and its faces, again X Σ is singular. If we add the ray generated by e 1, now the affine toric variety associated with the cone generated by e 1 and e 2 is smooth, but the one given by the cone generated by e 1 and 3e 1 2e 2 is still singular. But if we add the ray generated by 2e 1 e 2, the cones in the new fan Σ are all smooth. We have a proper birational morphism π : X Σ X Σ ; 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 X Σ, 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 Z 2 ) generators v 2 and v = me 1 ke 2, with m 2, 0 < k < m, and m and k coprime. We add the ray generated by e 1. The variety associated to new cone