Toric Varieties. David Cox John Little Hal Schenck

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1 Toric Varieties David Cox John Little Hal Schenck DEPARTMENT OF MATHEMATICS, AMHERST COLLEGE, AMHERST, MA address: DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, COLLEGE OF THE HOLY CROSS, WORCESTER, MA address: DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS AT URBANA- CHAMPAIGN, URBANA, IL address:

2 c 2010, David Cox, John Little and Hal Schenck

3 Preface The study of toric varieties is a wonderful part of algebraic geometry that has deep connections with polyhedral geometry. Our book is an introduction to this rich subject that assumes only a modest knowledge of algebraic geometry. There are elegant theorems, unexpected applications, and, as noted by Fulton [58], toric varieties have provided a remarkably fertile testing ground for general theories. The Current Version. The January 2010 version consists of seven chapters: Chapter 1: Affine Toric Varieties Chapter 2: Projective Toric Varieties Chapter 3: Normal Toric Varieties Chapter 4: Divisors on Toric Varieties Chapter 5: Homogeneous Coordinates Chapter 6: Line Bundles on Toric Varieties Chapter 7: Projective Toric Morphisms These are the chapters included in the version you downloaded. The book also has a list of notation, a bibliography, and an index, all of which will appear in more polished form in the published version of the book. Two versions are available on-line. We recommend using postscript version since it has superior quality. Changes to the January 2009 Version. The new version fixes some typographical errors and includes a few new examples, new exercises and some rewritten proofs. The Rest of the Book. Five chapters are in various stages of completion: Chapter 8: The Canonical Divisor of a Toric Variety Chapter 9: Sheaf Cohomology of Toric Varieties iii

4 iv Preface Chapter 10: Toric Surfaces Chapter 11: Resolutions and Singularities Chapter 12: The Topology of Toric Varieties When the book is completed in August 2010, there will be three final chapters: Chapter 13: The Riemann-Roch Theorem Chapter 14: Geometric Invariant Theory Chapter 15: The Toric Minimal Model Program Prerequisites. The text assumes the material covered in basic graduate courses in algebra, topology, and complex analysis. In addition, we assume that the reader has had some previous experience with algebraic geometry, at the level of any of the following texts: Ideals, Varieties and Algorithms by Cox, Little and O Shea [35] Introduction to Algebraic Geometry by Hassett [79] Elementary Algebraic Geometry by Hulek [91] Undergraduate Algebraic Geometry by Reid [146] Computational Algebraic Geometry by Schenck [153] An Invitation to Algebraic Geometry by Smith, Kahanpää, Kekäläinen and Traves [158] Readers who have studied more sophisticated texts such as Harris [76], Hartshorne [77] or Shafarevich [152] certainly have the background needed to read our book. We should also mention that Chapter 9 uses some basic facts from algebraic topology. The books by Hatcher [80] and Munkres [128] are useful references. Background Sections. Since we do not assume a complete knowledge of algebraic geometry, Chapters 1 9 each begin with a background section that introduces the definitions and theorems from algebraic geometry that are needed to understand the chapter. The remaining chapters do not have background sections. For some of the chapters, no further background is necessary, while for others, the material more sophisticated and the requisite background will be provided by careful references to the literature. The Structure of the Text. We number theorems, propositions and equations based on the chapter and the section. Thus 3.2 refers to section 2 of Chapter 3, and Theorem and equation (3.2.6) appear in this section. The end (or absence) of a proof is indicated by, and the end of an example is indicated by. For the Instructor. We do not yet have a clear idea of how many chapters can be covered in a given course. This will depend on both the length of the course and the level of the students. One reason for posting this preliminary version on

5 Preface v the internet is our hope that you will teach from the book and give us feedback about what worked, what didn t, how much you covered, and how much algebraic geometry your students knew at the beginning of the course. Also let us know if the book works for students who know very little algebraic geometry. We look forward to hearing from you! For the Student. The book assumes that you will be an active reader. This means in particular that you should do tons of exercises this is the best way to learn about toric varieties. For students with a more modest background in algebraic geometry, reading the book requires a commitment to learn both toric varieties and algebraic geometry. It will be a lot of work, but it s worth the effort. This is a great subject. What s Missing. Right now, we do not discuss the history of toric varieties, nor do we give detailed notes about how results in the text relate to the literature. We would be interesting in hearing from readers about whether these items should be included. Please Give Us Feedback. We urge all readers to let us know about: Typographical and mathematical errors. Unclear proofs. Omitted references. Topics not in the book that should be covered. Places where we do not give proper credit. As we said above, we look forward to hearing from you! January 2010 David Cox John Little Hal Schenck

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7 Contents Preface Notation iii xi Part I: Basic Theory of Toric Varieties 1 Chapter 1. Affine Toric Varieties Background: Affine Varieties Introduction to Affine Toric Varieties Cones and Affine Toric Varieties Properties of Affine Toric Varieties 35 Appendix: Tensor Products of Coordinate Rings 48 Chapter 2. Projective Toric Varieties Background: Projective Varieties Lattice Points and Projective Toric Varieties Lattice Points and Polytopes Polytopes and Projective Toric Varieties Properties of Projective Toric Varieties 85 Chapter 3. Normal Toric Varieties Background: Abstract Varieties Fans and Normal Toric Varieties The Orbit-Cone Correspondence Toric Morphisms Complete and Proper 139 vii

8 viii Contents Appendix: Nonnormal Toric Varieties 149 Chapter 4. Divisors on Toric Varieties Background: Valuations, Divisors and Sheaves Weil Divisors on Toric Varieties Cartier Divisors on Toric Varieties The Sheaf of a Torus-Invariant Divisor 187 Chapter 5. Homogeneous Coordinates Background: Quotients in Algebraic Geometry Quotient Constructions of Toric Varieties The Total Coordinate Ring Sheaves on Toric Varieties Homogenization and Polytopes 230 Chapter 6. Line Bundles on Toric Varieties Background: Sheaves and Line Bundles Ample Divisors on Complete Toric Varieties The Nef and Mori Cones The Simplicial Case 290 Appendix: Quasicoherent Sheaves on Toric Varieties 301 Chapter 7. Projective Toric Morphisms Background: Quasiprojective Varieties and Projective Morphisms Polyhedra and Toric Varieties Projective Morphisms and Toric Varieties Projective Bundles and Toric Varieties 324 Appendix: More on Projective Morphisms 334 Chapter 8. The Canonical Divisor of a Toric Variety Background: Reflexive Sheaves and Differential Forms One-Forms on Toric Varieties Differential Forms on Toric Varieties Fano Toric Varieties 371 Chapter 9. Sheaf Cohomology of Toric Varieties Background: Cohomology Cohomology of Toric Divisors Vanishing Theorems I 402

