Toric VarietiesDavid CoxJohn LittleHal SchenckDEPARTMENT OF MATHEMATICS, AMHERST COLLEGE, AMHERST, MA01002E-mail address: dac@math.amherst.eduDEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, COLLEGE OFTHE HOLY CROSS, WORCESTER, MA 01610E-mail address: little@mathcs.holycross.eduDEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN, URBANA, IL 61801E-mail address: schenck@math.uiuc.educ (2010, David Cox, John Little and Hal SchenckContentsPreface viiNotation xiiiPart I. Basic Theory of Toric Varieties 1Chapter 1. Afne Toric Varieties 31.0. Background: Afne Varieties 31.1. Introduction to Afne Toric Varieties 101.2. Cones and Afne Toric Varieties 231.3. Properties of Afne Toric Varieties 35Appendix: Tensor Products of Coordinate Rings 48Chapter 2. Projective Toric Varieties 492.0. Background: Projective Varieties 492.1. Lattice Points and Projective Toric Varieties 542.2. Lattice Points and Polytopes 622.3. Polytopes and Projective Toric Varieties 742.4. Properties of Projective Toric Varieties 86Chapter 3. Normal Toric Varieties 933.0. Background: Abstract Varieties 933.1. Fans and Normal Toric Varieties 1053.2. The Orbit-Cone Correspondence 1143.3. Toric Morphisms 1253.4. Complete and Proper 139iiiiv ContentsAppendix: Nonnormal Toric Varieties 150Chapter 4. Divisors on Toric Varieties 1554.0. Background: Valuations, Divisors and Sheaves 1554.1. Weil Divisors on Toric Varieties 1704.2. Cartier Divisors on Toric Varieties 1764.3. The Sheaf of a Torus-Invariant Divisor 189Chapter 5. Homogeneous Coordinates on Toric Varieties 1955.0. Background: Quotients in Algebraic Geometry 1955.1. Quotient Constructions of Toric Varieties 2055.2. The Total Coordinate Ring 2185.3. Sheaves on Toric Varieties 2265.4. Homogenization and Polytopes 231Chapter 6. Line Bundles on Toric Varieties 2456.0. Background: Sheaves and Line Bundles 2456.1. Ample and Basepoint Free Divisors on Complete Toric Varieties 2626.2. Polytopes and Projective Toric Varieties 2776.3. The Nef and Mori Cones 2866.4. The Simplicial Case 298Appendix: Quasicoherent Sheaves on Toric Varieties 309Chapter 7. Projective Toric Morphisms 3137.0. Background: Quasiprojective Varieties and Projective Morphisms 3137.1. Polyhedra and Toric Varieties 3187.2. Projective Morphisms and Toric Varieties 3287.3. Projective Bundles and Toric Varieties 335Appendix: More on Projective Morphisms 345Chapter 8. The Canonical Divisor of a Toric Variety 3478.0. Background: Reexive Sheaves and Differential Forms 3478.1. One-Forms on Toric Varieties 3588.2. Differential Forms on Toric Varieties 3658.3. Fano Toric Varieties 379Chapter 9. Sheaf Cohomology of Toric Varieties 3879.0. Background: Sheaf Cohomology 3879.1. Cohomology of Toric Divisors 398Contents v9.2. Vanishing Theorems I 4099.3. Vanishing Theorems II 4209.4. Applications to Lattice Polytopes 4309.5. Local Cohomology and the Total Coordinate Ring 443Part II. Topics in Toric Geometry 457Chapter 10. Toric Surfaces 45910.1. Singularities of Toric Surfaces and Their Resolutions 45910.2. Continued Fractions and Toric Surfaces 46710.3. Gr obner Fans and McKay Correspondences 48510.4. Smooth Toric Surfaces 49510.5. Riemann-Roch and Lattice Polygons 502Chapter 11. Toric Resolutions and Toric Singularities 51311.1. Resolution of Singularities 51311.2. Other Types of Resolutions 52511.3. Rees Algebras and Multiplier Ideals 53411.4. Toric Singularities 546Chapter 12. The Topology of Toric Varieties 56112.1. The Fundamental Group 56112.2. The Moment Map 56812.3. Singular Cohomology of Toric Varieties 57712.4. The Cohomology Ring 59212.5. The Chow Ring and Intersection Cohomology 612Chapter 13. Toric Hirzebruch-Riemann-Roch 62313.1. Chern Characters, Todd Classes, and HRR 62413.2. Brions Equalities 63213.3. Toric Equivariant Riemann-Roch 64113.4. The Volume Polynomial 65413.5. The Khovanskii-Pukhlikov Theorem 663Appendix: Generalized Gysin Maps 672Chapter 14. Toric GIT and the Secondary Fan 67714.1. Introduction to Toric GIT 67714.2. Toric GIT and Polyhedra 68514.3. Toric GIT and Gale Duality 699vi Contents14.4. The Secondary Fan 712Chapter 15. Geometry of the Secondary Fan 72515.1. The Nef and Moving Cones 72515.2. Gale Duality and Triangulations 73415.3. Crossing a Wall 74715.4. Extremal Contractions and Flips 76215.5. The Toric Minimal Model Program 772Appendix A. The History of Toric Varieties 787A.1. The First Ten Years 787A.2. The Story Since 1980 794Appendix B. Computational Methods 797B.1. The Rational Quartic 798B.2. Polyhedral Computations 799B.3. Normalization and Normaliz 803B.4. Sheaf Cohomology and Resolutions 805B.5. Sheaf Cohomology on the Hirzebruch Surface H2 806B.6. Resolving Singularities 808B.7. Intersection Theory and Hirzebruch-Riemann-Roch 809B.8. Anticanonical Embedding of a Fano Toric Variety 810Appendix C. Spectral Sequences 811C.1. Denitions and Basic Properties 811C.2. Spectral Sequences Appearing in the Text 814Bibliography 817Index 831PrefaceThe study of toric varieties is a wonderful part of algebraic geometry. There areelegant theorems and deep connections with polytopes, polyhedra, combinatorics,commutative algebra, symplectic geometry, and topology. Toric varieties also haveunexpected applications in areas as diverse as physics, coding theory, algebraicstatistics, and geometric modeling. Moreover, as noted by Fulton [105], toricvarieties have provided a remarkably fertile testing ground for general theories.At the same time, the concreteness of toric varieties provides an excellent contextfor someone encountering the powerful techniques of modern algebraic geometryfor the rst time. Our book is an introduction to this rich subject that assumes onlya modest background yet leads to the frontier of this active area of research.Brief Summary. The text covers standard material on toric varieties, including:(a) Convex polyhedral cones, polytopes, and fans.(b) Afne, projective, and abstract toric varieties.(c) Complete toric varieties and proper toric morphisms.(d) Weil and Cartier divisors on toric varieties.(e) Cohomology of sheaves on toric varieties.(f) The classical theory of toric surfaces.(g) The topology of toric varieties.(h) Intersection theory on toric varieties.These topics are discussed in earlier texts on the subject, such as [93], [105] and[218]. One difference is that we provide more details, with numerous examples,gures, and exercises to illustrate the concepts being discussed. We also providebackground material when needed. In addition, we cover a large number of topicspreviously available only in the research literature.viiviii PrefaceThe Fifteen Chapters. To give you a better idea of what is in the book, we nowhighlight a few topics from each chapter.Chapters 1, 2 and 3 cover the basic material mentioned in items (a)(c) above.The toric varieties encountered include: The afne toric variety YA of a nite set A M Zn(Chapter 1). The afne toric variety U of a polyhedral cone NRRn(Chapter 1). The projective toric variety XA of a nite set A M Zn(Chapter 2). The projective toric variety XP of a lattice polytope P MRRn(Chapter 2). The abstract toric variety X of a fan in NRRn(Chapter 3).Chapter 4 introduces Weil and Cartier divisors on toric varieties. We computethe class group and Picard group of a toric variety and dene the sheaf OX(D)associated to a Weil divisor D on a toric variety X.Chapter 5 shows that the classical construction Pn=

