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Notation

, M identity map (of set M), AbGroups category of abelian groups, aff(X) affine hull of the set X, AT

k Chow group of T -stable k-cycles, Ak(V ) Chow group of k-cycles on variety V , , A∗(V ) Chow group on variety V , AnnM annihilator of module MAssM set of associated prime ideals of module M, A∗(V ) operational Chow group of variety V , A∗(V ) ring generated by Chern classes, βiu(N) graded Betti number of module N , BL(C) c(L ) first Chern class of line bundle L , C field of complex numbersC[[M]] ring of formal power series with support in monoid M, C[[Zr]] group of formal Laurent series in r indeterminates, CDivT group of T -stable Cartier divisors, CDiv(V ) group of Cartier divisors on variety V , cpold , cconed , ch(E ) Chern character of vector bundle V , χ(V ,E ) Euler characteristic of vector bundle E on variety V , CH∗(V ) Chow ring of smooth variety V , chV Chern character from K(V ) to CH∗(V )Q, c i(E ) ith Chern class of vector bundle E , Cl(M) group of classes of divisorial monomial ideals, Cl(R) class group of ring R, clV cycle map on variety V , Coh(V) category of coherent sheaves on variety V , Col(P) set of column vectors of lattice polytope P, conv(X) closed convex hull of X, conv(X) convex hull of X, C(P) cone over polyhedron P, cΠ multiple of polyhedral complex Π,

Notation

CR(M) Carathéodory rank of monoid M, c∗ monoidal Frobenius endomorphism of monoid (algebra), C∗, X∗ dual cone, conical set, CW ′(W ) normal cone ofW ′ inW , deg Δ degree of triangulation Δ, Δpq empty simplex in dimension , Δσ simplex associated with permutation σ , ∗depthM depthM depth of (graded) module M, ∗dimM dimN Krull dimension of module N , div( f ) principal Cartier divisor of rational function f , div(I) divisor of ideal I, Div(M) group of divisorial monomial ideals, Div(R) group of divisors of ring R, div(r) divisor of rational function R, DivT group of T -stable Weil divisors, ∂P (relative) boundary of P, D(u) E(F) group of elementary automorphisms based on facet F, eHK,R(N) Hilbert–Kunz multiplicity of module N , eai j standard elementary matrix, e(I,R) multiplicity of ring R with respect to ideal I, e(N) multiplicity of module N , En(R), E(R) group of elementary matrices over ring R (of order n, stable), , E(P,−), E+ (P,−) Ehrhart function (of interior) of polytope P, EP(t), E+P (t) Ehrhart series (of interior) of polytope P, E ∣U restriction of vector bundle E to set U , eλv elementary automorphism, ∗Ext graded Ext functor, F∗C face of C∗ dual to face F of C, F [e] set of faces of dimension e of fan F , F[k] set of cones of codimension k in fan F , f∗ direct image functor induced by morphism f , f∗(M) extended module, f ∗ inverse image functor induced by morphism f , G(V ) Grothendieck group of coherent sheaves on variety V , Γk(A) group of graded k-algebra automorphisms of A, Γk(P) group of k-algebra automorphisms of k[P], γd(μ) Γw Γ(X,F) module of global sections ofF over X, [g, h] ghg−h− , Gk(A) group of k-algebra automorphisms of A, GLn(R) general linear group of order n over ring R, GL(R) stable general linear group of ring R, gp(M) group of differences of monoid M, grade(I,M) length of maximal M-regular sequence in I, gradeP grade of quasipolynomial P,

Notation

G identity component of algebraic group G, H(R) ring of continuous functions SpecR → Z, Hα hyperplane defined by affine form α, H>α ,H<α ,H> ,H< open halfspaces (defined by affine form α), H+α ,H−α ,H+ ,H− closed halfspaces (defined by affine form α), HBM