9 Contents ix 9.3. Vanishing Theorems II Applications to Lattice Polytopes Local Cohomology and the Total Coordinate Ring 434 Appendix: Introduction to Spectral Sequences 447 Topics in Toric Geometry 379 Chapter 10. Toric Surfaces Singularities of Toric Surfaces and Their Resolutions Continued Fractions and Toric Surfaces Gröbner Fans and McKay Correspondences Smooth Toric Surfaces Riemann-Roch and Lattice Polygons 489 Chapter 11. Resolutions and Singularities Toric Resolution of Singularities Other Types of Resolutions Rees Algebras and Multiplier Ideals Toric Singularities 530 Chapter 12. The Topology of Toric Varieties Homotopy of Toric Varieties The Moment Map Singular Cohomology of Toric Varieties The Cohomology Ring Complements 583 Chapter 13. The Riemann-Roch Theorem 595 Chapter 14. Geometric Invariant Theory 597 Chapter 15. The Toric Minimal Model Program 599 Bibliography 601 Index 609

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11 Notation Basic Notions Z, Q, R, C integers, rational numbers, real numbers, complex numbers N semigroup of nonnegative integers {0,1,2,... } im, ker image and kernel lim direct limit lim Rings and Varieties inverse limit C[x 1,...,x n ] polynomial ring in n variables C[[x 1,...,x n ]] formal power series ring in n variables C[x 1 ±1 n ] ring of Laurent polynomials V(I) affine or projective variety of an ideal I(V) ideal of an affine or projective variety C[V] coordinate ring of an affine or projective variety C[V] d graded piece in degree d when V is projective C(V) field of rational functions when V is irreducible Spec(R) affine variety of coordinate ring R Proj(S) projective variety of graded ring S V f subset of an affine variety V where f 0 R f, R S, R p localization of R at f, a multiplicative set S, a prime ideal p R integral closure of the integral domain R xi

12 xii Notation R O V,p, m V,p T p (V) dimv, dim p V S X Y R C S X S Y V Sing(X) completion of local ring R local ring of a variety at a point and its maximal ideal Zariski tangent space of a variety at a point dimension of a variety and dimension at a point Zariski closure of S in a variety product of varieties tensor product of rings over C fiber product of varieties affine cone of a projective variety diagonal map X X X singular locus of a variety Semigroups I L ZA Z A NA S S σ = S σ,n C[S] H lattice ideal of lattice L Z s lattice generated by A elements s i=1 a im i ZA with s i=1 a i = 0 affine semigroup generated by A affine semigroup affine semigroup σ M semigroup algebra of S Hilbert basis of S σ when σ is strongly convex Cones and Fans Cone(S) σ Span(σ) dim σ σ Relint(σ) Int(σ) convex cone generated by S rational convex polyhedral cone in N R subspace spanned by σ dimension of σ dual cone of σ relative interior of σ interior of σ when Span(σ) = N R σ set of m M R with m,σ = 0 τ σ, τ σ τ is a face or proper face of σ τ face of σ dual to τ σ, equal to σ τ H m hyperplane in N R defined by m, = 0, m M R \ {0} H m + half-space in N R defined by m, 0, m M R \ {0}

13 Notation xiii Σ Σ(r) u ρ Σ max N σ N(σ) M(σ) Star(σ) Σ (σ) ind(σ) fan in N R r-dimensional cones of Σ minimal generator of ρ N, ρ Σ(1) maximal cones of Σ sublattice Z(σ N) = Span(σ) N quotient lattice N/N σ dual lattice of N(σ), equal to σ M star of σ, a fan in N(σ) star subdivision of Σ along σ index of a simplicial cone Polyhedra Conv(S) convex hull of S P polytope or polyhedron dim P dimension of P H u,b hyperplane in M R defined by,u = b, u N R \ {0} H + u,b half-space in M R defined by,u b, u N R \ {0} Q P, Q P Q is a face or proper face of P P dual or polar of a polytope n standard n-simplex A+B Minkowski sum k P multiple of a polytope or polyhedron C(P) cone over a polytope or polyhedron σ Q cone of a face Q P Σ P normal fan of a polytope or polyhedron support function of a polytope or polyhedron ϕ P Toric Varieties M, χ m character lattice of a torus and character of m M N, λ u lattice of one-parameter subgroups of a torus and one-parameter subgroup of u N T N torus N Z C = Hom Z (M,C ) associated to N and M M R, M Q vector spaces M Z R, M Z Q built from M N R, N Q vector spaces N Z R, N Z Q built from N m,u pairing of m M or M R with u N or N R Y A, X A affine and projective toric variety of A M

14 xiv Notation U σ = U σ,n X Σ = X Σ,N X P X D affine toric variety of a cone σ N R toric variety of a fan projective toric variety of a lattice polytope or polyhedron toric variety of a basepoint free divisor φ, φ R lattice homomorphism of a toric morphism φ : X Σ1 X Σ2 and its real extension γ σ distinguished point of U σ O(σ) orbit of σ Σ V(σ) = O(σ) D ρ = O(ρ) D F U P U Σ Specific Varieties closure of orbit of σ Σ, toric variety of Star(σ) torus-invariant prime divisor on X Σ of ρ Σ(1) torus-invariant prime divisor on X P of facet F P affine toric variety of recession cone of a polyhedron affine toric variety of a fan with convex support C n, P n affine and projective n-dimensional space P(q 0,...,q n ) weighted projective space C multiplicative group of nonzero complex numbers C \ {0} (C ) n standard n-dimensional torus Ĉ d, C d Bl 0 (C n ) Bl V(τ) (X Σ ) H r S a,b Divisors rational normal cone and curve blowup of C n at the origin blowup of X Σ along V(τ), toric variety of Σ (τ) Hirzebruch surface rational normal scroll O X,D ν D div( f) D E D 0 Div 0 (X) Div(X) Div TN (X Σ ) CDiv(X) local ring of a variety at a prime divisor discrete valuation of a prime divisor D principal divisor of a rational function linear equivalence of divisors effective divisor group of principal divisors on X group of Weil divisors on X group of torus-invariant Weil divisors on X Σ group of Cartier divisors on X