Cn+1`0

/Ccan begeneralized to any toric variety X. The homogeneous coordinate ring C[x0, . . . , xn]of Pnalso has a toric generalization, called the total coordinate ring of X.Chapter 6 relates Cartier divisors to invertible sheaves on X. We introduceample, basepoint free, and nef divisors and discuss their relation to convexity. Thestucture of the nef cone and its dual, the Mori cone, are described in detail, as isthe intersection pairing between divisors and curves.Chapter 7 extends the relation between polytopes and projective toric varietiesto a relation between polyhedra and projective toric morphisms : XU. Wealso discuss projective bundles over a toric variety and use these to classify smoothprojective toric varieties of Picard number 2.Chapter 8 relates Weil divisors to reexive sheaves of rank one and denesZariski p-forms. For p = dim X, this gives the canonical sheaf X and canonicaldivisor KX. In the toric case we describe these explicitly and study the relation be-tween reexive polytopes and Gorenstein Fano toric varieties, meaning that KXis ample. We nd the 16 reexive polygons in R2(up to equivalence) and note therelation [PM[ +[PN[ = 12 for a reexive polygon P and its dual P.Chapter 9 is about sheaf cohomology. We give two methods for computingsheaf cohomology on a toric variety and prove a dizzying array of cohomologyvanishing theorems. Applications range from showing that normal toric varietiesare Cohen-Macaulay to the Dehn-Sommerville equations for a simple polytope andcounting lattice points in multiples of a polytope via the Ehrhart polynomial.Chapter 10 studies toric surfaces, where we add a few twists to this classicalsubject. After using Hirzebruch-Jung continued fractions to compute the minimalresolution of a toric surface singularity, we discuss the toric meaning of ordinarycontinued fractions. We then describe unexpected connections with Gr obner fansand the McKay correspondence. Finally, we use the Riemann-Roch theorem on aPreface ixsmooth complete toric surface to explain the mysterious appearance of the number12 in Chapter 8 when counting lattice points in reexive polygons.Chapter 11 proves the existence of toric resolutions of singularities for toricvarieties of all dimensions. This is more complicated than for surfaces because ofthe existence of toric ips and ops. We consider simple normal crossing, crepant,log, and embedded resolutions and study how Rees algebras and multiplier idealscan be applied in the resolution problem. We also discuss toric singularities andshow that a fan is simplicial if and only if X has at worst nite quotient singular-ities and hence is rationally smooth. We also explain what canonical and terminalsingularities mean in the toric context.Chapters 12 and 13 describe the singular and equivariant cohomology of acomplete simplicial toric variety X and prove the Hirzebruch-Riemann-Roch andequivariant Riemann-Roch theorems when X is smooth. We compute the funda-mental group of X and study the moment map, with a brief mention of topologicalmodels of toric varieties and connections with symplectic geometry. We describethe Chow ring and intersection cohomology of a complete simplicial toric variety.After proving Riemann-Roch, we give applications to the volume polynomial andlattice point enumeration in polytopes.Chapters 14 and 15 explore the rich connections that link geometric invarianttheory, the secondary fan, the nef and moving cones, Gale duality, triangulations,wall crossings, ips, extremal contractions, and the toric minimal model program.Appendices. The book ends with three appendices: Appendix A: The History of Toric Varieties. Appendix B: Computational Methods. Appendix C: Spectral Sequences.Appendix A surveys the history of toric geometry since its origins in the early1970s. It is fun to see how the concepts and terminology evolved. Appendix Bdiscusses some of the software packages for toric geometry and gives examplesto illustrate what they can do. Appendix C gives a brief introduction to spectralsequences and describes the spectral sequences used in Chapters 9 and 12.Prerequisites. We assume that the reader is familiar with the material covered inbasic graduate courses in algebra and topology, and to a somewhat lesser degree,complex analysis. In addition, we assume that the reader has had some previousexperience with algebraic geometry, at the level of any of the following texts: Ideals, Varieties and Algorithms by Cox, Little and OShea [69]. Introduction to Algebraic Geometry by Hassett [133]. Elementary Algebraic Geometry by Hulek [151]. Undergraduate Algebraic Geometry by Reid [238].x Preface Computational Algebraic Geometry by Schenck [246]. An Invitation to Algebraic Geometry by Smith, Kahanp a a, Kek al ainen andTraves [253].Chapters 9 and 12 assume knowledge of some basic algebraic topology. The booksby Hatcher [135] and Munkres [210] are useful references here.Readers who have studied more sophisticated algebraic geometry texts such asHarris [130], Hartshorne [131], or Shafarevich [245] certainly have the backgroundneeded to read our book. For readers with a more modest background, an importantprerequisite is a willingness to absorb a lot of algebraic geometry.Background Sections. Since we do not assume a complete knowledge of algebraicgeometry, Chapters 19 each begin with a background section that introduces thedenitions and theorems from algebraic geometry that are needed to understandthe chapter. References where proofs can be found are provided. The remainingchapters do not have background sections. For some of those chapters, no furtherbackground is necessary, while for others, the material is more sophisticated andthe requisite background is given by careful references to the literature.What Is Omitted. We work exclusively with varieties dened over the complexnumbers C. This means that we do not consider toric varieties over arbitrary elds(see [92] for a treatment of this topic), nor do we consider toric stacks (see [39] foran introduction). Moreover, our viewpoint is primarily algebro-geometric. Thus,while we hint at some of the connections with symplectic geometry and topology inChapter 12, we do not do justice to this side of the story. Even within the algebraicgeometry of toric varieties, there are many topics we have had to omit, though weprovide some references that should help readers who want to explore these areas.We have also omitted any discussion of how toric varieties are used in physics andapplied mathematics. Some pointers to the literature are given in our discussion ofthe recent history of toric varieties in A.2 of Appendix A.The Structure of the Text. We number theorems, propositions, and equationsbased on the chapter and the section. Thus 3.2 refers to Section 2 of Chapter 3,and Theorem 3.2.6, equation (3.2.6) and Exercise 3.2.6 all appear in this section.Denitions, theorems, propositions, lemmas, remarks, and examples are numberedtogether in one sequence within each section.Some individual chapters have appendices. Within a chapter appendix thesame numbering system is used, except that the section number is a capital A.This means that Theorem 3.A.3 is in the appendix to Chapter 3. On the other hand,the three appendices at the end of the book are treated in the numbering system aschapters A, B, and C. Thus Denition C.1.1 is in the rst section of Appendix C.The end (or absence) of a proof is indicated by , and the end of an exampleis indicated by .Preface xiFor the Instructor. There is much more material here than you can cover in anyone-semester graduate course, probably more than you can cover in a full yearin most cases. So choices will be necessary depending on the background andthe interests of the student audience. We think it is reasonable to expect to covermost of Chapters 16, 8 and 9 in a one-semester course where the students havea minimal background in algebraic geometry. More material can be covered, ofcourse, if the students know more algebraic geometry. If time permits, you canuse toric surfaces (Chapter 10) to illustrate the power of the basic material andintroduce more advanced topics such as the resolution of singularities (Chapter 11)and the Riemann-Roch theorem (Chapter 13).Finally, we emphasize that the exercises are extremely important. We havefound that when the students work in groups and present their solutions, their en-gagement with the material increases. We encourage instructors to consider usingthis strategy.For the Student. The book assumes that you will be an active reader. This meansin particular that you should do tons of exercisesthis is the best way to learnabout toric varieties. If you have a modest background in algebraic geometry, thenreading the book requires a commitment to learn both toric varieties and algebraicgeometry. It will be a lot of work but is worth the effort. This is a great subject.Send Us Feedback. We greatly appreciate hearing from instructors, students, orgeneral readers about what worked and what didnt. Please notify one or all of usabout any typographical or mathematical errors you might nd.Acknowledgements. We would like to thank to Dan Abramovich, Matt Baker,Matthias Beck, TomBraden, Volker Braun, Sandra Di Rocco, Dan Edidin, MatthiasFranz, Dan Grayson, Paul Hacking, J urgen Hausen, Al Kasprzyk, Diane Maclagan,Anvar Mavlyutov, Uwe Nagel, Andrey Novoseltsev, Sam Payne, Matthieu Ram-baud, Raman Sanyal, Thorsten Rahn, Greg Smith, Bernd Sturmfels, Zach Teitler,Michele Vergne, Mark Walker, G unter Ziegler, and Dylan Zwick for many helpfulconversations and emails as we worked on the book.We would also like to thank all the participants of the 2009 MSRI SummerGraduate Student Workshop on Toric Varieties and especially Dustin Cartwrightand Daniel Erman, our able assistants.Finally we thank Ina Mette for her support and advice and Barbara Beeton forher help with LATEX.September 2010 David CoxJohn LittleHal SchenckNotationThe notation used in the book is organized by topic. The number in parentheses atthe end of an entry indicates the chapter in which the notation rst appears.Basic SetsZ, Q, R, C integers, rational numbers, real numbers, complex numbersN semigroup of nonnegative integers 0, 1, 2, . . . The TorusCmultiplicative group of nonzero complex numbers C` 0 (1)(C)nstandard n-dimensional torus (1)M, mcharacter lattice of a torus and character of m M (1)N, ulattice of one-parameter subgroups of a torus andone-parameter subgroup of u N (1)TN torus N ZC= HomZ(M, C) associated to N and M (1)MR, MQ vector spaces MZR, MZQ built from M (1)NR, NQ vector spaces N ZR, N ZQ built from N (1)'m, u` pairing of m M or MR with u N or NR (1)Hyperplanes and Half-SpacesHm hyperplane in NR dened by 'm, ` = 0, m MR` 0 (1)H+m half-space in NR dened by 'm, ` 0, m MR` 0 (1)Hu,b hyperplane in MR dened by ', u` = b, u NR` 0 (2)H+u,b half-space in MR dened by ', u` b, u NR` 0 (2)xiiixiv NotationConesCone(S) convex cone generated by S (1) rational convex polyhedral cone in NR (1)Span() subspace spanned by (1)dim dimension of (1)dual cone of (1)Relint() relative interior of (1)Int() interior of when Span() = NR (1)set of m MR with 'm, ` = 0 (1) _, is a face or proper face of (1)face of dual to , equals (1)Rays 1-dimensional strongly convex cone (a ray) in NR (1)u minimal generator of N, a rational ray in NR (1)(1) rays of a strongly convex cone in NR (1)LatticesZA lattice generated by A (1)ZA elementssi=1aimi ZA withsi=1ai = 0 (2)N sublattice Z( N) = Span() N (3)N() quotient lattice N/N (3)M() dual lattice of N(), equals M (3)Fans fan in NR (2,3)(r) r-dimensional cones of (3)max maximal cones of (3)Star() star of , a fan in N() (3)() star subdivision of for (3)(v) star subdivision of for v [[ N primitive (11)Polytopes and Polyhedran standard n-simplex in Rn(2)Conv(S) convex hull of S (1)dim P dimension of a polyhedron P (2)Q _P, Q P Q is a face or proper face of P (2)Pdual or polar of a polytope (2)A+B Minkowski sum (2)k P multiple of a polytope or polyhedron (2)Notation xvCones Built From PolyhedraCv Cone(Pv) for a vertex v of a polytope or polyhedron (2)Q cone of a face Q _P in the normal fan P (2)P normal fan of a polytope or polyhedron P (2)C(P) cone over a polytope or polyhedron (1)SP semigroup algebra of C(P) (MZ) (7)Combinatorics and Lattice Points of Polytopesfi number of i-dimesional faces of P (9)hpni=p(1)ip