i (X, R) Borel–Moore homology of space X with coefficients in ring R, Hilb(M) Hilbert basis of monoid M, HKR(N ,−) Hilbert–Kunz function of module N , H(N ,−) Hilbert function of module N (over ring or monoid), , HN (t) Hilbert series of module N (over ring or monoid), , ∗Hom graded Hom functor, HomOV (F ,G ) sheaf (or vector bundle) of homomorphisms, ht I height of ideal I, htL ,H(y) height of y over hyperplane H with respect to lattice L, I ∶ J {x ∶ xJ ⊂ I}, I integral closure of ideal I, I(Δ) monomial ideal defined by (abstract) simplicial complex Δ, IE toric ideal, I− inverse of ideal I, int(M) interior of monoid M, int(P) (relative) interior of P, in(V) initial subspace, join(P,Q) join of polyhedra P and Q, J(R) Jacobson radical of ring R, Jw K(R) Grothendieck group of projective modules over ring R, K(R) K(V ) Grothendieck group of vector bundles on variety V , K(R) Bass–Whitehead group of ring R, K(R) Milnor group of ring R, Ki(R, I) relative K-group, Ki(R) ith K-group of ring R, k[Δ] Stanley–Reisner ring of Δ with coefficients in k, kM k-vector space generated by Mk[P] polytopal algebra over k defined by polytope P, k[P] normalization of k[P], L (D) line bundle associated to Cartier divisor D, LΔ L ∩ aff(Δ), L (Δ) affine lattice spanned by vert(Δ), ld x base logarithm of x, LE canonical line bundle on P(E ), linkΠ(X) link of X in polyhedral complex Π, L [[M]] L (P) line bundle associated with lattice polytope P, L( f ),L∞( f ) Laurent expansions of function f , maxΠ set of facets of polyhedral complex Π, M normalization of monoid M, c(M/M) conductor of normalization of monoid M in M, MΓ ,MΓ,E

Notation

MN integral closure of monoid M in overmonoid N , M[−N] localization of monoid M with respect to subset N , Mod(A-G) category of A-G-modules, Mod(OV ) category of OV -modules, M(P) monoid associated with lattice polytope P, M∗ int(M) ∪ {}, M(u) graded module M shifted by u, μL(C) multiplicity of cone C with respect to lattice L, μL(Δ) multiplicity of simplex Δ with respect to lattice L, μ(Π) multiplicity of lattice polytopal complex, (μΠ)μ , ν mixed triangulation of polytopal complex Π, μR(N) minimal number of generators of module N , M∣W N set of positive integersN(a) Newton polytope of element a, nil(R) nilradical of ring RN (P) normal fan of polytope P, NU (W ) normal bundle to subvarietyU inW , O(G) coordinate ring of algebraic group G, ΩR module of absolute differentials of ring R, ΩV /k sheaf of k-differentials on variety V , O(n) openstarΠ(X) open star neighborhood of X in polyhedral complex Π, ordU order function along codimension subvarietyU , OV structure sheaf of variety V , OU ,V local ring of V along subvarietyU ⊂ V , OV (X) ring of sections of OV over open subset X ⊂ V , O(V ) = OV (V ), PA(Π) space of piecewise affine functions, par(Δ, p) par(v , . . . , vr) semi-open parallelotope spanned by vectors v , . . . , vr , P closure of P, Φ(N) P(I) ∣Π∣ support of polyhedral complex Π, Pic(R), Pic(V ) Picard group of ring R, variety V , , Π(e) e-skeleton of polyhedral complex Π, πF face projection, Π(F), Π(P) polytope algebra defined by fanF , polytope Π, P d

k projective space of dimension d over field k, P(L ) polyhedron associated to line bundle L , PLZ(F) group of piecewise Z-linear functions on fan F , P(E ) projectivization of vector bundle E , PQF(R) Princ(M) group of principal monomial ideals, Princ(R) group of principal divisors of ring R, prU projection onto factorU of product U ×V , P(u) Pv base facet of column vector v of lattice polytope P,