15 Notation xv CDiv TN (X Σ ) group of torus-invariant Cartier divisors on X Σ Cl(X) divisor class group of a normal variety X Pic(X) Picard group of a normal variety X Supp(D) support of a divisor D U restriction of a divisor to an open set {(U i, f i )} local data of a Cartier divisor on X {m σ } σ Σ Cartier data of a torus-invariant Cartier divisor on X Σ P D polyhedron of a torus-invariant divisor Σ D fan associated to a basepoint free divisor D P Cartier divisor of a polytope or polyhedron ϕ D support function of a Cartier divisor SF(Σ, N) support functions integral with respect to N Intersection Products deg(d) D C D D, C C N 1 (X), N 1 (X) Nef(X) NE(X) NE(X) Pic(X) R r(p) degree of a divisor on a curve intersection product of Cartier divisor and complete curve numerically equivalent Cartier divisors and complete curves (CDiv(X)/ ) Z R and (Z 1 (X)/ ) Z R cone in N 1 (X) generated by nef divisors cone in N 1 (X) generated by complete curves Mori cone, equal to the closure of NE(X) Pic(X) Z R primitive relation of a primitive collection Sheaves and Bundles O X OX O X (D) K X F U Γ(U,F) I Y M M O XΣ (α) structure sheaf of a variety X sheaf of invertible elements of O X sheaf of a Weil divisor D on X constant sheaf of rational functions when X is irreducible restriction of a sheaf to an open set sections of a sheaf over an open set ideal sheaf of a subvariety Y X sheaf on Spec(R) of an R-module M sheaf on X Σ of a graded S-module M sheaf on X Σ of the graded S-module S(α)

16 xvi Notation F p stalk of a sheaf at a point F OX G tensor product of sheaves of O X -modules H om OX (F,G ) sheaf of homomorphisms F dual sheaf of F, equal to H om OX (F,O X ) π : V X vector bundle π : V L X rank 1 vector bundle of a line bundle L f L pullback of a line bundle φ L,W map to projective space determined by W Γ(X,L ) D complete linear system of D Σ D fan that gives V L for L = O XΣ (D) P(V), P(E ) projective bundle of vector bundle or locally free sheaf Quotients and Homogeneous Coordinates ring of invariants of G acting on R V/G good geometric quotient V//G good categorical quotient S total coordinate ring of X Σ x ρ variable in S corresponding to ρ Σ(1) S β graded piece of S in degree β Cl(X Σ ) deg(x α ) degree in Cl(X Σ ) of a monomial in S xˆσ monomial generator of B corresponding to σ Σ B(Σ) irrelevant ideal of S, generated by the xˆσ Z(Σ) exceptional set, equal to V(B(Σ)) G group Hom Z (Cl(X Σ ),C ) used in the quotient construction R G x m x m,d x F x m,p x v,p Laurent monomial ρ x m,uρ ρ, m M homogenization of χ m, m P D M facet variable of a facet F P P-monomial associated to m P M vertex monomial associated to vertex v P M M graded S-module M(α) shift of M by α Cl(X Σ )

17 Part I: Basic Theory of Toric Varieties Chapters 1 to 9 introduce the theory of toric varieties. This part of the book assumes only a minimal amount of algebraic geometry, at the level of Ideals, Varieties and Algorithms [35]. Each chapter begins with a background section that develops the necessary algebraic geometry. 1

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19 Chapter 1 Affine Toric Varieties 1.0. Background: Affine Varieties We begin with the algebraic geometry needed for our study of affine toric varieties. Our discussion assumes Chapters 1 5 and 9 of [35]. Coordinate Rings. An ideal I S = C[x 1,...,x n ] gives an affine variety V(I) = {p C n f(p) = 0 for all f I} and an affine variety V C n gives the ideal I(V) = { f S f(p) = 0 for all p V }. By the Hilbert Basis Theorem, an affine variety V is defined by the vanishing of finitely many polynomials in S, and for any ideal I, the Nullstellensatz tells us that I(V(I)) = I = { f S f l I for some l 1} since C is algebraically closed. The most important algebraic object associated to V is its coordinate ring C[V] = S/I(V). Elements of C[V] can be interpreted as the C-valued polynomial functions on V. Note that C[V] is a C-algebra, meaning that its vector space structure is compatible with its ring structure. Here are some basic facts about coordinate rings: C[V] is an integral domain I(V) is a prime ideal V is irreducible. Polynomial maps (also called morphisms) φ : V 1 V 2 between affine varieties correspond to C-algebra homomorphisms φ : C[V 2 ] C[V 1 ], where φ (g) = g φ for g C[V 2 ]. Two affine varieties are isomorphic if and only if their coordinate rings are isomorphic C-algebras. 3

20 4 Chapter 1. Affine Toric Varieties A point p of an affine variety V gives the maximal ideal { f C[V] f(p) = 0} C[V], and all maximal ideals of C[V] arise this way. Coordinate rings of affine varieties can be characterized as follows (Exercise 1.0.1). Lemma A C-algebra R is isomorphic to the coordinate ring of an affine variety if and only if R is a finitely generated C-algebra with no nonzero nilpotents, i.e., if f R satisfies f l = 0 for some l 1, then f = 0. To emphasize the close relation between V and C[V], we sometimes write (1.0.1) V = Spec(C[V]). This can be made canonical by identifying V with the set of maximal ideals of C[V] via the fourth bullet above. More generally, one can take any commutative ring R and define the affine scheme Spec(R). The general definition of Spec uses all prime ideals of R, not just the maximal ideals as we have done. Thus some authors would write (1.0.1) as V = Specm(C[V]), the maximal spectrum of C[V]. Readers wishing to learn about affine schemes should consult [48] and [77]. The Zariski Topology. An affine variety V C n has two topologies we will use. The first is the classical topology, induced from the usual topology on C n. The second is the Zariski topology, where the Zariski closed sets are subvarieties of V (meaning affine varieties of C n contained in V ) and the Zariski open sets are their complements. Since subvarieties are closed in the classical topology (polynomials are continuous), Zariski open subsets are open in the classical topology. Given a subset S V, its closure S in the Zariski topology is the smallest subvariety of V containing S. We call S the Zariski closure of S. It is easy to give examples where this differs from the closure in the classical topology. Affine Open Subsets and Localization. Some Zariski open subsets of an affine variety V are themselves affine varieties. Given f C[V]\{0}, let V f = {p V f(p) 0} V. Then V f is Zariski open in V and is also an affine variety, as we now explain. Let V C n have I(V) = f 1,..., f s and pick g C[x 1,...,x n ] representing f. Then V f = V \ V(g) is Zariski open in V. Now consider a new variable y and let W = V( f 1,..., f s,1 gy) C n C. Since the projection map C n C C n maps W bijectively onto V f, we can identify V f with the affine variety W C n C. When V is irreducible, the coordinate ring of V f is easy to describe. Let C(V) be the field of fractions of the integral domain C[V]. Recall that elements of C(V) give rational functions on V. Then let (1.0.2) C[V] f = {g/ f l C(V) g C[V], l 0}.

