WIXLIB

NotationxivUσ Uσ,NXΣ XΣ,NXPXDφ, φRγσO(σ)V (σ) O(σ)Dρ O(ρ)DFUPUΣaffine toric variety of a cone σ NRtoric variety of a fanprojective toric variety of a lattice polytope or polyhedrontoric variety of a basepoint free divisorlattice homomorphism of a toric morphism φ : XΣ1 XΣ2and its real extensiondistinguished point of Uσorbit of σ Σclosure of orbit of σ Σ, toric variety of Star(σ)torus-invariant prime divisor on XΣ of ρ Σ(1)torus-invariant prime divisor on XP of facet F Paffine toric variety of recession cone of a polyhedronaffine toric variety of a fan with convex supportSpecific VarietiesCn , P nP(q0 , . . . , qn )C (C )nCbd , CdBl0 (Cn )BlV (τ ) (XΣ )HrSa,baffine and projective n-dimensional spaceweighted projective spacemultiplicative group of nonzero complex numbers C \ {0}standard n-dimensional torusrational normal cone and curveblowup of Cn at the originblowup of XΣ along V (τ ), toric variety of Σ (τ )Hirzebruch surfacerational normal scrollDivisorsOX,DνDdiv( f )D ED 0Div0 (X )Div(X )DivTN (XΣ )CDiv(X )local ring of a variety at a prime divisordiscrete valuation of a prime divisor Dprincipal divisor of a rational functionlinear equivalence of divisorseffective divisorgroup of principal divisors on Xgroup of Weil divisors on Xgroup of torus-invariant Weil divisors on XΣgroup of Cartier divisors on X

NotationCDivTN (XΣ )Cl(X )Pic(X )Supp(D)D U{(Ui , fi )}{mσ }σ ΣPDΣDDPϕDSF(Σ, N)xvgroup of torus-invariant Cartier divisors on XΣdivisor class group of a normal variety XPicard group of a normal variety Xsupport of a divisorrestriction of a divisor to an open setlocal data of a Cartier divisor on XCartier data of a torus-invariant Cartier divisor on XΣpolyhedron of a torus-invariant divisorfan associated to a basepoint free divisorCartier divisor of a polytope or polyhedronsupport function of a Cartier divisorsupport functions integral with respect to NIntersection Productsdeg(D)D ·CD D′, C C′N 1 (X ), N1 (X )Nef (X )NE(X )NE(X )Pic(X )Rr(P)degree of a divisor on a curveintersection product of Cartier divisor and complete curvenumerically equivalent Cartier divisors and complete curves(CDiv(X )/ ) Z R and (Z1 (X )/ ) Z Rcone in N 1 (X ) generated by nef divisorscone in N1 (X ) generated by complete curvesMori cone, equal to the closure of NE(X )Pic(X ) Z Rprimitive relation of a primitive collectionSheaves and BundlesOXOX OX (D)KXF UΓ(U , F )IYeMeMOXΣ (α)structure sheaf of a variety Xsheaf of invertible elements of OXsheaf of a Weil divisor D on Xconstant sheaf of rational functions when X is irreduciblerestriction of a sheaf to an open setsections of a sheaf over an open setideal sheaf of a subvariety Y Xsheaf on Spec(R) of an R-module Msheaf on XΣ of a graded S-module Msheaf on XΣ of the graded S-module S(α)

NotationxviFpF OX GH omOX (F , G )F π :V Xπ : VL Xf LφL ,W D Σ DP(V ), P(E )stalk of a sheaf at a pointtensor product of sheaves of OX -modulessheaf of homomorphismsdual sheaf of F , equal to H omOX (F , OX )vector bundlerank 1 vector bundle of a line bundle Lpullback of a line bundlemap to projective space determined by W Γ(X , L )complete linear system of Dfan that gives VL for L OXΣ (D)projective bundle of vector bundle or locally free sheafQuotients and Homogeneous CoordinatesRGV /GV //GSxρSβdeg(x α )x σ̂B(Σ)Z(Σ)Gxhmixhm,DixFxhm,Pixhv,PiMM(α)ring of invariants of G acting on Rgood geometric quotientgood categorical quotienttotal coordinate ring of XΣvariable in S corresponding to ρ Σ(1)graded piece of S in degree β Cl(XΣ )degree in Cl(XΣ ) of a monomial in Smonomial generator of B corresponding to σ Σirrelevant ideal of S, generated by the x σ̂exceptional set, equal to V(B(Σ))group HomZ (Cl(XΣ ), C ) used in the quotient constructionQ hm,u iLaurent monomial ρ xρ ρ , m Mhomogenization of χ m , m PD Mfacet variable of a facet F PP-monomial associated to m P Mvertex monomial associated to vertex v P Mgraded S-moduleshift of M by α Cl(XΣ )