ip

fi, equals Betti number b2p(XP) when P simple (9)L(P) number of lattice points of a lattice polytope (9)L(P) number of interior lattice points of a lattice polytope (9)EhrP() Ehrhart polynomial of a lattice polytope (9)EhrpP() p-Ehrhart polynomial of a lattice polytope (9)SemigroupsS, C[S] afne semigroup and its semigroup algebra (1)NA afne semigroup generated by A (1)S = S,N afne semigroup M (1)H Hilbert basis of S (1)RingsRf, RS, Rp localization of R at f, a multiplicative set S, a prime ideal p (1)Rintegral closure of the integral domain R (1)R completion of local ring R (1)RCS tensor product of rings over C (1)RGring of invariants of G acting on R (1,5)R[a] Rees algebra of an ideal a R (11)R[]Veronese subring of a graded ring R (14)Specic RingsC[x1, . . . , xn] polynomial ring in n variables (1)C[[x1, . . . , xn]] formal power series ring in n variables (1)C[x11 , . . . , x1n ] ring of Laurent polynomials (1)I(V) ideal of an afne or projective variety (1,2)C[V] coordinate ring of an afne or projective variety (1,2)C[V]d graded piece in degree d when V is projective (2)C(V) eld of rational functions when V is irreducible (1)OV,p, mV,p local ring of a variety at a point and its maximal ideal (1)xvi NotationVarietiesV(I) afne or projective variety of an ideal (1,2)Vf subset of an afne variety V where f = 0 (1)S Zariski closure of S in a variety (1,3)Tp(X) Zariski tangent space of a variety at a point (1,3)dim X, dimpX dimension of a variety and dimension at a point (1,3)Spec(R) afne variety of coordinate ring R (1)Proj(S) projective variety of graded ring S (7)X Y product of varieties (1,3)X SY ber product of varieties (3)X afne cone of a projective variety X (2)Toric VarietiesYA, XA afne and projective toric variety of A M (1,2)U =U,N afne toric variety of a cone NR (1)X = X,N toric variety of a fan in NR (3)XP projective toric variety of a lattice polytope or polyhedron (2.7) lattice homomorphismof a toric morphism : X1 X2 (1,3)R real extension of (1) distinguished point of U (3)O() torus orbit corresponding to (3)V() = O() closure of orbit of , toric variety of Star() (3)UP afne toric variety of recession cone of a polyhedron (7)U afne toric variety of a fan with convex support (7)Specic VarietiesCn, Pnafne and projective n-dimensional space (1,2)P(q0, . . . , qn) weighted projective space (2)Cd, Cd rational normal cone and curve (1,2)Bl0(Cn) blowup of Cnat the origin (3)BlV()(X) blowup of X along V(), toric variety of () (3)Hr Hirzebruch surface (3)Sa,b rational normal scroll (3)Total Coordinate RingS total coordinate ring of X (5)x variable in S corresponding to (1) (5)S graded piece of S in degree Cl(X) (5)deg(x) degree in Cl(X) of a monomial in S (5)Notation xviix monomial/ (1)x for (5)B() irrelevant ideal of S, generated by the x (5)xmLaurent monomialxm,u , m M (5)xm,Dhomogenization of m, m PDM (5)xF facet variable of a facet F P (5)xm,PP-monomial associated to m PM (5)xv,Pvertex monomial associated to vertex v PM (5)M graded S-module (5)M() shift of M by Cl(X) (5)Quotient ConstructionX/G good geometric quotient (5)X//G good categorical quotient (5)Z() exceptional set in quotient construction, equals V(B()) (5)G group in quotient construction, equals HomZ(Cl(X), C) (5)DivisorsOX,D local ring of a variety at a prime divisor (4)D discrete valuation of a prime divisor D (4)div( f ) principal divisor of a rational function (4)D E linear equivalence of divisors (4)D 0 effective divisor (4)Div0(X) group of principal divisors on X (4)Div(X) group of Weil divisors on X (4)CDiv(X) group of Cartier divisors on X (4)Cl(X) divisor class group of a normal variety X (4)Pic(X) Picard group of a normal variety X (4)Pic(X)R Pic(X) ZR (6)Supp(D) support of a divisor (4)D[U restriction of a divisor to an open set (4)(Ui, fi) local data of a Cartier divisor on X (4)[D[ complete linear system of D (6)D, D round down and round up of a Q-divisor (9)Torus-Invariant DivisorsD = O() torus-invariant prime divisor on X of ray (1) (4)DF torus-invariant prime divisor on XP of facet F P (4)DivTN(X) group of torus-invariant Weil divisors on X (4)CDivTN(X) group of torus-invariant Cartier divisors on X (4)m Cartier data of a torus-invariant Cartier divisor on X (4)xviii NotationDP Cartier divisor of a polytope or polyhedron (4,7)PD polyhedron of a torus-invariant divisor (4)XD toric variety of a basepoint free divisor (6)D fan of XD (6)D pullback of a Cartier divisor (6)Support FunctionsD support function of a Cartier divisor (4)P support function of a polytope or polyhedron (4)SF() support functions for (4)SF(, N) support functions for integral with respect to N (4)Sheaves(U, F) sections of a sheaf over an open set (4)F[U restriction of a sheaf to an open set (4)Fp stalk of a sheaf at a point (6)F OXG tensor product of sheaves of OX-modules (6)HomOX(F, G ) sheaf of homomorphisms (6)Fdual sheaf of F, equals HomOX(F, OX) (6)fF direct image sheafSpecic SheavesOX structure sheaf of a variety X (3)OX sheaf of invertible elements of OX (4)KX constant sheaf of rational functions for X irreducible (6)OX(D) sheaf of a Weil divisor D on X (4)IY ideal sheaf of a subvariety Y X (3)M sheaf on Spec(R) of an R-module M (4)M sheaf on X of the graded S-modules M (5)OX() sheaf of the S-module S() (5)Vector Bundles and Locally Free SheavesL, E invertible sheaf (line bundle) and locally free sheaf (6) : V X vector bundle (6) : VL X rank 1 vector bundle of an invertible sheaf L (6)fL pullback of an invertible sheaf (6)L,W map to projective space determined by W (X, L) (6)P(V), P(E ) projective bundle of vector bundle or locally free sheaf (7)D fan for rank 1 vector bundle VL for L =OX(D) (7)Notation xixIntersection Theorydeg(D) degree of a divisor on a smooth complete curve (6)D C intersection product of Cartier divisor and complete curve (6)D D, C Cnumerically equivalent Cartier divisors and complete curves (6)N1(X), N1(X) (CDiv(X)/) ZR and (proper 1-cycles on X/) ZR (6)Nef(X) cone in N1(X) generated by nef divisors (6)Mov(X) moving cone of a variety X in N1(X) (15)Eff(X) pseudoeffective cone of a variety X in N1(X) (15)NE(X) cone in N1(X) generated by complete curves (6)NE(X) Mori cone, equals the closure of NE(X) (6)Differential Forms and SheavesR/C module of K ahler differentials of a C-algebra R (8)1X, TX cotangent and tangent sheaves of a variety X (8)IY/I2Y , NY/X conormal and normal sheaves of Y X (8)pX, pX sheaves of p-forms and Zariski p-forms on X (8)KX, X canonical divisor and canonical sheaf nX, n = dim X (8)1X(logD) sheaf of 1-forms with logarithmic poles on D (8)Sheaf CohomologyH0(X, F) global sections (X, F) of a sheaf F on X (9)Hp(X, F) p-th sheaf cohomology group of a sheaf F on X (9)RpfF higher direct image sheaf (9)ExtpOX(G, F) Ext groups of sheaves of OX-modules G, F (9)C(U , F) Cech complex for sheaf cohomology (9)(F) Euler characteristic of F, equalsp(1)pdim Hp(X, F) (9)Sheaf Cohomology of a Toric VarietyHp(X, L)m graded piece of sheaf cohomology of L =OX(D) for m M (9)VD,m, VsuppD,m subsets of [[ used to compute Hp(X, L)m (9)Local CohomologyHpI (M) p-th local cohomology of an R-module M for the ideal I R (9)C(f, M) Cech complex for local cohomology when I ='f` (9)ExtpR(N, M) Ext groups of R-modules N, M (9)Resolution of SingularitiesXsing singular locus of a variety (11)Exc() exceptional locus of a resolution of singularities (11)xx Notation.(c I) multiplier ideal sheaf (11)(X, D) log pair, D =iaiDi, ai [0, 1] Q (11)Singularities of Toric Varietiesmult() multiplicity of a simplicial cone, equals [N : Zu1+ +Zud] (6,11)P parallelotope of a simplicial cone, equals iiui[ 0 i < 1 (11) polytope related to canonical and terminal singularities of U (11) the polyhedron Conv( N ` 0) (10, 11)can fan over bounded faces of , reduces to canonical singularities (11)Topology of a Toric VarietyN sublattice of N generated by [[ N (12)1(X) fundamental group of X, isomorphic to N/N (12)SN real torus N ZS1= HomZ(M, S1) (S1)n(12)(X)0 nonnegative real points of a toric variety (12)f , algebraic and symplectic moment maps XPMR (12) symplectic moment map C(1)Cl(X)R (12)Singular Homology and CohomologyHi(X, R) ith singular cohomology of X with coefcients in a ring R (9)Hi(X, R) i-th reduced cohomology of X (9)Hic(X, R) ith cohomology of X with compact supports (12)Hi(X, R) ith singular homology of X (12)HBMi (X, R) ith Borel-Moore homology of X (13)bi(X) ith Betti number of X, equals dim Hi(X, Q) (12)e(X) Euler characteristic of X, equalsi(1)ibi(X) (9,10,12), cap and cup products (12)H(X, R) cohomology ringpHp(X, R) under cup product (12)[W] cohomology class of a subvariety W in H2n2k(X, Q) (12,13)[W]r rened cohomology class of W in H2n2k(X, X `W, Q) (12,13)f! generalized Gysin map (13)

X integral

X : H(X, Q) Q, equals Gysin map of X pt (12,13)Equivariant Cohomology for a Group ActionEG a contractible space on which G acts freely (12)BG the quotient EG/G (12)EGGX quotient of EGX modulo relation (e g, x) (e, g x) (12)HG(X, R) equivariant cohomology ring, equals H(EGGX, R) (12)G, (G)Q integral and rational equivariant cohomology ring of a point (12)XGxed point set for action of G on X (12)Notation xxiEquivariant Cohomology for a Torus ActionSymZ(M) symmetric algebra of M over Z (12)SymQ(M) rational symmetric algebra on M, equals SymZ(M) ZQ (12)s isomorphisms : SymQ(M) (T)Q (12)[D]T equivariant cohomology class of a T-invariant divisor D (12)