21 1.0. Background: Affine Varieties 5 In Exercise you will prove that Spec(C[V] f ) is the affine variety V f. Example The n-dimensional torus is the affine open subset (C ) n = C n \ V(x 1 x n ) C n, with coordinate ring C[x 1,...,x n ] x1 x n = C[x ±1 1,...,x±1 n ]. Elements of this ring are called Laurent polynomials. The ring C[V] f from (1.0.2) is an example of localization. In Exercises and you will show how to construct this ring for all affine varieties, not just irreducible ones. The general concept of localization is discussed in standard texts in commutative algebra such as [3, Ch. 3] and [47, Ch. 2]. Normal Affine Varieties. Let R be an integral domain with field of fractions K. Then R is normal, or integrally closed, if every element of K which is integral over R (meaning that it is a root of a monic polynomial in R[x]) actually lies in R. For example, any UFD is normal (Exercise 1.0.5). Definition An irreducible affine variety V is normal if its coordinate ring C[V] is normal. For example, C n is normal since its coordinate ring C[x 1,...,x n ] is a UFD and hence normal. Here is an example of a non-normal affine variety. Example Let C = V(x 3 y 2 ) C 2. This is an irreducible plane curve with a cusp at the origin. It is easy to see that C[C] = C[x,y]/ x 3 y 2. Now let x and ȳ be the cosets of x and y in C[C] respectively. This gives ȳ/ x C(C). A computation shows that ȳ/ x / C[C] and that (ȳ/ x) 2 = x. Consequently C[C] and hence C are not normal. We will see below that C is an affine toric variety. An irreducible affine variety V has a normalization defined as follows. Let C[V] = {α C(V) : α is integral over C[V]}. We call C[V] the integral closure of C[V]. One can show that C[V] is normal and (with more work) finitely generated as a C-algebra (see [47, Cor ]). This gives the normal affine variety V = Spec(C[V] ) We call V the normalization of V. The natural inclusion C[V] C[V] = C[V ] corresponds to a map V V. This is the normalization map.

22 6 Chapter 1. Affine Toric Varieties Example We saw in Example that the curve C C 2 defined by x 3 = y 2 has elements x,ȳ C[C] such that ȳ/ x / C[C] is integral over C[C]. In Exercise you will show that C[ȳ/ x] C(C) is the integral closure of C[C] and that the normalization map is the map C C defined by t (t 2,t 3 ). At first glance, the definition of normal does not seem very intuitive. Once we enter the world of toric varieties, however, we will see that normality has a very nice combinatorial interpretation and that the nicest toric varieties are the normal ones. We will also see that normality leads to a nice theory of divisors. In Exercise you will prove some properties of normal domains that will be used in 1.3 when we study normal affine toric varieties. Smooth Points of Affine Varieties. In order to define a smooth point of an affine variety V, we first need to define local rings and Zariski tangent spaces. When V is irreducible, the local ring of V at p is O V,p = { f/g C(V) f,g C[V] and g(p) 0}. Thus O V,p consists of all rational functions on V that are defined at p. Inside of O V,p we have the maximal ideal m V,p = {φ O V,p φ(p) = 0}. In fact, m V,p is the unique maximal ideal of O V,p, so that O V,p is a local ring. Exercises and explain how to define O V,p when V is not irreducible. The Zariski tangent space of V at p is defined to be T p (V) = Hom C (m V,p /m 2 V,p,C). In Exercise you will verify that dimt p (C n ) = n for every p C n. According to [77, p. 32], we can compute the Zariski tangent space of a point in an affine variety as follows. Lemma Let V C n be an affine variety and let p V. Also assume that I(V) = f 1,..., f s C[x 1,...,x n ]. For each i, let d p ( f i ) = f i x 1 (p)x f i x n (p)x n. Then the Zariski tangent T p (V) is isomorphic to the subspace of C n defined by the equations d p ( f 1 ) = = d p ( f s ) = 0. In particular, dimt p (V) n. Definition A point p of an affine variety V is smooth or nonsingular if dimt p (V) = dim p V, where dim p V is the maximum of the dimensions of the irreducible components of V containing p. The point p is singular if it is not smooth. Finally, V is smooth if every point of V is smooth.