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 [35]. Each chapter begins with a background section that develops the necessary algebraic geometry.1

Chapter 1Affine Toric Varieties§1.0. Background: Affine VarietiesWe 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[x1 , . . . , xn ] gives an affine varietyV(I) {p Cn f (p) 0 for all f I}and an affine variety V Cn gives the idealI(V ) { f S f (p) 0 for all p V }.By the Hilbert Basis Theorem, an affine variety V is defined by the vanishing offinitely many polynomials in S, and for any ideal I, the Nullstellensatz tells us thatI(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 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) φ : V1 V2 between affine varietiescorrespond to C-algebra homomorphisms φ : C[V2 ] C[V1 ], where φ (g) g φ for g C[V2 ]. Two affine varieties are isomorphic if and only if their coordinate rings areisomorphic C-algebras.3

Chapter 1. Affine Toric Varieties4 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 1.0.1. A C-algebra R is isomorphic to the coordinate ring of an affinevariety 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 ofC[V ] via the fourth bullet above. More generally, one can take any commutativering R and define the affine scheme Spec(R). The general definition 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 affine schemes should consult [48] and [77].The Zariski Topology. An affine variety V Cn has two topologies we will use.The first 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 affine varieties of Cn contained 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.Affine Open Subsets and Localization. Some Zariski open subsets of an affinevariety V are themselves affine varieties. Given f C[V ] \ {0}, letV f {p V f (p) 6 0} V .Then V f is Zariski open in V and is also an affine variety, as we now explain.Let V Cn have I(V ) h f1 , . . . , fs i and pick g C[x1 , . . . , xn ] representing f .Then V f V \ V(g) is Zariski open in V . Now consider a new variable y and letW V( f1 , . . . , fs , 1 gy) Cn C. Since the projection map Cn C Cn mapsW bijectively onto V f , we can identify V f with the affine variety W Cn 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}.

§1.0. Background: Affine Varieties5In Exercise 1.0.3 you will prove that Spec(C[V ] f ) is the affine variety V f .Example 1.0.2. The n-dimensional torus is the affine open subset(C )n Cn \ V(x1 · · · xn ) Cn ,with coordinate ringC[x1 , . . . , xn ]x1 ···xn C[x1 1 , . . . , xn 1 ].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 affine varieties, not justirreducible ones. The general concept of localization is discussed in standard textsin 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 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).Definition 1.0.3. An irreducible affine variety V is normal if its coordinate ringC[V ] is normal.For example, Cn is normal since its coordinate ring C[x1 , . . . , xn ] is a UFD andhence normal. Here is an example of a non-normal affine variety.Example 1.0.4. Let C V(x 3 y 2 ) C2 . This is an irreducible plane curve with acusp at the origin. It is easy to see that C[C] C[x, y]/hx 3 y 2 i. Now let x̄ and ȳ bethe cosets of x and y in C[C] respectively. This gives ȳ/x̄ C(C). A computationshows that ȳ/x̄ / C[C] and that (ȳ/x̄)2 x̄. Consequently C[C] and hence C are notnormal.We will see below that C is an affine toric variety. An irreducible affine variety V has a normalization defined as follows. LetC[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. 13.13]). Thisgives the normal affine varietyV ′ 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.

Chapter 1. Affine Toric Varieties6Example 1.0.5. We saw in Example 1.0.4 that the curve C C2 defined by x 3 y 2has elements x̄, ȳ C[C] such that ȳ/x̄ / C[C] is integral over C[C]. In Exercise 1.0.6 you will show that C[ȳ/x̄] C(C) is the integral closure of C[C] andthat the normalization map is the map C C defined by t 7 (t 2 ,t 3 ). At first glance, the definition 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 affine toric varieties.Smooth Points of Affine Varieties. In order to define a smooth point of an affinevariety V , we first need to define 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) 6 0}.Thus OV ,p consists of all rational functions on V that are defined at p. Inside ofOV ,p we have the maximal idealmV ,p {φ OV ,p φ(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 define OV ,p when V is not irreducible.The Zariski tangent space of V at p is defined to beTp (V ) HomC (mV ,p /mV2 ,p , C).In Exercise 1.0.8 you will verify that dim Tp (Cn ) n for every p Cn . Accordingto [77, p. 32], we can compute the Zariski tangent space of a point in an affinevariety as follows.Lemma 1.0.6. Let V Cn be an affine variety and let p V . Also assume thatI(V ) h f1 , . . . , fs i C[x1 , . . . , xn ]. For each i, letd p ( fi ) fi fi(p) x1 · · · (p) xn . x1 xnThen the Zariski tangent Tp (V ) is isomorphic to the subspace of Cn defined by theequations d p ( f1 ) · · · d p ( fs ) 0. In particular, dim Tp (V ) n. Definition 1.0.7. A point p of an affine variety V is smooth or nonsingular ifdim Tp (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.