Xeq equivariant integral

Xeq : HT(X, Q) (T)Q (13)HT(X, Q) completionk=0HkT(X, Q) of equivariant cohomology of X (13) completion of the equivariant cohomology of a point (13)Chow Groups and the Chow RingAk(X) Chow group of k-cycles modulo rational equivalence (12)Ak(X) codimension k cycles modulo rational equivalence (12)A(X) integral Chow ring of X smooth and complete (12)A(X)Q rational Chow ring of X quasismooth and complete (12)Intersection CohomologyIHpi (X) ith intersection homology of X for perversity p (12)IHi(X) ith intersection cohomology of X for middle perversity (12)IHi(X)Q ith rational intersection cohomology of X (12)Cohomology Ring of a Complete Simplicial Toric VarietyI Stanley-Reisner ideal of the fan , ideal in Q[x1, . . . , xr] (12). ideal 'ri=1'm, ui`xi[ m M` Q[x1, . . . , xr] (12)RQ() Jurkiewicz-Danilov ring Q[x1, . . . , xr]/(I +.) H(X, Q) (12)SRQ() Stanley-Reisner ring Q[x1, . . . , xr]/I HT(X, Q) (12)Hirzebruch-Riemann-Rochci(E ) ith Chern class of a locally free sheaf E (13)ch(L) Chern character of a line bundle L (13)Td(X) Todd class of the variety X (13)Bk kth Bernoulli number (13)ci = ci(TX) ith Chern class of the tangent bundle (13)Ti ith Todd polynomial in the ci (13)K(X) Grothendieck group of classes of coherent sheaves on X (13)T(L) equivariant Euler characteristic (13)T(L) local contribution of (n) to T(L) (13)chT(L) equivariant Chern character of L (13)TdT(X) equivariant Todd class of X (13)Todd(x) formal Todd differential operator for the variable x (13)xxii NotationBrions EqualitiesZ[M] integral semigroup algebra of M (13)Z[[M]] formal semigroup module of M, formal sumsmMamm(13)Z[[M]]Sum summable elements in Z[[M]] (13)o( f ) sum of an element f Z[[M]]Sum (13) (L)mMni=0dimHi(X, L)mm Z[[M]] (13)Geometric Invariant TheoryG character group of algebraic subgroup G (C)r(14)L sheaf of sections of rank 1 line bundle on Crfor character G (14)(Cr)ss, (Cr)s semistable and stable points for (14)R graded ringd=0(Cr, Ld )G(14)Cr//G GIT quotient of Crby G for , equals Proj(R) = (Cr)ss//G (14)B() irrelevant ideal of (14)Z() exceptional set of , equals V(B()) (14)P, Pa polyhedra in Rrand MR for = a(14)Fi,, Fi,a ith virtual facet of P, Pa (14)The Secondary Fan, lists of r vectors in GR and NR (14)C, C cones generated by and (14),I, ,I GKZ cones determined by , I (14)B(, I) irrelevant ideal determined by , I (14)GKZ secondary fan of (14)MovGKZ moving cone of the secondary fan (15)PGKZ secondary polytope, normal fan is GKZ (15)Toric Minimal Model Program1 extremal ray of the Mori cone (15)D 1 intersection product D C for [C] 1` 0 (15)fD birational transform of a divisor by a birational map (15)J, J+ index sets determined by a wall relation (15), + fans determined by a wall relation (15), + toric morphisms determined by a wall relation (15)Miscellaneousd multiplicative group of dth roots of unity in C (1)[a1, . . . , as] ordinary continued fraction of a rational number (10)[[b1, . . . , br]] Hirzebruch-Jung continued fraction of a rational number (10)Part I. Basic Theory ofToric VarietiesChapters 1 to 9 introduce the theory of toric varieties. This part of thebook assumes only a minimal amount of algebraic geometry, at the levelof Ideals, Varieties and Algorithms [69]. Each chapter begins with a back-ground section that develops the necessary algebraic geometry.1Chapter 1Afne Toric Varieties1.0. Background: Afne VarietiesWe begin with the algebraic geometry needed for our study of afne toric varieties.Our discussion assumes Chapters 15 and 9 of [69].Coordinate Rings. An ideal I S =C[x1, . . . , xn] gives an afne varietyV(I) =p Cn[ f (p) = 0 for all f Iand an afne variety V Cngives the idealI(V) =f S [ f (p) = 0 for all p V.By the Hilbert basis theorem, an afne variety V is dened by the vanishing ofnitely many polynomials in S, and for any ideal I, the Nullstellensatz tells us thatI(V(I)) =I = f S [ f I for some 1 since C is algebraically closed.The most important algebraic object associated to V is its coordinate ringC[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 compatiblewith 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) : V1V2 between afne varietiescorrespond to C-algebra homomorphisms : C[V2] C[V1], where (g) =g for g C[V2]. Two afne varieties are isomorphic if and only if their coordinate rings areisomorphic C-algebras.34 Chapter 1. Afne Toric Varieties A point p of an afne variety V gives the maximal idealf C[V] [ f (p) = 0 C[V],and all maximal ideals of C[V] arise this way.Coordinate rings of afne varieties can be characterized as follows (Exercise 1.0.1).Lemma 1.0.1. A C-algebra R is isomorphic to the coordinate ring of an afnevariety if and only if R is a nitely generated C-algebra with no nonzero nilpotents,i.e., if f R satises f= 0 for some 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 ofC[V] via the fourth bullet above. More generally, one can take any commutativering R and dene the afne scheme Spec(R). The general denition of Spec usesall prime ideals of R, not just the maximal ideals as we have done. Thus someauthors would write (1.0.1) as V = Specm(C[V]), the maximal spectrum of C[V].Readers wishing to learn about afne schemes should consult [90] and [131].The Zariski Topology. An afne variety V Cnhas two topologies we will use.The rst is the classical topology, induced from the usual topology on Cn. Thesecond is the Zariski topology, where the Zariski closed sets are subvarieties of V(meaning afne varieties of Cncontained in V) and the Zariski open sets are theircomplements. Since subvarieties are closed in the classical topology (polynomialsare continuous), Zariski open subsets are open in the classical topology.Given a subset S V, its closure S in the Zariski topology is the smallestsubvariety of V containing S. We call S the Zariski closure of S. It is easy to giveexamples where this differs from the closure in the classical topology.Afne Open Subsets and Localization. Some Zariski open subsets of an afnevariety V are themselves afne varieties. Given f C[V] `0, letVf =p V [ f (p) = 0 V.Then Vf is Zariski open in V and is also an afne variety, as we now explain.Let V Cnhave I(V) =' f1, . . . , fs` and pick g C[x1, . . . , xn] representing f .Then Vf =V `V(g) is Zariski open in V. Now consider a new variable y and letW = V( f1, . . . , fs, 1gy) CnC. Since the projection map CnC CnmapsW bijectively onto Vf, we can identify Vf with the afne variety W CnC.When V is irreducible, the coordinate ring of Vf is easy to describe. Let C(V)be the eld 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 C(V) [ g C[V], 0.1.0. Background: Afne Varieties 5In Exercise 1.0.3 you will prove that Spec(C[V]f) is the afne variety Vf.Example 1.0.2. The n-dimensional torus is the afne open subset(C)n=Cn`V(x1 xn) Cn,with coordinate ringC[x1, . . . , xn]x1xn =C[x11 , . . . , x1n ].Elements of this ring are called Laurent polynomials. The ring C[V]f from (1.0.2) is an example of localization. In Exercises 1.0.2and 1.0.3 you will show how to construct this ring for all afne varieties, not justirreducible ones. The general concept of localization is discussed in standard textsin commutative algebra such as [10, Ch. 3] and [89, Ch. 2].Normal Afne Varieties. Let R be an integral domain with eld of fractions K.Then R is normal, or integrally closed, if every element of K which is integral overR (meaning that it is a root of a monic polynomial in R[x]) actually lies in R. Forexample, any UFD is normal (Exercise 1.0.5).Denition 1.0.3. An irreducible afne variety V is normal if its coordinate ringC[V] is normal.For example, Cnis normal since its coordinate ring C[x1, . . . , xn] is a UFD andhence normal. Here is an example of a nonnormal afne variety.Example 1.0.4. Let C =V(x3y2) C2. This is an irreducible plane curve with acusp at the origin. It is easy to see that C[C] =C[x, y]/'x3y2`. Now let x and y bethe cosets of x and y in C[C] respectively. This gives y/ x C(C). A computationshows that y/ x / C[C] and that ( y/ x)2= x. Consequently C[C] and hence C are notnormal. We will see below that C is an afne toric variety. An irreducible afne variety V has a normalization dened as follows. LetC[V]= C(V) : 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) nitely generated as a C-algebra (see [89, Cor. 13.13]). Thisgives the normal afne varietyV= Spec(C[V])We call Vthe normalization of V. The natural inclusion C[V] C[V]= C[V]corresponds to a map VV. This is the normalization map.Example 1.0.5. We saw in Example 1.0.4 that the curve C C2dened by x3=y2has elements x, y C[C] such that y/ x / C[C] is integral over C[C]. In Exer-cise 1.0.6 you will show that C[ y/ x] C(C) is the integral closure of C[C] andthat the normalization map is the map C C dened by t (t2, t3). 6 Chapter 1. Afne Toric VarietiesAt rst glance, the denition of normal does not seem very intuitive. Once weenter the world of toric varieties, however, we will see that normality has a verynice combinatorial interpretation and that the nicest toric varieties are the normalones. We will also see that normality leads to a nice theory of divisors.In Exercise 1.0.7 you will prove some properties of normal domains that willbe used in 1.3 when we study normal afne toric varieties.Smooth Points of Afne Varieties. In order to dene a smooth point of an afnevariety V, we rst need to dene local rings and Zariski tangent spaces. When Vis irreducible, the local ring of V at p isOV,p =f /g C(V) [ f , g C[V] and g(p) = 0.Thus OV,p consists of all rational functions on V that are dened at p. Inside ofOV,p we have the maximal idealmV,p = OV,p[ (p) = 0.In fact, mV,p is the unique maximal ideal of OV,p, so that OV,p is a local ring.Exercises 1.0.2 and 1.0.4 explain how to dene OV,p when V is not irreducible.The Zariski tangent space of V at p is dened to beTp(V) = HomC(mV,p/m2V,p, C).In Exercise 1.0.8 you will verify that dim Tp(Cn) = n for every p Cn. Accordingto [131, p. 32], we can compute the Zariski tangent space of a point in an afnevariety as follows.Lemma 1.0.6. Let V Cnbe an afne variety and let p V. Also assume thatI(V) =' f1, . . . , fs` C[x1, . . . , xn]. For each i, letdp( fi) = fix1(p)x1 + + fixn(p)xn.Then the Zariski tangent space Tp(V) is isomorphic to the subspace of Cndenedby the equations dp( f1) = = dp( fs) = 0. In particular, dim Tp(V) n. Denition 1.0.7. A point p of an afne variety V is smooth or nonsingular ifdim Tp(V) = dimpV, where dimpV is the maximum of the dimensions of the irre-ducible 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.Points lying in the intersection of two or more irreducible components of V arealways singular (see [69, Thm. 8 of Ch. 9, 6]).Since dim Tp(Cn) = n for every p Cn, we see that Cnis smooth. For anirreducible afne variety V Cnof dimension d, x p V and write I(V) =1.0. Background: Afne Varieties 7' f1, . . . , fs`. Using Lemma 1.0.6, it is straightforward to show that V is smoothat p if and only if the Jacobian matrix(1.0.3) Jp( f1, . . . , fs) =

fixj(p)

1is,1jnhas rank nd (Exercise 1.0.9). Here is a simple example.Example 1.0.8. As noted in Example 1.0.4, the plane curve C dened by x3= y2has I(C) ='x3y2` C[x, y]. A point p = (a, b) C has JacobianJp = (3a2, 2b),so the origin is the only singular point of C. Since Tp(V) =HomC(mV,p/m2V,p, C), we see that V is smooth at p when dimVequals the dimension of mV,p/m2V,p as a vector space over OV,p/mV,p. In terms ofcommutative algebra, this means that p V is smooth if and only if OV,p is aregular local ring. See [10, p. 123] or [89, 10.3].We can relate smoothness and normality as follows.Proposition 1.0.9. A smooth irreducible afne variety V is normal.Proof. In 3.0 we will see that C[V] =pVOV,p. By Exercise 1.0.7, C[V] isnormal once we prove that OV,p is normal for all p V. Hence it sufces to showthat OV,p is normal whenever p is smooth.This follows from some powerful results in commutative algebra: OV,p is aregular local ring when p is a smooth point of V (see above), and every regularlocal ring is a UFD (see [89, Thm. 19.19]). Then we are done since every UFD isnormal. A direct proof that OV,p is normal at a smooth point p V is sketched inExercise 1.0.10. The converse of Propostion 1.0.9 can fail. We will see in 1.3 that the afnevariety V(xy zw) C4is normal, yet V(xy zw) is singular at the origin.Products of Afne Varieties. Given afne varieties V1 and V2, there are severalways to show that the cartesian product V1V2 is an afne variety. The most directway is to proceed as follows. Let V1 Cm= Spec(C[x1, . . . , xm]) and V2 Cn=Spec(C[y1, . . . , yn]). Take I(V1) =' f1, . . . , fs` and I(V2) ='g1, . . . , gt`. Since the fiand gj depend on separate sets of variables, it follows thatV1V2 = V( f1, . . . , fs, g1, . . . , gt) Cm+nis an afne variety.A fancier method is to use the mapping properties of the product. This willalso give an intrinsic description of its coordinate ring. Given V1 and V2 as above,8 Chapter 1. Afne Toric VarietiesV1V2 should be an afne variety with projections i : V1V2 Vi such thatwhenever we have a diagramW 1

2

V1V21

2

V1V2where i : W Vi are morphisms from an afne variety W, there should be a uniquemorphism (the dotted arrow) that makes the diagram commute, i.e., i = i.For the coordinate rings, this means that whenever we have a diagramC[V2]2

