Notation xvii Cones Built From Polyhedra Cv Cone(P − v) for a vertex v of a polytope or polyhedron (2) σQ cone of a face Q P in the normal fan ΣP (2) ΣP normal fan of a polytope or polyhedron P (2) C(P) cone over a polytope or polyhedron (1) SP semigroup algebra of C(P) ∩ (M ×Z) (7) Combinatorics and Lattice Points of Polytopes fi number of i-dimesional faces of P (9) hp ∑n i=p (−1)i−p ( i p ) 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) EhrP( p ) p-Ehrhart polynomial of a lattice polytope (9) Semigroups S, C[S] affine semigroup and its semigroup algebra (1) NA affine semigroup generated by A (1) Sσ = Sσ,N affine semigroup σ∨ ∩ M (1) H Hilbert basis of Sσ (1) Rings R f , RS, Rp localization of R at f, a multiplicative set S, a prime ideal p (1) R integral closure of the integral domain R (1) R completion of local ring R (1) R ⊗C S tensor product of rings over C (1) RG ring 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) Specific Rings C[x1,...,xn] polynomial ring in n variables (1) C[[x1,...,xn]] formal power series ring in n variables (1) C[x±1,...,x±1] 1 n ring of Laurent polynomials (1) I(V) ideal of an affine or projective variety (1,2) C[V ] coordinate ring of an affine or projective variety (1,2) C[V ]d graded piece in degree d when V is projective (2) C(V) field 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)

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