23 1.0. Background: Affine Varieties 7 Points lying in the intersection of two or more irreducible components of V are always singular ([35, Thm. 8 of Ch. 9, 6]). Since dimt p (C n ) = n for every p C n, we see that C n is smooth. For an irreducible affine variety V C n of dimension d, fix p V and write I(V) = f 1,..., f s. Using Lemma 1.0.6, it is straightforward to show that V is smooth at p if and only if the Jacobian matrix ( fi ) (1.0.3) J p ( f 1,..., f s ) = (p) x j 1 i s,1 j n has rank n d (Exercise 1.0.9). Here is a simple example. Example As noted in Example 1.0.4, the plane curve C defined by x 3 = y 2 has I(C) = x 3 y 2 C[x,y]. A point p = (a,b) C has Jacobian J p = (3a 2, 2b), so the origin is the only singular point of C. Since T p (V) = Hom C (m V,p /mv,p 2,C), we see that V is smooth at p when dimv equals the dimension of m V,p /mv,p 2 as a vector space over O V,p/m V,p. In terms of commutative algebra, this means that p V is smooth if and only if O V,p is a regular local ring. See [3, p. 123] or [47, 10.3]. We can relate smoothness and normality as follows. Proposition A smooth irreducible affine variety V is normal. Proof. In 3.0 we will see that C[V] = p V O V,p. By Exercise 1.0.7, C[V] is normal once we prove that O V,p is normal for all p V. Hence it suffices to show that O V,p is normal whenever p is smooth. This follows from some powerful results in commutative algebra: O V,p is a regular local ring when p is a smooth point of V (see above), and every regular local ring is a UFD (see [47, Thm ]). Then we are done since every UFD is normal. A direct proof that O V,p is normal at a smooth point p V is sketched in Exercise The converse of Propostion can fail. We will see in 1.3 that the affine variety V(xy zw) C 4 is normal, yet V(xy zw) is singular at the origin. Products of Affine Varieties. Given affine varieties V 1 and V 2, there are several ways to show that the cartesian product V 1 V 2 is an affine variety. The most direct way is to proceed as follows. Let V 1 C m = Spec(C[x 1,...,x m ]) and V 2 C n = Spec(C[y 1,...,y n ]). Take I(V 1 ) = f 1,..., f s and I(V 2 ) = g 1,...,g t. Since the f i and g j depend on separate sets of variables, it follows that is an affine variety. V 1 V 2 = V( f 1,..., f s,g 1,...,g t ) C m+n

24 8 Chapter 1. Affine Toric Varieties A fancier method is to use the mapping properties of the product. This will also give an intrinsic description of its coordinate ring. Given V 1 and V 2 as above, V 1 V 2 should be an affine variety with projections π i : V 1 V 2 V i such that whenever we have a diagram W φ 1 φ 2 ν V 1 V 2 π 1 V 2 π 2 V 1 where φ i :W V i are morphisms from an affine variety W, there should be a unique morphism ν (the dotted arrow) that makes the diagram commute, i.e., π i ν = φ i. For the coordinate rings, this means that whenever we have a diagram C[V 2 ] C[V 1 ] π 1 π 2 φ 2 C[V 1 V 2 ] φ 1 ν C[W] with C-algebra homomorphisms φ i : C[V i ] C[W], there should be a unique C- algebra homomorphism ν (the dotted arrow) that makes the diagram commute. By the universal mapping property of the tensor product of C-algebras, C[V 1 ] C C[V 2 ] has the mapping properties we want. Since C[V 1 ] C C[V 2 ] is a finitely generated C-algebra with no nilpotents (see the appendix to this chapter), it is the coordinate ring C[V 1 V 2 ]. For more on tensor products, see [3, pp ] or [47, A2.2]. Example Let V be an affine variety. Since C n = Spec(C[y 1,...,y n ]), the product V C n has coordinate ring C[V] C C[y 1,...,y n ] = C[V][y 1,...,y n ]. If V is contained in C m with I(V) = f 1,..., f s C[x 1,...,x m ], it follows that I(V C n ) = f 1,..., f s C[x 1,...,x m,y 1,...,y n ]. For later purposes, we also note that the coordinate ring of V (C ) n is C[V] C C[y ±1 1,...,y±1 n ] = C[V][y±1 1,...,y±1 n ]. Given affine varieties V 1 and V 2, we note that the Zariski topology on V 1 V 2 is usually not the product of the Zariski topologies on V 1 and V 2.

25 1.0. Background: Affine Varieties 9 Example Consider C 2 = C C. By definition, a basis for the product of the Zariski topologies consists of sets U 1 U 2 where U i are Zariski open in C. Such a set is the complement of a union of collections of horizontal and vertical lines in C 2. This makes it easy to see that Zariski closed sets in C 2 such as V(y x 2 ) cannot be closed in the product topology. Exercises for Prove Lemma Hint: You will need the Nullstellensatz Let R be a commutative C-algebra. A subset S R is a multipliciative subset provided 1 S, 0 / S, and S is closed under multiplication. The localization R S consists of all formal expressions g/s, g R, s S, modulo the equivalence relation g/s h/t u(tg sh) = 0 for some u S. (a) Show that the usual formulas for adding and multiplying fractions induce well-defined binary operations that make R S into C-algebra. (b) If R has no nonzero nilpotents, then prove that the same is true for R S. For more on localization, see [3, Ch. 3] or [47, Ch. 2] Let R be a finitely generated C-algebra without nilpotents as in Lemma and let f R be nonzero. Then S = {1, f, f 2,...} is a multiplicative set. The localization R S is denoted R f and is called the localization of R at f. (a) Show that R f is a finitely generated C-algebra without nilpotents. (b) Show that R f satisfies Spec(R f ) = Spec(R) f. (c) Show that R f is given by (1.0.2) when R is an integral domain Let V be an affine variety with coordinate ring C[V]. Given a point p V, let S = {g C[V] g(p) 0}. (a) Show that S is a multiplicative set. The localization C[V] S is denoted O V,p and is called the local ring of V at p. (b) Show that every φ O V,p has a well-defined value φ(p) and that is the unique maximal ideal of O V,p. m V,p = {φ O V,p φ(p) = 0} (c) When V is irreducible, show that O V,p agrees with the definition given in the text Prove that a UFD is normal In the setting of Example 1.0.5, show that C[ȳ/ x] C(C) is the integral closure of C[C] and that the normalization C C is defined by t (t 2,t 3 ) In this exercise, you will prove some properties of normal domains needed for 1.3. (a) Let R be a normal domain with field of fractions K and let S R be a multiplicative subset. Prove that the localization R S is normal. (b) Let R α, α A, be normal domains with the same field of fractions K. Prove that the intersection α A R α is normal Prove that dimt p (C n ) = n for all p C n.