2C[V1]1

1 C[V1V2]

C[W]with C-algebra homomorphisms i : C[Vi] C[W], there should be a unique C-algebra homomorphism (the dotted arrow) that makes the diagram commute. Bythe universal mapping property of the tensor product of C-algebras, C[V1] CC[V2]has the mapping properties we want. Since C[V1] CC[V2] is a nitely generatedC-algebra with no nilpotents (see the appendix to this chapter), it is the coordinatering C[V1V2]. For more on tensor products, see [10, pp. 2427] or [89, A2.2].Example 1.0.10. Let V be an afne variety. Since Cn= Spec(C[y1, . . . , yn]), theproduct V Cnhas coordinate ringC[V] CC[y1, . . . , yn] =C[V][y1, . . . , yn].If V is contained in Cmwith I(V) =' f1, . . . , fs` C[x1, . . . , xm], it follows thatI(V Cn) =' f1, . . . , fs` C[x1, . . . , xm, y1, . . . , yn].For later purposes, we also note that the coordinate ring of V (C)nisC[V] CC[y11 , . . . , y1n ] =C[V][y11 , . . . , y1n ]. Given afne varieties V1 and V2, we note that the Zariski topology on V1V2is usually not the product of the Zariski topologies on V1 and V2.Example 1.0.11. Consider C2= CC. By denition, a basis for the product ofthe Zariski topologies consists of sets U1U2 where Ui are Zariski open in C. Sucha set is the complement of a union of collections of horizontal and vertical lines1.0. Background: Afne Varieties 9in C2. This makes it easy to see that Zariski closed sets in C2such as V(y x2)cannot be closed in the product topology. Exercises for 1.0.1.0.1. Prove Lemma 1.0.1. Hint: You will need the Nullstellensatz.1.0.2. Let R be a commutative C-algebra. A subset S R is a multipliciative subset pro-vided 1 S, 0 / S, and S is closed under multiplication. The localization RS consists of allformal expressions g/s, g R, s S, modulo the equivalence relationg/s h/t u(tg sh) = 0 for some u S.(a) Show that the usual formulas for adding and multiplying fractions induce well-denedbinary operations that make RS into C-algebra.(b) If R has no nonzero nilpotents, then prove that the same is true for RS.For more on localization, see [10, Ch. 3] or [89, Ch. 2].1.0.3. Let R be a nitely generated C-algebra without nilpotents as in Lemma 1.0.1 andlet f R be nonzero. Then S =1, f , f2, . . . is a multiplicative set. The localization RS isdenoted Rf and is called the localization of R at f .(a) Show that Rf is a nitely generated C-algebra without nilpotents.(b) Show that Rf satises Spec(Rf) = Spec(R)f.(c) Show that Rf is given by (1.0.2) when R is an integral domain.1.0.4. Let V be an afne variety with coordinate ring C[V]. Given a point p V, letS =g C[V] [ g(p) = 0.(a) Show that S is a multiplicative set. The localization C[V]S is denoted OV,p and iscalled the local ring of V at p.(b) Show that every OV,p has a well-dened value (p) and thatmV,p = OV,p[ (p) = 0is the unique maximal ideal of OV,p.(c) When V is irreducible, show that OV,p agrees with the denition given in the text.1.0.5. Prove that a UFD is normal.1.0.6. In the setting of Example 1.0.5, show that C[ y/ x] C(C) is the integral closure ofC[C] and that the normalization C C is dened by t (t2, t3).1.0.7. In this exercise, you will prove some properties of normal domains needed for 1.3.(a) Let R be a normal domain with eld of fractions K and let S R be a multiplicativesubset. Prove that the localization RS is normal.(b) Let R, A, be normal domains with the same eld of fractions K. Prove that theintersectionAR is normal.1.0.8. Prove that dim Tp(Cn) = n for all p Cn.1.0.9. Use Lemma 1.0.6 to prove the claim made in the text that smoothness is determinedby the rank of the Jacobian matrix (1.0.3).10 Chapter 1. Afne Toric Varieties1.0.10. Let V be irreducible and suppose that p V is smooth. The goal of this exerciseis to prove that OV,p is normal using standard results from commutative algebra. Set n =dimV and consider the ring of formal power series C[[x1, . . . , xn]]. This is a local ring withmaximal ideal m ='x1, . . . , xn`. We will use three facts: C[[x1, . . . , xn]] is a UFD by [280, p. 148] and hence normal by Exercise 1.0.5. Since p V is smooth, [207, 1C] proves the existence of a C-algebra homomorphismOV,pC[[x1, . . . , xn]] that induces isomorphismsOV,p/mV,pC[[x1, . . . , xn]]/mfor all 0. This implies that the completion (see [10, Ch. 10])OV,p = limOV,p/mV,pis isomorphic to a formal power series ring, i.e., OV,p C[[x1, . . . , xn]]. Such an iso-morphism captures the intuitive idea that at a smooth point, functions should havepower series expansions in local coordinates x1, . . . , xn. If I OV,p is an ideal, thenI ==1(I +mV,p).This theorem of Krull holds for any ideal I in a Noetherian local ring A and followsfrom [10, Cor. 10.19] with M = A/I.Now assume that p V is smooth.(a) Use the third bullet to show that OV,pC[[x1, . . . , xn]] is injective.(b) Suppose that a, b OV,p satisfy b[a in C[[x1, . . . , xn]]. Prove that b[a in OV,p. Hint:Use the second bullet to show a bOV,p+mV,p and then use the third bullet.(c) Prove that OV,p is normal. Hint: Use part (b) and the rst bullet.This argument can be continued to show that OV,p is a UFD. See [207, (1.28)]1.0.11. Let V andW be afne varieties and let S V be a subset. Prove that SW =S W.1.0.12. Let V and W be irreducible afne varieties. Prove that V W is irreducible. Hint:Suppose V W =Z1Z2, where Z1, Z2 are closed. Let Vi =v V [ vW Zi. Provethat V =V1V2 and that Vi is closed in V. Exercise 1.0.11 will be useful.1.1. Introduction to Afne Toric VarietiesWe rst discuss what we mean by torus and then explore various constructionsof afne toric varieties.The Torus. The afne variety (C)nis a group under component-wise multiplica-tion. A torus T is an afne variety isomorphic to (C)n, where T inherits a groupstructure from the isomorphism.The term torus is taken from the language of linear algebraic groups. Wewill use (without proof) basic results about tori that can be found in standard textson algebraic groups such as [37], [152], and [256]. See also [36, Ch. 3] for aself-contained treatment of tori.1.1. Introduction to Afne Toric Varieties 11We begin with characters and one-parameter subgroups.A character of a torus T is a morphism : T Cthat is a group homo-morphism. For example, m = (a1, . . . , an) Zngives a character m: (C)nCdened by(1.1.1) m(t1, . . . , tn) = ta11 tann .One can show that all characters of (C)narise this way (see [152, 16]). Thus thecharacters of (C)nform a group isomorphic to Zn.For an arbitrary torus T, its characters form a free abelian group M of rankequal to the dimension of T. It is customary to say that m M gives the characterm: T C.We will need the following result concerning tori (see [152, 16] for a proof).Proposition 1.1.1.(a) Let T1 and T2 be tori and let : T1T2 be a morphism that is a group homo-morphism. Then the image of is a torus and is closed in T2.(b) Let T be a torus and let H T be an irreducible subvariety of T that is asubgroup. Then H is a torus. Assume that a torus T acts linearly on a nite dimensional vector space W overC, where the action of t T on w W is denoted t w. A basic result is that thelinear maps w t w are diagonalizable and can be simultaneously diagonalized.We describe this as follows. Given m M, dene the eigenspaceWm =w W [ t w = m(t)w for all t T.If Wm= 0, then every w Wm`0 is a simultaneous eigenvector for all t T,with eigenvalue given by m(t). See [256, Thm. 3.2.3] for a proof of the following.Proposition 1.1.2. In the above situation, we have W =mMWm. A one-parameter subgroup of a torus T is a morphism : CT that is agroup homomorphism. For example, u = (b1, . . . , bn) Zngives a one-parametersubgroup u: C(C)ndened by(1.1.2) u(t) = (tb1, . . . , tbn).All one-parameter subgroups of (C)narise this way (see [152, 16]). It followsthat the group of one-parameter subgroups of (C)nis naturally isomorphic to Zn.For an arbitrary torus T, the one-parameter subgroups form a free abelian group Nof rank equal to the dimension of T. As with the character group, an element u Ngives the one-parameter subgroup u: CT.There is a natural bilinear pairing ' , ` : MN Z dened as follows. (Intrinsic) Given a character mand a one-parameter subgroup u, the com-position mu: CCis a character of C, which is given by t tforsome Z. Then 'm, u` = .12 Chapter 1. Afne Toric Varieties (Concrete) If T =(C)nwith m=(a1, . . . , an) Zn, u =(b1, . . . , bn) Zn, thenone computes that(1.1.3) 'm, u` =ni=1aibi,i.e., the pairing is the usual dot product.It follows that the characters and one-parameter subgroups of a torus T formfree abelian groups M and N of nite rank with a pairing ' , ` : MN Z thatidenties N with HomZ(M, Z) and M with HomZ(N, Z). In terms of tensor prod-ucts, one obtains a canonical isomorphism N ZCT via ut u(t). Henceit is customary to write a torus as TN.From this point of view, picking an isomorphism TN (C)ninduces dualbases of M and N, i.e., isomorphisms M Znand N Znthat turn charactersinto Laurent monomials (1.1.1), one-parameter subgroups into monomial curves(1.1.2), and the pairing into dot product (1.1.3).The Denition of Afne Toric Variety. We now dene the main object of study ofthis chapter.Denition 1.1.3. An afne toric variety is an irreducible afne variety V contain-ing a torus TN (C)nas a Zariski open subset such that the action of TN on itselfextends to an algebraic action of TN on V. (By algebraic action, we mean an actionTNV V given by a morphism.)Obvious examples of afne toric varieties are (C)nand Cn. Here are someless trivial examples.Example 1.1.4. The plane curve C = V(x3y2) C2has a cusp at the origin.This is an afne toric variety with torusC`0 =C(C)2=(t2, t3) [ t C C,where the isomorphism is t (t2, t3). Example 1.0.4 shows that C is a nonnormaltoric variety. Example 1.1.5. The variety V = V(xy zw) C4is a toric variety with torusV (C)4=(t1, t2, t3, t1t2t13 ) [ ti C (C)3,where the isomorphism is (t1, t2, t3) (t1, t2, t3, t1t2t13 ). We will see later that V isnormal. Example 1.1.6. Consider the surface in Cd+1parametrized by the map : C2Cd+1dened by (s, t) (sd, sd1t, . . . , std1, td). Thus is dened using all degree dmonomials in s, t.1.1. Introduction to Afne Toric Varieties 13Let the coordinates of Cd+1be x0, . . . , xd and let I C[x0, . . . , xd] be the idealgenerated by the 22 minors of the matrix

x0 x1 xd2 xd1x1 x2 xd1 xd

,so I = 'xixj+1xi+1xj[ 0 i < j d 1`. In Exercise 1.1.1 you will verify that(C2) = V(I), so that Cd = (C2) is an afne variety. You will also prove thatI(Cd) = I, so that I is the ideal of all polynomials vanishing on Cd. It follows that Iis prime since V(I) is irreducible by Proposition 1.1.8 below. The afne surface Cdis called the rational normal cone of degree d and is an example of a determinantalvariety. We will see below that I is a toric ideal.It is straightforward to show that Cd is a toric variety with torus((C)2) = Cd(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 afne toric varieties.Lattice Points. In this book, a lattice is a free abelian group of nite rank. Thusa lattice of rank n is isomorphic to Zn. For example, a torus TN has lattices M (ofcharacters) and N (of one-parameter subgroups).Given a torus TN with character lattice M, a set A = m1, . . . , ms M givescharacters mi: TN C. Then consider the map(1.1.4) A : TN Csdened byA(t) =

m1(t), . . . , ms(t)