§1.0. Background: Affine Varieties7Points lying in the intersection of two or more irreducible components of V arealways singular ([35, Thm. 8 of Ch. 9, §6]).Since dim Tp (Cn ) n for every p Cn , we see that Cn is smooth. For anirreducible affine variety V Cn of dimension d, fix p V and write I(V ) h f1 , . . . , fs i. Using Lemma 1.0.6, it is straightforward to show that V is smoothat p if and only if the Jacobian matrix f i(1.0.3)J p ( f1 , . . . , fs ) (p) x j1 i s,1 j nhas rank n d (Exercise 1.0.9). Here is a simple example.Example 1.0.8. As noted in Example 1.0.4, the plane curve C defined by x 3 y 2has I(C) hx 3 y 2 i 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 /mV2 ,p , C), we see that V is smooth at p when dimVequals the dimension of mV ,p /mV2 ,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 [3, p. 123] or [47, 10.3].We can relate smoothness and normality as follows.Proposition 1.0.9. A smooth irreducible affine variety V is normal.TProof. In §3.0 we will see that C[V ] p V OV ,p . By Exercise 1.0.7, C[V ] isnormal once we prove that OV ,p is normal for all p V . Hence it suffices 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 [47, 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 affinevariety V(xy zw) C4 is normal, yet V(xy zw) is singular at the origin.Products of Affine Varieties. Given affine varieties V1 and V2 , there are severalways to show that the cartesian product V1 V2 is an affine 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 ) h f1 , . . . , fs i and I(V2 ) hg1 , . . . , gt i. Since the fiand g j depend on separate sets of variables, it follows thatV1 V2 V( f1 , . . . , fs , g1 , . . . , gt ) Cm nis an affine variety.

Chapter 1. Affine Toric Varieties8A 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,V1 V2 should be an affine variety with projections πi : V1 V2 Vi such thatwhenever we have a diagramWφ1ν#φ2V1 V2"π1/ V1π2 V2where φi : W Vi are morphisms from an affine variety W , there should be a uniquemorphism ν (the dotted arrow) that makes the diagram commute, i.e., πi ν φi .For the coordinate rings, this means that whenever we have a diagramC[V2 ]π2 C[V1 ]π1 φ 2 / C[V1 V2 ]ν φ 1% . C[W ]with C-algebra homomorphisms φ i : C[Vi ] C[W ], there should be a unique Calgebra homomorphism ν (the dotted arrow) that makes the diagram commute. Bythe universal mapping property of the tensor product of C-algebras, C[V1 ] C C[V2 ]has the mapping properties we want. Since C[V1 ] C C[V2 ] is a finitely generatedC-algebra with no nilpotents (see the appendix to this chapter), it is the coordinatering C[V1 V2 ]. For more on tensor products, see [3, pp. 24–27] or [47, A2.2].Example 1.0.10. Let V be an affine variety. Since Cn Spec(C[y1 , . . . , yn ]), theproduct V Cn has coordinate ringC[V ] C C[y1 , . . . , yn ] C[V ][y1 , . . . , yn ].If V is contained in Cm with I(V ) h f1 , . . . , fs i C[x1 , . . . , xm ], it follows thatI(V Cn ) h f1 , . . . , fs i C[x1 , . . . , xm , y1 , . . . , yn ].For later purposes, we also note that the coordinate ring of V (C )n isC[V ] C C[y1 1 , . . . , yn 1 ] C[V ][y1 1 , . . . , yn 1 ]. Given affine varieties V1 and V2 , we note that the Zariski topology on V1 V2is usually not the product of the Zariski topologies on V1 and V2 .

§1.0. Background: Affine Varieties9Example 1.0.11. Consider C2 C C. By definition, a basis for the product ofthe Zariski topologies consists of sets U1 U2 where Ui are Zariski open in C. Sucha set is the complement of a union of collections of “horizontal” and “vertical” linesin C2 . This makes it easy to see that Zariski closed sets in C2 such as V(y x 2 )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 provided 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-definedbinary 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 [3, Ch. 3] or [47, Ch. 2].1.0.3. Let R be a finitely generated C-algebra without nilpotents as in Lemma 1.0.1 andlet f R be nonzero. Then S {1, f , f 2 , . . . } is a multiplicative set. The localization RS isdenoted R f and is called the localization of R at f .(a) Show that R

Toric Varieties David Cox John Little Hal Schenck DEPARTMENT OF MATHEMATICS, AMHERST COLLEGE, AMHERST, MA 01002 E-mail address: dac@cs.amherst.edu DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE, COLLEGE OF THE HOLY CROSS, WORCESTER, MA 01610 E-mail address: little@mathcs.holycross.edu DEPARTMENT OF MATHEMATICS, UNIVERSITY OF ILLINOIS AT URBANA- CHAMPAIGN, URBANA, IL 61801