Notation

Q,Q+ (nonnegative) rational numbersQF(R) field of fractions of integral domain RR,R+ (nonnegative) real numbersR+X conical set generated by X, RM R-vector space generated by M(R) Serre condition, R[M] algebra over monoidal complex M, Rad(I) radical of ideal I (in monoid or ring), rankM rank of monoid, R (often) normalization of ring R, R(c) Veronese subring of ring R, RD ring of invariants of R with respect to action of D, rec(P) recession cone of polyhedron P, reg(M) Castelnuovo–Mumford regularity of module M, RS integral closure of ring R in overring S, Rings category of (commutative) rings, R[M] algebra of monoid M with coefficients in R, R(p) homogeneous localization of R, Rred R/nil(R)R(V ) field of rational functions of variety V , (S) Serre condition, sdiv(C) SF(Π,Π′) set of support functions of subdivision Π′ of Π, si(E ) ith Segre class of vector bundle E , σ , σi (often) standard map, support form, Σ(P)inv Σ(M), Σ(P) automorphism group of monoid M, polytope P, , SK(R) SLn(R), SL(R) special linear group over ring R (of order n, stable), sn(M) seminormalization of monoid M, snS(R), sn(R) seminormalization of ring R in overring S, absolute, SpecR prime spectrum of ring RstarΠ(X) star of X in polyhedral complex Π, St(R) Steinberg group of ring R, supp(a), supp B support of element a, subset B in graded group, SuppM support of module M, Sur(E ,E) set of homomorphisms surjective on fibers, τC , τC orbit (closure) corresponding to cone C, τV natural transformation G → (A∗)Q, td(E ) Todd class of vector bundle E , Tmk(A) group of termic automorphisms of monoid algebra A, T n(k),T n algebraic n-torus (over k), TV tangent bundle of variety V , tZ+ monoid multiplicatively generated by t, UC(P), UCL(P) , Uε(x) open ε-neighborhood of xU(M) group of units of monoid M, Umr(A) set of unimodular rows of length r over A, U(R) group of units of ring R

Notation

Var(k) category of k-varieties, Vχ weight space of V with respect to character χ, Vect(V ) category of vector bundles on variety V , Vect(V , F , S) vert(P) set of vertices of P, V (F) toric variety corresponding to fanF , volL volume function defined by lattice L, V∗ space of linear forms on V , VV trivial G-bundle associated with G-module V , w Watermanmap, Witt(Λ) ring of Witt vectors over Λ, ⌈x⌉ min{z ∈ Z ∶ z ≥ x}, ⌊x⌋ max{z ∈ Z ∶ z ≤ x}, X(G) character group of algebraic group G, xi(E ) ith Chern root of vector bundle E , xai j Steinberg symbol, [x, y] line segment between x and y,

Z,Z+ (nonnegative) integersZTk group of T -stable k-cycles,

ZTk group of T -stable k-cycles,

Zk(V ) group of k-cycles on variety V , , Z(L, )

Index

admissiblepair of functors, sequence of polytopes,

affine(linear) combination, form, function, rational,

hull, lattice, lattice structure, map, , integral, rational,

piecewise, rational structure, semigroup, see affine monoidsubspace, , dimension of, rational,

affine monoid, see alsomonoid, –algebra, seemonoid algebrahomogeneous, interior of, normal, see normal monoidpolytopal, see polytopal (affine) monoidseminormal, see seminormal monoidtight,

affinely independent, A-G-module, finitely generated, projective,

algebraic group, –

(homo)morphism of, abelian, character (group), connected, diagonalizable, linear, linearly reductive, rational action of, rational representation of, semi-invariant of, weight space of,

algebraic torus, tautological action of,

barycentric coordinates, base facet, see column vector, base facet ofbasicconfiguration, degree, grading, simplex,

Bass functor, Bass’ cancellation, Bass–Whitehead group, big Witt vectors, , binomial, Birkhoff polytope, Borel’s theorem, Borel–Moore homology, bottom, , , , boundary, complex, relative,

Index

subcomplex,

cancellation problem, canonicalorder, simplex, tiling, triangulation,

∗canonical module, –and simplicial affine monoid, Hilbert quasipolynomial of, Hilbert series of, –is conic, of normal monoid algebra,

Carathéodory rank, , asymptotic, bound for, of direct sum, of Segre product, virtual,

Carathéodory’s theorem, Cartan homomorphism, cartesian diagram, Cartier divisor, on toric variety, principal,

Castelnuovo–Mumford regularity, c-divisiblehull,

c-divisible monoid, ceiling, cell, cellular decomposition, change of rings, character group, of finite abelian group,

Cherncharacter, classfirst, higher,

root, Chowgroup of k-cycles, , ring,

class group, and factoriality, automorphisms fixing, of monoid,

of monoid algebra, –of positively graded ring,

class ring, Cohen–Macaulaydivisorial ideal, finite number of divisor classes, module, –Hilbert series of,