Cs.Denition 1.1.7. Given a nite set A M, the afne toric variety YA is denedto be the Zariski closure of the image of the map A from (1.1.4).This denition is justied by the following proposition.Proposition 1.1.8. Given A M as above, let ZA M be the sublattice gener-ated by A. Then YA is an afne toric variety whose torus has character latticeZA. In particular, the dimension of YA is the rank of ZA.Proof. The map (1.1.4) can be regarded as a mapA : TN (C)sof tori. By Proposition 1.1.1, the image T = A(TN) is a torus that is closed in(C)s. The latter implies that YA (C)s= T since YA is the Zariski closure ofthe image. It follows that the image is Zariski open in YA. Furthermore, T isirreducible (it is a torus), so the same is true for its Zariski closure YA.14 Chapter 1. Afne Toric VarietiesWe next consider the action of T. Since T (C)s, an element t T acts onCsand takes varieties to varieties. ThenT = t T t YAshows that t YA is a variety containing T. Hence YA t YA by the denition ofZariski closure. Replacing t with t1leads to YA = t YA, so that the action of Tinduces an action on YA. We conclude that YA is an afne toric variety.It remains to compute the character lattice of T, which we will temporarilydenote by M. Since T = A(TN), the map A gives the commutative diagramTNA

(C)sT?

where denotes a surjective map and an injective map. This diagram of toriinduces a commutative diagram of character latticesM ZsbA

M.0P

Since A : ZsM takes the standard basis e1, . . . , es to m1, . . . , ms, the image ofA is ZA. By the diagram, we obtain M ZA. Then we are done since thedimension of a torus equals the rank of its character lattice. In concrete terms, x a basis of M, so that we may assume M =Zn. Then the svectors in A Zncan be regarded as the columns of an ns matrix A with integerentries. In this case, the dimension of YA is simply the rank of the matrix A.We will see below that every afne toric variety is isomorphic to YA for somenite subset A of a lattice.Toric Ideals. Let YA Cs= Spec(C[x1, . . . , xs]) be the afne toric variety com-ing from a nite set A = m1, . . . , ms M. We can describe the ideal I(YA) C[x1, . . . , xs] as follows. As in the proof of Proposition 1.1.8, A induces a map ofcharacter latticesA : ZsMthat sends the standard basis e1, . . . , es to m1, . . . , ms. Let L be the kernel of thismap, so that we have an exact sequence0 L ZsM.In down to earth terms, elements = (1, . . . , s) of L satisfy si=1imi = 0 andhence record the linear relations among the mi.1.1. Introduction to Afne Toric Varieties 15Given = (1, . . . , s) L, set+ =i>0iei and =i0xii i onC[x1, . . . , xs] and an isomorphism TN (C)n. Thus we may assume M = Znandthe map : (C)nCsis given by Laurent monomials tmiin variables t1, . . . , tn.If IL = I(YA), then we can pick f I(YA) ` IL with minimal leading monomialx=si=1xaii . Rescaling if necessary, xbecomes the leading term of f .Since f (tm1, . . . , tms) is identically zero as a polynomial in t1, . . . , tn, there mustbe cancellation involving the term coming from x. In other words, f must containa monomial x=si=1xbii < xsuch thatsi=1(tmi)ai=si=1(tmi)bi.This implies thatsi=1aimi =si=1bimi,so that =si=1(aibi)ei L. Then xx IL by the second descriptionof IL. It follows that f x+xalso lies in I(YA) ` IL and has strictly smallerleading term. This contradiction completes the proof. Given A M, there are several ways to compute the ideal I(YA) = IL ofProposition 1.1.9. In simple cases, the rational implicitization algorithm of [69,Ch. 3, 3] can be used. One can also compute IL using a basis of L and idealquotients (Exercise 1.1.3). For more on computing IL, see [264, Ch. 12].Inspired by Proposition 1.1.9, we make the following denition.Denition 1.1.10. Let L Zsbe a sublattice.(a) The ideal IL =

xx[ , Nsand L

is called a lattice ideal.(b) A prime lattice ideal is called a toric ideal.16 Chapter 1. Afne Toric VarietiesSince toric varieties are irreducible, the ideals appearing in Proposition 1.1.9are toric ideals. Examples of toric ideals include:Example 1.1.4 : 'x3y2` C[x, y]Example 1.1.5 : 'xz yw` C[x, y, z, w]Example 1.1.6 : 'xixj+1xi+1xj[ 0 i < j d 1` C[x0, . . . , xd].(The latter is the ideal of the rational normal cone Cd Cd+1.) In each example,we have a prime ideal generated by binomials. As we now show, such ideals areautomatically toric.Proposition 1.1.11. An ideal I C[x1, . . . , xs] is toric if and only if it is prime andgenerated by binomials.Proof. One direction is obvious. So suppose that I is prime and generated by bino-mials xixi. Then observe that V(I) (C)sis nonempty (it contains (1, . . . , 1))and is a subgroup of (C)s(easy to check). Since V(I) Csis irreducible, it fol-lows that V(I) (C)sis an irreducible subvariety of (C)sthat is also a subgroup.By Proposition 1.1.1, we see that T = V(I) (C)sis a torus.Projecting on the ith coordinate of (C)sgives a character T (C)sC,which by our usual convention we write as mi: T Cfor mi M. It followseasily that V(I) = YA for A = m1, . . . , ms, and since I is prime, we have I =I(YA) by the Nullstellensatz. Then I is toric by Proposition 1.1.9. We will later see that all afne toric varieties arise from toric ideals. For moreon toric ideals and lattice ideals, the reader should consult [204, Ch. 7].Afne Semigroups. A semigroup is a set S with an associative binary operationand an identity element. To be an afne 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 nite set A S givesNA =mAamm [ am NS. The semigroup is nitely generated, meaning that there is a nite set A Ssuch that NA = S. The semigroup can be embedded in a lattice M.The simplest example of an afne semigroup is Nn Zn. More generally, givena lattice M and a nite set A M, we get the afne semigroup NA M. Up toisomorphism, all afne semigroups are of this form.Given an afne semigroup S M, the semigroup algebra C[S] is the vectorspace over C with S as basis and multiplication induced by the semigroup structure1.1. Introduction to Afne Toric Varieties 17of S. To make this precise, we think of M as the character lattice of a torus TN, sothat m M gives the character m. ThenC[S] =

mScmm[ cm C and cm = 0 for all but nitely many m,with multiplication induced bym m= m+m.If S =NA for A =m1, . . . , ms, then C[S] =C[m1, . . . , ms].Here are two basic examples.Example 1.1.12. The afne semigroup NnZngives the polynomial ringC[Nn] =C[x1, . . . , xn],where xi = eiand e1, . . . , en is the standard basis of Zn. Example 1.1.13. If e1, . . . , en is a basis of a lattice M, then M is generated byA = e1, . . . , en as an afne semigroup. Setting ti = eigives the Laurentpolynomial ringC[M] =C[t11 , . . . , t1n ].Using Example 1.0.2, one sees that C[M] is the coordinate ring of the torus TN. Afne semigroup rings give rise to afne toric varieties as follows.Proposition 1.1.14. Let S M be an afne semigroup. Then:(a) C[S] is an integral domain and nitely generated as a C-algebra.(b) Spec(C[S]) is an afne toric variety whose torus has character lattice ZS, andif S =NA for a nite set A M, then Spec(C[S]) =YA.Proof. As noted above, A =m1, . . . , ms implies C[S] =C[m1, . . . , ms], so C[S]is nitely generated. Since C[S] C[M] follows from S M, we see that C[S] isan integral domain by Example 1.1.13.Using A =m1, . . . , ms, we get the C-algebra homomorphism : C[x1, . . . , xs] C[M]where ximi C[M]. This corresponds to the morphismA : TN Csfrom (1.1.4), i.e., = (A)in the notation of 1.0. One checks that the kernelof is the toric ideal I(YA) (Exercise 1.1.4). The image of is C[m1, . . . , ms] =C[S], and then the coordinate ring of YA is(1.1.5) C[YA] =C[x1, . . . , xs]/I(YA)=C[x1, . . . , xs]/Ker() Im() =C[S].18 Chapter 1. Afne Toric VarietiesThis proves that Spec(C[S]) =YA. Since S =NA implies ZS =ZA, the torus ofYA = Spec(C[S]) has the desired character lattice by Proposition 1.1.8. Here is an example of this proposition.Example 1.1.15. Consider the afne semigroup S Z generated by 2 and 3, sothat S = 0, 2, 3, . . . . To study the semigroup algebra C[S], we use (1.1.5). If weset A = 2, 3, then A(t) = (t2, t3) and the toric ideal is I(YA) = 'x3y2` byExample 1.1.4. HenceC[S] =C[t2, t3] C[x, y]/'x3y2`and the afne toric variety YA is the curve x3= y2from Example 1.1.4. Equivalence of Constructions. Before stating our main result, we need to studythe action of the torus TN on the semigroup algebra C[M]. The action of TN onitself given by multiplication induces an action on C[M] as follows: if t TN andf C[M], then t f C[M] is dened by p f (t1 p) for p TN. The minussign will be explained in 5.0.The following lemma will be used several times in the text.Lemma 1.1.16. Let A C[M] be a subspace stable under the action of TN. ThenA =

mAC m.Proof. Let A=mAC mand note that A A. For the opposite inclusion,pick f = 0 in A. Since A C[M], we can writef =mBcmm,where BM is nite and cm= 0 for all m B. Then f BA, whereB = Span(m[ m B) C[M].An easy computation shows that t m= m(t1)m. It follows that B andhence BA are stable under the action of TN. Since BA is nite-dimensional,Proposition 1.1.2 implies that BA is spanned by simultaneous eigenvectors of TN.This is taking place in C[M], where simultaneous eigenvectors are characters. Itfollows that BAis spanned by characters. Then the above expression for f BAimplies that m A for m B. Hence f A, as desired. We can now state the main result of this section, which asserts that our variousapproaches to afne toric varieties all give the same class of objects.Theorem 1.1.17. Let V be an afne variety. The following are equivalent:(a) V is an afne toric variety according to Denition 1.1.3.(b) V =YA for a nite set A in a lattice.1.1. Introduction to Afne Toric Varieties 19(c) V is an afne variety dened by a toric ideal.(d) V = Spec(C[S]) for an afne semigroup S.Proof. The implications (b) (c) (d) (a) follow from Propositions 1.1.8,1.1.9 and 1.1.14. For (a) (d), let V be an afne toric variety containing thetorus TN with character lattice M. Since the coordinate ring of TN is the semigroupalgebra C[M], the inclusion TNV induces a map of coordinate ringsC[V] C[M].This map is injective since TN is Zariski dense in V, so that we can regard C[V] asa subalgebra of C[M].Since the action of TN on V is given by a morphism TNV V, we see thatif t TN and f C[V], then p f (t1 p) is a morphism on V. It follows thatC[V] C[M] is stable under the action of TN. By Lemma 1.1.16, we obtainC[V] =

mC[V]C m.Hence C[V] =C[S] for the semigroup S =m M [ m C[V].Finally, since C[V] is nitely generated, we can nd f1, . . . , fs C[V] withC[V] = C[ f1, . . . , fs]. Expressing each fi in terms of characters as above gives anite generating set of S. It follows that S is an afne semigroup. Here is one way to think about the above proof. When an irreducible afnevariety V contains a torus TN as a Zariski open subset, we have the inclusionC[V] C[M].Thus C[V] consists of those functions on the torus TN that extend to polynomialfunctions on V. Then the key insight is that V is a toric variety precisely when thefunctions that extend are determined by the characters that extend.Example 1.1.18. We have seen that V = V(xy zw) C4is a toric variety withtoric ideal 'xyzw` C[x, y, z, w]. Also, the torus is (C)3via the map (t1, t2, t3) (t1, t2, t3, t1t2t13 ). The lattice points used in this map can be represented as thecolumns of the matrix(1.1.6)

