Index 835 good categorical quotient, 198–202, 684 Gordan’s Lemma, 30 Gorenstein Fano variety, see variety, Gorenstein Fano Gorenstein ring, 662 Grobner ¨ basis, 485, 798 marked, 485 reduced, 485 universal, 485 Grobner ¨ fan, 486, 747, 804 resolution of singularities, 490 graded module, 227, 248, 310 shift of, 227 graded polynomial ring, 219, 697 Grassmannian, 254, 260, 701 Grothendieck-Riemann-Roch, 631, 632 Gysin map, 676 Hom sheaf, 249, 255 h-polynomial, 622 hard Lefschetz theorem, 620, 622, 795 Hausdorff topological space, 102 height of a prime ideal, 157 higher direct image, 393 Hilbert basis, 32, 33, 473, 474, 801, 805 Hilbert basis theorem, 3 Hilbert polynomial, 657, 805 of a projective variety, 432 of a sheaf, 431, 440 Hirzebruch surface, 112, 175, 191, 263, 267, 273, 303, 338, 368, 408, 425, 454, 498, 509, 595, 600, 602, 611, 631, 640, 664, 698, 706, 733, 735, 789, 808, 809, 811 Hirzebruch-Riemann-Roch, 625, 630, 631, 658, 793, 811 proof of, 651 proof via Khovanskii-Pukhlikov, 671 simplicial, 655 Hodge decomposition, 440 Hodge index theorem, 794 homogeneous coordinate ring, see coordinate ring, homogeneous homogeneous coordinates, 49 homogenization, 192 homomorphism of sheaves, 166, 246 injective, 246 sujective, 246 Hopf fibration, 575 Iitaka dimension, 427 image sheaf, 247 index of a Q-Gorenstein toric variety, 555 index of a simplicial cone, see multiplicity of a simplicial cone injective resolution, 387, 388 injective sheaf, 387 injectivity lemma, 411, 412, 424, 430 inner normal fan, see normal fan integer programming, 798 integral closure, 5 of an ideal, 537, 540 integral of a cohomology class, 592, 625 equivariant, see equivariant integral integrally closed ideal, 536 integrally closed ring, 5 intersection cohomology, 620, 622, 795 Betti numbers, 622 intersection homology, 619 intersection product, 288, 593 on a toric surface, 497 on a toric variety, 289, 300 inverse limit, 182 inverse system, 182 invertible sheaf, 168, 254, 255 inward-pointing facet normal, see facet, normal irreducible components, 97 irreducible variety, 97 irrelevant ideal, 207 of a character, 700 of a GKZ cone, 722 Ishida complex, 378, 423 isomorphism of varieties, 96 Jacobian matrix, 7, 351 Jurkiewicz-Danilov ring, 593, 595 theorem, 594, 595, 606 k-cycle, see algebraic cycle Kahler ¨ differentials, 351, 352, 354 Kunneth ¨ formula, 597 Kawamata log terminal singularities, see klt singularities kernel sheaf, 247 Khovanskii-Pukhlikov theorem, 668, 669 proof of, 670 relation to HRR, 671 Kleinschmidt’s classification theorem, 341, 740, 796 klt singularities, 557, 783 Koszul complex, 403, 408 Krull principal ideal theorem, 161, 169 LattE, 799 lattice, 13 lattice polyhedron, 319 lattice equivalence, 381, 461 lattice ideal, 15 lattice polytope, 66 empty, 555 Laurent polynomial, 5 LDP polygon, 386 index of, 386 left exact, 389
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