26 10 Chapter 1. Affine Toric Varieties Use Lemma to prove the claim made in the text that smoothness is determined by the rank of the Jacobian matrix (1.0.3) Let V be irreducible and suppose that p V is smooth. The goal of this exercise is to prove that O V,p is normal using standard results from commutative algebra. Set n = dimv and consider the ring of formal power series C[[x 1,...,x n ]]. This is a local ring with maximal ideal m = x 1,...,x n. We will use three facts: C[[x 1,...,x n ]] is a UFD by [174, p. 148] and hence normal by Exercise Since p V is smooth, [125, 1C] proves the existence of a C-algebra homomorphism O V,p C[[x 1,...,x n ]] that induces isomorphisms O V,p /m l V,p C[[x 1,...,x n ]]/m l for all l 0. This implies that the completion [3, Ch. 10] Ô V,p = lim O V,p /m l V,p is isomorphic to a formal power series ring, i.e., Ô V,p C[[x 1,...,x n ]]. Such an isomorphism captures the intuitive idea that at a smooth point, functions should have power series expansions in local coordinates x 1,...,x n. If I O V,p is an ideal, then I = l=1 (I +ml V,p). This theorem of Krull holds for any ideal I in a Noetherian local ring A and follows from [3, Cor ] with M = A/I. Now assume that p V is smooth. (a) Use the third bullet to show that O V,p C[[x 1,...,x n ]] is injective. (b) Suppose that a,b O V,p satisfy b a in C[[x 1,...,x n ]]. Prove that b a in O V,p. Hint: Use the second bullet to show a bo V,p +m l V,p and then use the third bullet. (c) Prove that O V,p is normal. Hint: Use part (b) and the first bullet. This argument can be continued to show that O V,p is a UFD. See [125, (1.28)] Let V and W be affine varieties and let S V be a subset. Prove that S W = S W Let V and W be irreducible affine varieties. Prove that V W is irreducible. Hint: Suppose V W = Z 1 Z 2, where Z 1,Z 2 are closed. Let V i = {v V {v} W Z i }. Prove that V = V 1 V 2 and that V i is closed in V. Exercise will be useful Introduction to Affine Toric Varieties We first discuss what we mean by torus and then explore various constructions of affine toric varieties. The Torus. The affine variety (C ) n is a group under component-wise multiplication. A torus T is an affine variety isomorphic to (C ) n, where T inherits a group structure from the isomorphism. Associated to T are its characters and oneparameter subgroups. We discuss each of these briefly.

27 1.1. Introduction to Affine Toric Varieties 11 A character of a torus T is a morphism χ : T C that is a group homomorphism. For example, m = (a 1,...,a n ) Z n gives a character χ m : (C ) n C defined by (1.1.1) χ m (t 1,...,t n ) = t a 1 1 tan n. One can show that all characters of (C ) n arise this way (see [92, 16]). Thus the characters of (C ) n form a group isomorphic to Z n. For an arbitrary torus T, its characters form a free abelian group M of rank equal to the dimension of T. It is customary to say that m M gives the character χ m : T C. We will need the following results concerning tori (see [92, 16] for proofs). Proposition (a) Let T 1 and T 2 be tori and let Φ : T 1 T 2 be a morphism that is a group homomorphism. Then the image of Φ is a torus and is closed in T 2. (b) Let T be a torus and let H T be an irreducible subvariety of T that is a subgroup. Then H is a torus. Now assume that a torus T acts linearly on a finite dimensional vector space W over C, where the action of t T on w W is denoted t w. Given m M, we get the eigenspace W m = {w W t w = χ m (t)w for all t T }. If W m {0}, then every w W m \ {0} is a simultaneous eigenvector for all t T, with eigenvalue given by χ m (t). Proposition In the above situation, we have W = m M W m. This proposition is a sophisticated way of saying that a family of commuting diagonalizable linear maps can be simultaneously diagonalized. A one-parameter subgroup of a torus T is a morphism λ : C T that is a group homomorphism. For example, u = (b 1,...,b n ) Z n gives a one-parameter subgroup λ u : C (C ) n defined by (1.1.2) λ u (t) = (t b 1,...,t bn ). All one-parameter subgroups of (C ) n arise this way (see [92, 16]). It follows that the group of one-parameter subgroups of (C ) n is naturally isomorphic to Z n. For an arbitrary torus T, the one-parameter subgroups form a free abelian group N of rank equal to the dimension of T. As with the character group, an element u N gives the one-parameter subgroup λ u : C T. There is a natural bilinear pairing, : M N Z defined as follows. (Intrinsic) Given a character χ m and a one-parameter subgroup λ u, the composition χ m λ u : C C is character of C, which is given by t t l for some l Z. Then m,u = l.

28 12 Chapter 1. Affine Toric Varieties (Concrete) If T = (C ) n with m = (a 1,...,a n ) Z n, u = (b 1,...,b n ) Z n, then one computes that n (1.1.3) m,u = a i b i, i.e., the pairing is the usual dot product. It follows that the characters and one-parameter subgroups of a torus T form free abelian groups M and N of finite rank with a pairing, : M N Z that identifies N with Hom Z (M,Z) and M with Hom Z (N,Z). In terms of tensor products, one obtains a canonical isomorphism N Z C T via u t λ u (t). Hence it is customary to write a torus as T N. From this point of view, picking an isomorphism T N (C ) n induces dual bases of M and N, i.e., isomorphisms M Z n and N Z n that turn characters into Laurent monomials (1.1.1), one-parameter subgroups into monomial curves (1.1.2), and the pairing into dot product (1.1.3). The Definition of Affine Toric Variety. We now define the main object of study of this chapter. Definition An affine toric variety is an irreducible affine variety V containing a torus T N (C ) n as a Zariski open subset such that the action of T N on itself extends to an algebraic action of T N on V. (By algebraic action, we mean an action T N V V given by a morphism.) Obvious examples of affine toric varieties are (C ) n and C n. Here are some less trivial examples. Example The plane curve C = V(x 3 y 2 ) C 2 has a cusp at the origin. This is an affine toric variety with torus i=1 C {0} = C (C ) 2 = {(t 2,t 3 ) t C } C, where the isomorphism is t (t 2,t 3 ). Example shows that C is a non-normal toric variety. Example The variety V = V(xy zw) C 4 is a toric variety with torus V (C ) 4 = {(t 1,t 2,t 3,t 1 t 2 t 1 3 ) t i C } (C ) 3, where the isomorphism is (t 1,t 2,t 3 ) (t 1,t 2,t 3,t 1 t 2 t 1 3 ). We will see later that V is normal. Example Consider the surface in C d+1 parametrized by the map Φ : C 2 C d+1 defined by (s,t) (s d,s d 1 t,...,st d 1,t d ). Thus Φ is defined using all degree d monomials in s,t.