Cohen–Macaulay ring, –see also Cohen–Macaulay moduleand initial ideal, Koszul property of Veronese subring,

columnstructure, vector, , base facet of, , invertible, ,

combinatorial type, compartment, complementary subspace, concave, function, strictly,

conductor, , , cone, –cross-section of, dual, duality theorem for, is finitely generated, multiplicity of, seemultiplicity of coneover polyhedron, , over polytope, –pointed, minimal system of generators of,

rational, simplicial, , –tight, unimodular, flag of,

congruence (relation), generating set of,

conicdivisor class, –versus Cohen–Macaulay class,

divisorial ideal, is Cohen–Macaulay,

conicalcomplex, cross-section of,

Index

simplicial, hull, set, dual, finitely generated,

connected component, connected in codimension , constructible sets, continuous functor, convex, (linear) combination, function, , hull, closed, of monomial ideal,

strictly, , subcomplex,

coset module, , coverof simplicial cone, –

cross-section, , cycle map,

Danilov–Stanley theorem, Dedekind domain, monoid algebra over,

degree homomorphism, depth, ∗depth, destructive element, determinantal ring, automorphism group of,

diagonalizable group, and finitely generated abelian groups,

dimensionKrull, of polyhedral complex, of polyhedron,

∗dimension, discrete Hodge algebra, divisorof ideal, principal,

divisor class group, see class groupdivisor complex, domain of linearity, d-restricted, dual, see object, dual

edge, of polyhedral complex,

Ehrhartfunction, , , polynomial, quasipolynomial, grade of,

reciprocity, , series, ,

elementaryaction on unimodular rows, automorphism, matrixgroup of, stable group of,

elementary divisor theorem, embedded torus, equivarianthomomorphism, Serre problem, vector bundle, trivial,

Euler characteristic, excision, extended module, , exterior order, extreme(integral) generator, ray, submonoid, , subset,

face, –and monomial prime ideal, dual, improper, interior, lattice, of polyhedral complex, projection, ring,

facet, –, of polyhedral complex,

fan, see also conical complexcomplete, full-dimensional, normal, of polytope, projective,

Index

quasiprojective, rational, simplicial, unimodular,

filtered union, filtration and decomposition of monomial

basis, finitely presented, floor, Fourier–Motzkin elimination, Frobeniuscoin problem, endomorphism,

Gauss’ theorem, g-conjecture, theorem, , , general linear group, stable,

generic fiber, geometric realization, ghost isomorphism, G-modulerational,

Gordan’s lemma, , , inhomogeneous version of,

Gorensteinmonoid, , , polytope, see lattice polytope, Gorensteinring, and initial ideal, Hilbert series of,

gradeof ideal, of quasipolynomial,

gradedalgebra, Betti numbers, , duality for,

component, Ext, free resolution, , –minimal,

group, ring, –convention on,

subgroup, Tor,

graded Noether normalization, grading, ,

positive, graph, Gröbnerbasis, , , , , , deformation,

Grothendieck groupcohomological, homological, of coherent sheaves, of vector bundles,

Grothendieck group K

of monoid algebra, –Grothendieck–Riemann–Roch theorem, for singular varieties,

group of k-cycles, group ringreducedness of, seminormality of, –

g-vector, Gysin homomorphism, map,

halfspace, closed, discrete, linear, open, rational,

Hamann’s criterion, height, above hyperplane, of ideal,

Hilbertfunction, –and triangulation,

quasipolynomial, coefficients of, of ∗canonical module,

series, Z-graded, , of ∗canonical module, is rational function, of Cohen–Macaulay module, of Gorenstein ring, of normal affine monoid,

Hilbert basis, of normal monoid, –

Hilbert’s syzygy theorem, Hilbert–Kunzfunction,

Index

multiplicity, Hilbert–Serre theorem, Hirzebruch–Riemann–Roch theorem, and lattice point counting, ,

Hochster’s theorem, Hochster–Roberts–Kempf theorem, hom-polytope, homogeneouscomponent, homomorphism, linear diophantine system, system of parameters,

homogenization, , of ideal,

homotopy invariance, , h-vector, see alsoHilbert series, of Gorenstein polytope, –

hyperplane, dissection, , has degree ,

(ICP), counterexample to,

idealbinomial, –divisor of, divisorial, fractional, monomial,

integral closure of, inverse, invertible, monomial, seemonomial idealprime, see prime idealtoric, see toric ideal

ideal in monoid, divisorial, prime, radical, radical of,

initialcoefficient, component, , ideal, monomial, subalgebra, subspace, , term,

integral Carathéodory property, see (ICP)

integral closure, see object, integral closure ofintegral element, , integrally closed, see object, integrally closedinterior, relative,

invariant, inverse image formula, invertibleideal, module,

irreducible,

Jacobson radical, join,

Karoubi square, k-cycle, Knudsen–Mumford triangulation, see

triangulation, Knudsen–MumfordKoszul algebra, and initial ideal, and lattice polygon, and multiple of lattice polytope,

