Index 839 semigroup homomorphism, 35, 116 semiprojective toric variety, 332, 693, 697, 714 semistable points, 681, 699, 700, 704, 709 separated variety, 102, 204 separating transcendence basis, 48 separation lemma, 28, 107 Serre duality, 395, 420 for a toric variety, 416, 793 MCM sheaves, 416 set-theoretic complete intersection, 22 sheaf, 95 constant, 255 generated by global sections, 249, 258, 346 locally constant, 255 locally free, 250 of 1-forms, 352 of OX -modules, 165, 245 of p-forms, 355 of a graded module, 228, 248, 310 of a torus-invariant divisor, 189 global sections of, 189, 190, 193 of sections of a vector bundle, 252 of Zariski p-forms, 355 sheaf cohomology, 387, 388, 391 long exact sequence, 389 toric, 402, 447, 811 toric surface, 406 sheafification, 247 sheet, see spectral sequence, sheet of simple normal crossings, see SNC divisor simple polytope, 64, 434 simplex, 64 simplicial cone, 30 multiplicity of, see multiplicity of a simplicial cone simplicial fan, 113 simplicial polytope, 64 simplicial toric variety, 113, 180 simplicialization, 518, 520 Singular, 799, 806 singular cohomology, 395 of a simplicial toric variety, 588 of a toric variety, 582 of an affine toric variety, 577 reduced, 397 relative, 405 ring, see singular cohomology ring with compact supports, 582 singular cohomology ring, 396 of a toric variety, 594, 595, 662, 793 of an affine toric variety, 577 singular locus, 163, 513 of a normal variety, 163 of a toric variety, 514 singular point, 6 of a toric surface, 459 singularities, see also toric singularities canonical, see canonical singularities finite quotient, see finite quotient singularities klt, see klt singularities log canonical, see log canonical singularities terminal, see terminal singularities small contraction, 764 Smith normal form, 173 smooth cone, 30, 40 smooth fan, 113 smooth Fano polytope, 385 smooth locus, 513 smooth point, 6, 101 smooth polytope, 87 smooth toric variety, 40, 87, 113, 179 smooth variety, 6, 101 SNC divisor, 360, 525 span of a cone, 24 Spec, 4, 317 spectral sequence, 393, 813 convergent, 814 degenerates, 815 differentials, 813 E1, 814 E2, 814 edge homomorphism, 816 Ext, 417, 817 first quadrant, 814 Leray, 393, 816 of a closed cover, 404, 817 of a filtered space, 582, 818 of an open cover, 394, 578, 817 Serre, 598, 599, 818 sheet of, 813 singular cohomology of a toric variety, 582, 587 spherical dual, 574 spherical variety, 663 splitting fan, 133, see also weak splitting, 137 stable points, 681, 704, 709 stalk of a sheaf, 246, 257 standard n-simplex Δn, 66 Stanley-Reisner ideal, 593, 600 Stanley-Reisner ring, 600, 601 star subdivision, 130, 132, 515, 518, 520, 750, 751 properties of, 515, 517 Stasheff polytope, 745 strictly convex, see support function, of a Cartier divisor, strictly convex strong Oda conjecture, 792 strongly convex cone, 28 structure sheaf, 95, 99 analytic, 547 sublattice of finite index, 44 subvariety, 97 sum function, 638 Sumihiro’s theorem, 108, 151, 791 supplementary cone, 478, 479 isomorphic to dual cone, 480

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