29 1.1. Introduction to Affine Toric Varieties 13 Let the coordinates of C d+1 be x 0,...,x d and let I C[x 0,...,x d ] be the ideal generated by the 2 2 minors of the matrix ( ) x0 x 1 x d 2 x d 1, x 1 x 2 x d 1 x d so I = x i x j+1 x i+1 x j 0 i < j d 1. In Exercise you will verify that Φ(C 2 ) = V(I), so that Ĉ d = Φ(C 2 ) is an affine variety. You will also prove that I(Ĉ d ) = I, so that I is the ideal of all polynomials vanishing on Ĉ d. It follows that I is prime since V(I) is irreducible by Proposition below. The affine surface Ĉ d is called the rational normal cone of degree d and is an example of a determinantal variety. We will see below that I is a toric ideal. It is straightforward to show that Ĉ d is a toric variety with torus Φ((C ) 2 ) = Ĉ d (C ) d+1 (C ) 2. We will study this example from the projective point of view in Chapter 2. We next explore three equivalent ways of constructing affine toric varieties. Lattice Points. In this book, a lattice is a free abelian group of finite rank. Thus a lattice of rank n is isomorphic to Z n. For example, a torus T N has lattices M (of characters) and N (of one-parameter subgroups). Given a torus T N with character lattice M, a set A = {m 1,...,m s } M gives characters χ m i : T N C. Then consider the map (1.1.4) Φ A : T N C s defined by Φ A (t) = ( χ m 1 (t),...,χ ms (t) ) C s. Definition Given a finite set A M, the affine toric variety Y A is defined to be the Zariski closure of the image of the map Φ A from (1.1.4). This definition is justified by the following proposition. Proposition Given A M as above, let ZA M be the sublattice generated by A. Then Y A is an affine toric variety whose torus has character lattice ZA. In particular, the dimension of Y A is the rank of ZA. Proof. The map (1.1.4) can be regarded as a map Φ A : T N (C ) s of tori. By Proposition 1.1.1, the image T = Φ A (T N ) is a torus that is closed in (C ) s. The latter implies that Y A (C ) s = T since Y A is the Zariski closure of the image. It follows that the image is Zariski open in Y A. Furthermore, T is irreducible (it is a torus), so the same is true for its Zariski closure Y A.

30 14 Chapter 1. Affine Toric Varieties We next consider the action of T. Since T (C ) s, an element t T acts on C s and takes varieties to varieties. Then T = t T t Y A shows that t Y A is a variety containing T. Hence Y A t Y A by the definition of Zariski closure. Replacing t with t 1 leads to Y A = t Y A, so that the action of T induces an action on Y A. We conclude that Y A is an affine toric variety. It remains to compute the character lattice of T, which we will temporarily denote by M. Since T = Φ A (T N ), the map Φ A gives the commutative diagram Φ A T N (C ) s where denotes a surjective map and an injective map. This diagram of tori induces a commutative diagram of character lattices bφ A T M M. Since Φ A : Z s M takes the standard basis e 1,...,e s to m 1,...,m s, the image of Φ A is ZA. By the diagram, we obtain M ZA. Then we are done since the dimension of a torus equals the rank of its character lattice. In concrete terms, fix a basis of M, so that we may assume M = Z n. Then the s vectors in A Z n can be regarded as the columns of an n s matrix A with integer entries. In this case, the dimension of Y A is simply the rank of the matrix A. We will see below that every affine toric variety is isomorphic to Y A for some finite subset A of a lattice. Toric Ideals. Let Y A C s = Spec(C[x 1,...,x s ]) be the affine toric variety coming from a finite set A = {m 1,...,m s } M. We can describe the ideal I(Y A ) C[x 1,...,x s ] as follows. As in the proof of Proposition 1.1.8, Φ A induces a map of character lattices Φ A : Z s M that sends the standard basis e 1,...,e s to m 1,...,m s. Let L be the kernel of this map, so that we have an exact sequence Z s 0 L Z s M. In down to earth terms, elements l = (l 1,...,l s ) of L satisfy s i=1 l im i = 0 and hence record the linear relations among the m i.

31 1.1. Introduction to Affine Toric Varieties 15 Given l = (l 1,...,l s ) L, set l + = l i >0 l i e i and l = l i <0l i e i. Note that l = l + l and that l +,l N s. It follows easily that the binomial x l + x l = l i >0 xl i i l i <0 x l i i vanishes on the image of Φ A and hence on Y A since Y A is the Zariski closure of the image. Proposition The ideal of the affine toric variety Y A C s is I(Y A ) = x l + x l l L = x α x β α,β N s and α β L. Proof. We leave it to the reader to prove equality of the two ideals on the right (Exercise 1.1.2). Let I L denote this ideal and note that I L I(Y A ). We prove the opposite inclusion following [166, Lem. 4.1]. Pick a monomial order > on C[x 1,...,x s ] and an isomorphism T N (C ) n. Thus we may assume M = Z n and the map Φ : (C ) n C s is given by Laurent monomials t m i in variables t 1,...,t n. If I L I(Y A ), then we can pick f I(Y A ) \ I L with minimal leading monomial x α = s i=1 xa i i. Rescaling if necessary, x α becomes the leading term of f. Since f(t m 1,...,t ms ) is identically zero as a polynomial in t 1,...,t n, there must be cancellation involving the term coming from x α. In other words, f must contain a monomial x β = s i=1 xb i i < x α such that s s (t m i ) a i = (t m i ) b i. This implies that i=1 s a i m i = i=1 i=1 s b i m i, so that α β = s i=1 (a i b i )e i L. Then x α x β I L by the second description of I L. It follows that f x α + x β also lies in I(Y A ) \ I L and has strictly smaller leading term. This contradiction completes the proof. Given a finite set A M, there are several methods to compute the ideal I(Y A ) = I L of Proposition For simple examples, the rational implicitization algorithm of [35, Ch. 3, 3] can be used. It is also possible to compute I L using a basis of L and ideal quotients (Exercise 1.1.3). Further comments on computing I L can be found in [166, Ch. 12]. Inspired by Proposition 1.1.9, we make the following definition. i=1