1 0 0 10 1 0 10 0 1 1

.The corresponding semigroup S Z3consists of the N-linear combinations of thecolumn vectors. Hence the elements of S are lattice points lying in the polyhe-dral region in R3pictured in Figure 1 on the next page. In this gure, the fourvectors generating S are shown in bold, and the boundary of the polyhedral regionis partially shaded. In the terminology of 1.2, this polyhedral region is a rational20 Chapter 1. Afne Toric Varieties(0,0,1)(0,1,0)(1,0,0)(1,1,1)Figure 1. Cone containing the lattice points corresponding to V =V(xy zw)polyhedral cone. In Exercise 1.1.5 you will show that S consists of all lattice pointslying in the cone in Figure 1. We will use this in 1.3 to prove that V is normal. Exercises for 1.1.1.1.1. As in Example 1.1.6, letI ='xixj+1xi+1xj[ 0 i < j d 1` C[x0, . . . , xd]and let Cd be the surface parametrized by(s, t) = (sd, sd1t, . . . , std1, td) Cd+1.(a) Prove that V(I) = (C2) Cd+1. Thus Cd = V(I).(b) Prove that I(Cd) is homogeneous.(c) Consider lex monomial order with x0 > x1 > > xd. Let f I(Cd) be homogeneousof degree and let r be the remainder of f on division by the generators of I. Provethat r can be writtenr = h0(x0, x1) +h1(x1, x2) + +hd1(xd1, xd)where hi is homogeneous of degree . Also explain why we may assume that thecoefcient of xi in hi is zero for 1 i d 1.(d) Use part (c) and r(sd, sd1t, . . . , std1, td) = 0 to show that r = 0.(e) Use parts (b), (c) and (d) to prove that I = I(Cd). Also explain why the generators ofI are a Gr obner basis for the above lex order.1.1.2. Let L Zsbe a sublattice. Prove that'x+x[ L` ='xx[ , Ns, L`.Note that when L, the vectors +, Nshave disjoint support (i.e., no coordinate ispositive in both), while this may fail for arbitrary , Nswith L.1.1. Introduction to Afne Toric Varieties 211.1.3. Let IL be a toric ideal and let 1, . . . , rbe a basis of the sublattice L Zs. DeneIL ='xi+xi[ i = 1, . . . , r`.Prove that IL = IL: 'x1 xs`. Hint: Given , Nswith L, write =ri=1aii, ai Z. This impliesx1 =ai >0

xi+xi

aiai 0 be irra-tional and consider the semigroupS =(a, b) N2[ b a Z2.Prove that S is not nitely generated. (The generators of S are related to continued frac-tions. For example, when = (1 +5)/2 is the golden ratio, the minimal generators ofS are (0, 1) and (F2n, F2n+1) for n = 1, 2, . . . , where Fn is the nth Fibonacci number. See[231] and [259]. Continued fractions will play an important role in Chapter 10.1.2. Cones and Afne Toric Varieties 231.1.15. Suppose that : M M is a group isomorphism. Fix a nite set A M andlet B = (A ). Prove that the toric varieties YA and YB are equivariantly isomorphic(meaning that the isomorphism respects the torus action).1.2. Cones and Afne Toric VarietiesWe begin with a brief discussion of rational polyhedral cones and then explain howthey relate to afne toric varieties.Convex Polyhedral Cones. Fix a pair of dual vector spaces MR and NR. Our dis-cussion of cones will omit most proofswe refer the reader to [105] for moredetails and [218, App. A.1] for careful statements. See also [51, 128, 241].Denition 1.2.1. A convex polyhedral cone in NR is a set of the form = Cone(S) =

uSuu [ u0NR,where S NR is nite. We say that is generated by S. Also set Cone() =0.A convex polyhedral cone is in fact convex, meaning that x, y impliesx +(1 )y for all 0 1, and is a cone, meaning that x impliesx for all 0. Since we will only consider convex cones, the cones satisfyingDenition 1.2.1 will be called simply polyhedral cones.Examples of polyhedral cones include the rst quadrant in R2or rst octant inR3. For another example, the cone Cone(e1, e2, e1 +e3, e2 +e3) R3is picturedin Figure 2. It is also possible to have cones that contain entire lines. For example,zyxFigure 2. Cone in R3generated by e1, e2, e1 +e3, e2 +e324 Chapter 1. Afne Toric VarietiesCone(e1, e1) R2is the x-axis, while Cone(e1, e1, e2) is the closed upper half-plane (x, y) R2[ y 0. As we will see below, these last two examples are notstrongly convex.We can also create cones using polytopes, which are dened as follows.Denition 1.2.2. A polytope in NR is a set of the formP = Conv(S) =