Koszul complex, duality of,

Koszul filtration, Krull dimension, Krusemeyer’s ring,

latticepoints, polytopal complex, embedded, multiplicity of,

structure, affine, on polytopal complex,

subdivision, width,

lattice polygonand Gröbner basis, and Koszul algebra, interior of, triangulation of,

lattice polytope, and Koszul property of multiple, Gaussian algorithm for, Gorenstein, integrally closed,

Index

isomorphism class of, multiple of, normal, , smooth, unimodular cover of multiple, –unimodular triangulation of multiple, see

triangulation, unimodularvery ample, ,

level ring, line bundle, ample, equivariant, generated by global sections, very ample,

line segment, halfopen, open,

linear diophantine system, , linear variety, linearization problem, , link, ∗localring,

∗localextension,

local ring of along a subvariety, local-global principle of Quillen, localizationexact sequence, , square,

locally free sheaf,

Macaulay sequence, Macaulay’s inequalities, magic square, ∗maximal ideal, Mayer–Vietoris sequence, , , Mennicke symbol, Milnorgroup, patching, for unimodular rows,

square, Minkowskisummand, weight,

Minkowski sum, vertex of,

Minkowski’s theorem,

mixed volume, module over monoid, monoidaffine, see affine monoidcancellative, c-divisible, convention on, dual, finitely generated, free, graded, group of differences, half-factorial, ideal in, see ideal in monoidintegral closure of, , integrally closed, interior of, localization of, module over, monoidal Frobenius endomorphism of,

normal, see normal monoidnormalization of, positive, saturated, saturation of, seminormal, see seminormal monoidseminormalization of, , simplicial, , torsionfree,

monoid algebra, , see also ringclass group of, see class groupfactorial, graded automorphism of, isomorphism problem for, graded version,

Krull dimension of, –, monoidal Frobenius endomorphism of,

normal, see normal monoid algebraover Dedekind domain, over principal ideal domain, polytopal, see polytopal algebrareducedness of, regular, seminormalization of, simplicial, , subintegral closure of,

Index

termic automorphism of, toric automorphism of, torus action on, –

monoid domain, –monoid ring, seemonoid algebramonoidal complex, algebra over, , free, unimodular,

monomial, , , monomial ideal, integral closure of, isomorphism theorem for, prime, –and face of cone,

radical, –monomial order, degree lexicographic, (degree) reverse lexicographic, lexicographic,

monomial preorder, morphismof fans,

Motzkin’s theorem, discrete analogue of,

M-sequence, see regular sequencemultiplicityof ring with respect to ideal, , and volume, , of a minimal prime ideal, of cone, of irreducible component of a subscheme,

of lattice polytopal complex, of polytopal algebra, of polytope, of ring or module , of simplex, , ,

Murthy–Horrocks theorem,

Nagata’s theorem, natural augmentation, Newton polytope, nilpotent extension, nondestructive element, nonface, normal bundle, normal cone, normal domain, –

and initial ideal, normal monoid, –, as pure submonoid, Gorenstein, see Gorenstein monoidHilbert series of, union of,

normal monoid algebra, –and Gorenstein property, as pure subalgebra, as ring of invariants, , ∗canonical module of, Cohen–Macaulay property of, –graded automorphism group of, Hilbert series of,

normalization, see object, normalization of

open star neighborhood, operational A∗-theory, orbifold, ordered group,

parallel, halfspace,

parallelotopesemi-open,

path, Picard group, –, , , of a variety,

Pick’s formula, PID, Poincaré series, polygon, polyhedral algebra, polyhedral complex, –embedded, rational, support of,

polyhedron, –face structure of, –finite generation of, –rational,

polytopal(affine) monoid, , algebra, graded automorphism group of, multiplicity of, see multiplicity ofpolytopal algebra

complex, see also polyhedral complex

Index

lattice, see lattice polytopal complexpolytope, –

see also polyhedron, lattice polytopedual, is convex hull of finite set, lattice, multiplicity of, seemultiplicity of polytopenormal fan of, rational, reflexive, simple, simplicial,