32 16 Chapter 1. Affine Toric Varieties Definition Let L Z s be a sublattice. (a) The ideal I L = x α x β α,β N s and α β L is called a lattice ideal. (b) A prime lattice ideal is called a toric ideal. Since toric varieties are irreducible, the ideals appearing in Proposition are toric ideals. Examples of toric ideals include: Example : x 3 y 2 C[x,y] Example : xz yw C[x,y,z,w] Example : x i x j+1 x i+1 x j 0 i < j d 1 C[x 0,...,x d ]. (The latter is the ideal of the rational normal cone Ĉ d C d+1.) In each example, we have a prime ideal generated by binomials. As we now show, such ideals are automatically toric. Proposition An ideal I C[x 1,...,x s ] is toric if and only if it is prime and generated by binomials. Proof. One direction is obvious. So suppose that I is prime and generated by binomials x α i x β i. Then observe that V(I) (C ) s is nonempty (it contains (1,...,1)) and is a subgroup of (C ) s (easy to check). Since V(I) C s is irreducible, it follows that V(I) (C ) s is an irreducible subvariety of (C ) s that is also a subgroup. By Proposition 1.1.1, we see that T = V(I) (C ) s is a torus. Projecting on the ith coordinate of (C ) s gives a character T (C ) s C, which by our usual convention we write as χ m i : T C for m i M. It follows easily that V(I) = Y A for A = {m 1,...,m s }, and since I is prime, we have I = I(Y A ) by the Nullstellensatz. Then I is toric by Proposition We will later see that all affine toric varieties arise from toric ideals. For more on toric ideals and lattice ideals, the reader should consult [123, Ch. 7]. Affine Semigroups. A semigroup is a set S with an associative binary operation and an identity element. To be an affine semigroup, we further require that: The binary operation on S is commutative. We will write the operation as + and the identity element as 0. Thus a finite set A S gives NA = { m A a mm a m N } S. The semigroup is finitely generated, meaning that there is a finite set A S such that NA = S. The semigroup can be embedded in a lattice M. The simplest example of an affine semigroup is N n Z n. More generally, given a lattice M and a finite set A M, we get the affine semigroup NA M. Up to isomorphism, all affine semigroups are of this form.

33 1.1. Introduction to Affine Toric Varieties 17 Given an affine semigroup S M, the semigroup algebra C[S] is the vector space over C with S as basis and multiplication induced by the semigroup structure of S. To make this precise, we think of M as the character lattice of a torus T N, so that m M gives the character χ m. Then { } C[S] = c m χ m c m C and c m = 0 for all but finitely many m, m S with multiplication induced by χ m χ m = χ m+m. If S = NA for A = {m 1,...,m s }, then C[S] = C[χ m 1,...,χ ms ]. Here are two basic examples. Example The affine semigroup N n Z n gives the polynomial ring C[N n ] = C[x 1,...,x n ], where x i = χ e i and e 1,...,e n is the standard basis of Z n. Example If e 1,...,e n is a basis of a lattice M, then M is generated by A = {±e 1,...,±e n } as an affine semigroup. Setting t i = χ e i gives the Laurent polynomial ring C[M] = C[t ±1 1,...,t±1 n ]. Using Example 1.0.2, one sees that C[M] is the coordinate ring of the torus T N. Affine semigroup rings give rise to affine toric varieties as follows. Proposition Let S M be an affine semigroup. (a) C[S] is an integral domain and finitely generated as a C-algebra. (b) Spec(C[S]) is an affine toric variety whose torus has character lattice ZS, and if S = NA for a finite set A M, then Spec(C[S]) = Y A. Proof. As noted above, A = {m 1,...,m s } implies C[S]=C[χ m 1,...,χ ms ], so C[S] is finitely generated. Since C[S] C[M] follows from S M, we see that C[S] is an integral domain by Example Using A = {m 1,...,m s }, we get the C-algebra homomorphism π : C[x 1,...,x s ] C[M] where x i χ m i C[M]. This corresponds to the morphism Φ A : T N C s from (1.1.4), i.e., π = (Φ A ) in the notation of 1.0. One checks that the kernel of π is the toric ideal I(Y A ) (Exercise 1.1.4). The image of π is C[χ m 1,...,χ ms ] = C[S], and then the coordinate ring of Y A is (1.1.5) C[Y A ] = C[x 1,...,x n ]/I(Y A ) = C[x 1,...,x n ]/Ker(π) Im(π) = C[S].

34 18 Chapter 1. Affine Toric Varieties This proves that Spec(C[S]) = Y A. Since S = NA implies ZS = ZA, the torus of Y A = Spec(C[S]) has the desired character lattice by Proposition Here is an example of this proposition. Example Consider the affine semigroup S Z generated by 2 and 3, so that S = {0,2,3,... }. To study the semigroup algebra C[S], we use (1.1.5). If we set A = {2,3}, then Φ A (t) = (t 2,t 3 ) and the toric ideal is I(Y A ) = x 3 y 2 by Example Hence C[S] = C[t 2,t 3 ] C[x,y]/ x 3 y 2 and the affine toric variety Y A is the curve x 3 = y 2. Equivalence of Constructions. Before stating our main result, we need to study the action of the torus T N on the semigroup algebra C[M]. The action of T N on itself given by multiplication induces an action on C[M] as follows: if t T N and f C[M], then t f C[M] is defined by p f(t 1 p) for p V. The minus sign will be explained in 5.0. The following lemma will be used several times in the text. Lemma Let A C[M] be a subspace stable under the action of T N. Then A = χ m AC χ m. Proof. Let A = χ m A C χm and note that A A. For the opposite inclusion, pick f 0 in A. Since A C[M], we can write f = m B c m χ m, where B M is finite and c m 0 for all m B. Then f B A, where B = Span(χ m m B) C[M]. An easy computation shows that t χ m = χ m (t 1 )χ m. It follows that B and hence B A are stable under the action of T N. Since B A is finite-dimensional, Proposition implies that B A is spanned by simultaneous eigenvectors of T N. This is taking place in C[M], where simultaneous eigenvectors are characters. It follows that B A is spanned by characters. Then the above expression for f B A implies that χ m A for m B. Hence f A, as desired. We can now state the main result of this section, which asserts that our various approaches to affine toric varieties all give the same class of objects. Theorem Let V be an affine variety. The following are equivalent: (a) V is an affine toric variety according to Definition (b) V = Y A for a finite set A in a lattice.