uSuu [ u0,uSu = 1NR,where S NR is nite. We say that P is the convex hull of S.Polytopes include all polygons in R2and bounded polyhedra in R3. As we willsee in later chapters, polytopes play a prominent role in the theory of toric varieties.Here, however, we simply observe that a polytope P NR gives a polyhedral coneC(P) NRR, called the cone of P and dened byC(P) = (u, 1) NRR[ u P, 0.If P = Conv(S), then we can also describe this as C(P) = Cone(S1). Figure 3shows what this looks when P is a pentagon in the plane.PFigure 3. The cone C(P) of a pentagon P R2The dimension dim of a polyhedral cone is the dimension of the smallestsubspace W = Span() of NR containing . We call Span() the span of .Dual Cones and Faces. As usual, the pairing between MR and NR is denoted ' , `.Denition 1.2.3. Given a polyhedral cone NR, its dual cone is dened by=m MR[ 'm, u` 0 for all u .Duality has the following important properties.Proposition 1.2.4. Let NR be a polyhedral cone. Then is a polyhedral conein MR and ()= . 1.2. Cones and Afne Toric Varieties 25Given m = 0 in MR, we get the hyperplaneHm =u NR[ 'm, u` = 0 NRand the closed half-spaceH+m =u NR[ 'm, u` 0 NR.Then Hm is a supporting hyperplane of a polyhedral cone NR if H+m ,and H+m is a supporting half-space. Note that Hm is a supporting hyperplane of if and only if m ` 0. Furthermore, if m1, . . . , ms generate , then it isstraightforward to check that(1.2.1) = H+m1 H+ms.Thus every polyhedral cone is an intersection of nitely many closed half-spaces.We can use supporting hyperplanes and half-spaces to dene faces of a cone.Denition 1.2.5. A face of a cone of the polyhedral cone is =Hm for somem , written _ . Using m = 0 shows that is a face of itself, i.e., _ .Faces = are called proper faces, written .The faces of a polyhedral cone have the following obvious properties.Lemma 1.2.6. Let = Cone(S) be a polyhedral cone. Then:(a) Every face of is a polyhedral cone.(b) An intersection of two faces of is again a face of .(c) A face of a face of is again a face of . You will prove the following useful property of faces in Exercise 1.2.1.Lemma 1.2.7. Let be a face of a polyhedral cone . If v, w and v +w ,then v, w . A facet of is a face of codimension 1, i.e., dim =dim 1. An edge of is a face of dimension 1. In Figure 4 on the next page we illustrate a 3-dimensionalcone with shaded facets and a supporting hyperplane (a plane in this case) that cutsout the vertical edge of the cone.Here are some properties of facets.Proposition 1.2.8. Let NRRnbe a polyhedral cone. Then:(a) If = H+m1 H+ms for mi , 1 i s, then = Cone(m1, . . . , ms).(b) If dim = n, then in (a) we can assume that the facets of are i = Hmi.(c) Every proper face is the intersection of the facets of containing . Note how part (b) of the proposition renes (1.2.1) when dim = dim NR.26 Chapter 1. Afne Toric Varietiessupportinghyperplane HHFigure 4. A cone R3and a hyperplane H supporting an edge HWhen working in Rn, dot product allows us to identify the dual with Rn. Fromthis point of view, the vectors m1, . . . , ms in part (a) of the proposition are facetnormals, i.e., perpendicular to the facets. This makes it easy to compute examples.Example 1.2.9. It is easy to see that the facet normals to the cone R3inFigure 2 are m1 = e1, m2 = e2, m3 = e3, m4 = e1+e2e3. Hence= Cone(e1, e2, e3, e1+e2e3) R3.This is the cone of Figure 1 at the end of 1.1 whose lattice points describe thesemigroup of the afne toric variety V(xy zw) (see Example 1.1.18). As we willsee, this is part of how cones describe normal afne toric varieties.Now consider , which is the cone in Figure 1. The reader can check that thefacet normals of this cone are e1, e2, e1 +e3, e2 +e3. Using duality and part (a) ofProposition 1.2.8, we obtain = ()= Cone(e1, e2, e1+e3, e2+e3).Hence we recover our original description of . See also Example B.2.1. In this example, facets of the cone correspond to edges of its dual. More gen-erally, given a face _ NR, we dene=m MR[ 'm, u` = 0 for all u =m [ 'm, u` = 0 for all u = .We call the dual face of because of the following proposition.Proposition 1.2.10. If is a face of a polyhedral cone and = , then:(a) is a face of .1.2. Cones and Afne Toric Varieties 27(b) The map is a bijective inclusion-reversing correspondence between thefaces of and the faces of .(c) dim +dim= n. Here is an example of Proposition 1.2.10 when dim < dim NR.Example 1.2.11. Let = Cone(e1, e2) R3. Figure 5 shows and . YouxyzxyzFigure 5. A 2-dimensional cone R3and its dual R3should check that the maximal face of , namely itself, gives the minimal faceof , namely the z-axis. Note also thatdim +dim= 3even though has dimension 2. Relative Interiors. As already noted, the span of a cone NR is the smallestsubspace of NR containing . Then the relative interior of , denoted Relint(), isthe interior of in its span. Exercise 1.2.2 will characterize Relint() as follows:u Relint() 'm, u` > 0 for all m `.When the span equals NR, the relative interior is just the interior, denoted Int().For an example of how relative interiors arise naturally, suppose that _ .This gives the dual face = of . Furthermore, if m , then oneeasily sees thatm Hm.In Exercise 1.2.2, you will show that if m , thenm Relint() = Hm.Thus the relative interior Relint() tells us exactly which supporting hyperplanesof cut out the face .28 Chapter 1. Afne Toric VarietiesStrong Convexity. Of the cones shown in Figures 15, all but in Figure 5 havethe nice property that the origin is a face. Such cones are called strongly convex.This condition can be stated several ways.Proposition 1.2.12. Let NRRnbe a polyhedral cone. Then: is strongly convex 0 is a face of contains no positive-dimensional subspace of NR () =0 dim = n. You will prove Proposition 1.2.12 in Exercise 1.2.3. One corollary is that if apolyhedral cone is strongly convex of maximal dimension, then so is . Thecones pictured in Figures 14 satisfy this condition.In general, a polyhedral cone always has a minimal face that is the largestsubspace W contained in . Furthermore: W = (). W = Hm whenever m Relint(). = /W NR/W is a strongly convex polyhedral cone.See Exercise 1.2.4.Separation. When two cones intersect in a face of each, we can separate the coneswith the following result, often called the separation lemma.Lemma 1.2.13 (Separation Lemma). Let 1, 2 be polyhedral cones in NR thatmeet along a common face = 12. Then = Hm1 = Hm2for any m Relint(1 (2)).Proof. Given A, B NR, we set AB =ab [ a A, b B. A standard resultfrom cone theory tells us that1 (2)= (12).Now x m Relint(1 (2)). The above equation and Exercise 1.2.4 implythat Hm cuts out the minimal face of 12, i.e.,Hm(12) = (12) (21).However, we also have(12) (21) = .One inclusion is obvious since = 12. For the other inclusion, write u (12) (21) asu = a1a2 = b2b1, a1, b1 1, a2, b2 2.1.2. Cones and Afne Toric Varieties 29Then a1+b1 = a2+b2 implies that this element lies in = 12. Since a1, b11, we have a1, b1 by Lemma 1.2.7, and a2, b2 follows similarly. Henceu = a1a2 , as desired.We conclude that Hm(12) = . Intersecting with 1, we obtainHm1 = ( ) 1 = ,where the last equality again uses Lemma 1.2.7 (Exercise 1.2.5). If instead weintersect with 2, we obtainHm(2) = ( ) (2) =,and Hm2 = follows. In the situation of Lemma 1.2.13 we call Hm a separating hyperplane.Rational Polyhedral Cones. Let N and M be dual lattices with associated vectorspaces NR = N ZR and MR = MZR. For Rnwe usually use the lattice Zn.Denition 1.2.14. A polyhedral cone NR is rational if = Cone(S) for somenite set S N.The cones appearing in Figures 1, 2 and 5 are rational. We note without proofthat faces and duals of rational polyhedral cones are rational. Furthermore, if =Cone(S) for S N nite and NQ = N ZQ, then(1.2.2) NQ =uSuu [ u0 in Q.One new feature is that a strongly convex rational polyedral cone has acanonical generating set, constructed as follows. Let be an edge of . Since isstrongly convex, is a ray, i.e., a half-line, and since is rational, the semigroupN is generated by a unique element u N. We call u the ray generator of. Figure 6 shows the ray generator of a rational ray in the plane. The dots arethe lattice N =Z2and the white ones are N. ray generator uFigure 6. A rational ray R2and its unique ray generator uLemma 1.2.15. A strongly convex rational polyhedral cone is generated by the raygenerators of its edges. 30 Chapter 1. Afne Toric VarietiesIt is customary to call the ray generators of the edges the minimal generatorsof a strongly convex rational polyhedral cone. Figures 1 and 2 show 3-dimensionalstrongly convex rational polyhedral cones and their ray generators.In a similar way, a rational polyhedral cone of maximal dimension has uniquefacet normals, which are the ray generators of the dual , which is strongly con-vex by Proposition 1.2.12.Here are some especially important strongly convex cones.Denition 1.2.16. Let NR be a strongly convex rational polyhedral cone.(a) is smooth or regular if its minimal generators form part of a Z-basis of N,(b) is simplicial if its minimal generators are linearly independent over R.The cone pictured in Figure 5 is smooth, while those in Figures 1 and 2 arenot even simplicial. Note also that the dual of a smooth (resp. simplicial) cone ofmaximal dimension is again smooth (resp. simplicial). Later in the section we willgive examples of simplicial cones that are not smooth.Semigroup Algebras and Afne Toric Varieties. Given a rational polyhedral cone NR, the lattice pointsS = M Mform a semigroup. A key fact is that this semigroup is nitely generated.Proposition 1.2.17 (Gordans Lemma). S = M is nitely generated andhence is an afne semigroup.Proof. Since is rational polyhedral, =Cone(T) for a nite set T M. ThenK = mT mm [ 0 m < 1 is a bounded region of MRRn, so that K M isnite since M Zn. Note that T (KM) S.We claim T (KM) generates S as a semigroup. To prove this, take w Sand write w =mT mm where m 0. Then m = m +m with m Nand 0 m < 1, so thatw =mTmm+mTmm.The second sum is in K M (remember w M). It follows that w is a nonnegativeinteger combination of elements of T (KM). Since afne semigroups give afne toric varieties, we get the following.Theorem 1.2.18. Let NR Rnbe a rational polyhedral cone with semigroupS = M. ThenU = Spec(C[S]) = Spec(C[M])is an afne toric variety. Furthermore,dimU = n the torus of U is TN = NZC is strongly convex.1.2. Cones and Afne Toric Varieties 31Proof. By Gordans Lemma and Proposition 1.1.14, U is an afne toric varietywhose torus has character lattice ZSM. To study ZS, note thatZS = SS =m1m2[ m1, m2 S.Now suppose that km ZS for some k > 1 and m M. Then km = m1m2 form1, m2 S = M. Since m1 and m2 lie in the convex set , we havem+m2 = 1km1+ k1k m2 .It follows that m = (m+m2) m2 ZS, so that M/ZS is torsion-free. Hence(1.2.3) the torus of U is TN ZS = M rankZS = n.Since is strongly convex if and only if dim =n (Proposition 1.2.12), it remainsto show thatdimU = n rankZS = n dim= n.The rst equivalence follows since the dimension of an afne toric variety is thedimension of its torus, which is the rank of its character lattice. We leave the proofof the second equivalence to the reader (Exercise 1.2.6). Remark 1.2.19.(a) For the rest of the book, we will always assume that NR is strongly convexsince we want TN to be the torus of the afne toric variety U.(b) The reader may ask why we focus on NR since U = Spec(C[M])makes MR seem more important. The answer will become clear once weunderstand how normal toric varieties are constructed from afne pieces. Thediscussion following Proposition 1.3.16 gives a rst hint of how this works.Example 1.2.20. Let = Cone(e1, e2, e1 +e3, e2 +e3) NR = R3with N = Z3.This is the cone pictured in Figure 2. By Example 1.2.9, is the cone picturedin Figure 1, and by Example 1.1.18, the lattice points in this cone are generatedby columns of matrix (1.1.6). It follows from Example 1.1.18 that U is the afnetoric variety V(xy zw). Example 1.2.21. Fix 0 r n and set = Cone(e1, . . . , er) Rn. Then= Cone(e1, . . . , er, er+1, . . . , en)and the corresponding afne toric variety isU = Spec(C[x1, . . . , xr, x1r+1, . . . , x1n ]) =Cr(C)nr(Exercise 1.2.7). This implies that if NRRnis a smooth cone of dimension r,then UCr(C)nr. Figure 5 from Example 1.2.11 shows r =2 and n =3. Example 1.2.22. Fix a positive integer d and let = Cone(de1e2, e2) R2.This has dual cone = Cone(e1, e1+de2). Figure 7 on the next page shows when d = 4. The afne semigroup S = Z2is generated by the lattice points32 Chapter 1. Afne Toric Varieties(1, i) for 0 i d. When d = 4, these are the white dots in Figure 7. (You willprove these assertions in Exercise 1.2.8.)Figure 7. The cone when d =4By 1.1, the afne toric variety U is the Zariski closure of the image of themap : (C)2Cd+1dened by(s, t) = (s, st, st2, . . . , std).This map has the same image as the map (s, t) (sd, sd1t, . . . , std1, td) used inExample 1.1.6. Thus U is isomorphic to the rational normal cone Cd Cd+1whose ideal is generated by the 22 minors of the matrix

x0 x1 xd2 xd1x1 x2 xd1 xd

.Note that the cones and are simplicial but not smooth. We will return to this example often. One thing evident in Example 1.1.6 is thedifference between cone generators and semigroup generators: the cone hastwo generators but the semigroup S = Z2has d +1.When NR has maximal dimension, the semigroup S = M has aunique minimal generating set constructed as follows. Dene an element m= 0 ofS to be irreducible if m = m+mfor m, m S implies m= 0 or m= 0.Proposition 1.2.23. Let NR be a strongly convex rational polyhedral cone ofmaximal dimension and let S = M. ThenH =m S[ m is irreduciblehas the following properties:(a) H is nite and generates S.(b) H contains the ray generators of the edges of .(c) H is the minimal generating set of S with respect to inclusion.1.2. Cones and Afne Toric Varieties 33Proof. Proposition 1.2.12 implies that is strongly convex, so we can nd anelement u N `0 such that 'm, u` N for all m S and 'm, u` = 0 if andonly if m = 0.Now suppose that m S is not irreducible. Then m = m+mwhere mandmare nonzero elements of S. It follows that'm, u` ='m, u` +'m, u`with 'm, u`, 'm, u` N`0, so that'm, u` 1, we can nd abasis e1, . . . , en of N with u = e1 (Exercise 1.3.5). This allows us to assume that = Cone(e1), so thatC[S] =C[x1, x12 , . . . , x1n ]by Example 1.2.21. But C[x1, . . . , xn] is normal (it is a UFD), so its localizationC[x1, . . . , xn]x2xn =C[x1, x12 , . . . , x1n ]is also normal by Exercise 1.0.7. This completes the proof. Example 1.3.6. We saw in Example 1.2.20 that V = V(xy zw) is the afne toricvariety U of the cone = Cone(e1, e2, e1+e3, e2+e3) pictured in Figure 2. ThenTheorem 1.3.5 implies that V is normal, as claimed in Example 1.1.5. Example 1.3.7. By Example 1.2.22, the rational normal cone Cd Cd+1is theafne toric variety of a strongly convex rational polyhedral cone and hence is nor-mal by Theorem 1.3.5.It is instructive to view this example using the parametrizationA(s, t) = (sd, sd1t, . . . , std1, td)from Example 1.1.6. Plotting the lattice points in A for d = 2 gives the whitesquares in Figure 9 (a) below. These generate the semigroup S = NA, and theproof of Theorem 1.3.5 gives the cone = Cone(e1, e2), which is the rst quad-rant in the gure. At rst glance, something seems wrong. The afne variety C2is normal, yet in Figure 9 (a) the semigroup generated by the white squares missessome lattice points in . This semigroup does not look saturated. How can theafne toric variety be normal?(a)(b)Figure 9. Lattice points for the rational normal cone bC2The problem is that we are using the wrong lattice! Proposition 1.1.8 tells usto use the lattice ZA, which gives the white dots and squares in Figure 9 (b). This1.3. Properties of Afne Toric Varieties 39gure shows that the white squares generate the semigroup of lattice points in .Hence S is saturated and everything is ne. This example points out the importance of working with the correct lattice.The Normalization of an Afne Toric Variety. The normalization of an afne toricvariety is easy to describe. Let V = Spec(C[S]) for an afne semigroup S, so thatthe torus of V has character lattice M = ZS. Let Cone(S) denote the cone of anynite generating set of S and set = Cone(S) NR. In Exercise 1.3.6 you willprove the following.Proposition 1.3.8. The above cone is a strongly convex rational polyhedral conein NR and the inclusion C[S] C[M] induces a morphism UV th