polytope algebra, positive, see object, positivepositively

Zr-graded, graded,

prime avoidance, prime element, , nonassociated,

prime idealin graded ring,

primitive, integral form,

projection formula, , for Chern classes,

projective bundle formula, projective module, , projectivization, p-seminormal, pureextension of rings, submonoid, subring, and Cohen–Macaulay property,

pyramid, apex of, base of,

pyramidaldecomposition, nondegenerate,

descent, , –extension, nondegenerate,

quasipolynomial, grade of,

Quillen–Suslin theorem, quotient

categorical, geometric,

rankof module, of projective module,

rational, see object, rationalrational equivalence, , recession cone, , reduced ringand initial ideal,

reduction to characteristic p, Rees’ lemma, reflexive module, regularlocal ring, ring, sequence, regular ∗local ring, subdivision, see subdivision, regulartriangulation, see triangulation, regular

regular embedding of a closed subvariety, regular sequencehomogeneous,

regularity test system, representation length, resolution property, ringfactorial, graded, see graded ringintegral closure of, integral extension of, integrally closed, normal, normalization of, seminormal, seminormalization of, subintegral closure of,

ring of invariants, of diagonalizable group,

Roberts’ theorem,

Segre class, Segre product, , semiexact functor, semigroup, seemonoidnumerical,

seminormal monoid, –seminormal monoid algebra, –,

Index

K of, nontriviality of K , Picard group of, , , projective module over,

seminormalization, see object, seminormal-ization of

separation of polyhedra, Serre’s condition(R), , (S), , shift, shrinking a monoid or cone, simplex, basic, empty, of dimension ,

multiplicity of, seemultiplicity of simplexunimodular,

simplicial complex, , abstract , exterior face ring, Stanley–Reisner ring of, see Stanley–

Reisner ringsimplicial growth, simplicial object, see object, simplicialskeleton, special linear group, stable,

splitting construction, squarefree divisor complex, Srinivas’ element, stablyfree, isomorphic,

standardembedding, map,

standard affine charts, Stanley–Reisner ring, , isomorphism theorem for,

star, steadfast, Steinberggroup, relations, symbols,

Steinitz’ theorem, stellar subdivision, see subdivision, stellarstrong Lefschetz theorem, ,

structurerational,

subcomplex, subdivision, –

see also triangulationcoherent, composite of, Delaunay, intersection of, lattice, projective, regular, –stellar, regularity of,

subintegralelementary, , extension of monoid, extension of ring, and action of elementary group,

subintegral closure, see object, subintegralclosure of

subvariety, supernormal, supportformof monoid,

function, hyperplane, of monoid,

of element, of subset,

Swan’s element, Swan–Weibel homotopy trick, symbolic power, system of parameters, homogeneous,

tameautomorphism, generation conjecture,

tangent vector bundle, term, termic, tight, see object, tightTodd class, toricdeformation, , face ring, ideal, –,

Index

toric variety, –automorphism group of, Chow cohomology of complete, Chow group of, Chow ring of complete smooth, cohomology ring of complete smooth, complete, divisors on, equivariant morphism of, Grothendieck group G of, Grothendieck group of complete smooth,

, has rational singularities, homogeneous coordinate ring of, line bundles on, nondegenerate, projective, , simplicial, smooth, with huge K ,

total Betti number, total degree, Traverso–Swan theorem, triangular automorphism, triangulation, –

see also subdivisionand Hilbertfunction,

core of, degree of, –existence of, , existence of regular, –full, , Hilbert, full,

Knudsen–Mumford, –lexicographic extension of, , mixed, of simplicial cone, placing extension of, pulling extension of, regular, –support function of, unimodular,

reverse lexicographic extension of, unimodularof cone, of fan, of multiple of complex, –

of multiple of polytope, –of polytope,

(UHC), , counterexample to, of direct sum, of Segre product,

(UHT), failure in dim ≥ , in dimension ,

unimodal, unimodular object, see object, unimodularunimodular row, unit, group of, , –

universal central extension,

variety, group action on, Q-factorial,

vector bundle, direct sum, dual of, equivariant, trivial,

homomorphism of, rank of, section, stably trivial, trivial,

Veronesesubring, , and Koszul algebra,

vertex, simple,

visible, V-manifold, volume, see alsomultiplicity, mixed, normalized,

Voronoicell, diagram,

Watermanmap, point,

Weierstraß preparation, weight space

Index

module structure on, weight vector, Weil divisor, on toric variety,

Q-Cartier,

Weyl chamber,

Whitney sum formula,