xii GLOSSARY OF NOTATION

τtx

linear form associated to the B´ ezoutian 72

dx

t

generalized Jacobian 74

ωV/k sheaf of regular differential forms

(also called canonical or dualizing sheaf) 84

Reg V set of regular points of V 84

cV/k fundamental class of a variety 85

F ∗, F ∗∗ dual (double dual) of a sheaf F 85

Pic V Picard group of a variety V 86

CV/W complementary module of a

finite morphism V → W 87

µx(X) multiplicity of X at x 96

∆(X) centroid of a zero-dimensional aﬃne scheme 96

pg(V ) geometric genus of a projective variety 103

Coh(V ) category of coherent sheaves on V 110

δF i i-duality isomorphism 111

Sing(X) singular locus of X 112

δ(X) singularity degree of X 113

c(X) conductor degree of X 113

|µ| µ0 + ··· + µd, µ = (µ0, . . . , µd) ∈ Nd+1 115

µ! µ0! ··· µd!, µ = (µ0, . . . , µd) ∈ Nd+1 115

∂µ

(

∂

∂X0

)µ0

··· (

∂

∂Xd

)µd

, µ = (µ0, . . . , µd) ∈

Nd+1

115

Xµ

X0

µ0

··· Xd

µd

, µ = (µ0, . . . , µd) ∈

Nd+1

116

PF (∂) differential operator associated

to F = {F0, . . . , Fd} 115

ev0 evaluation at zero 116

polysol(I(∂)) polynomial solutions of the differential equations

associated to an ideal I 116

P(V ) projective space associated to a vector space V 117

X(∆) toric variety associated to a lattice polytope ∆ 119

∆f (X, Z) representative of B´ ezoutian of f = {f1, . . . , fd} 120

degX f degree of f in a set of variables X 123

(f) radical of the ideal (f) 127

Ω0 homogeneous d-form used for residues 128, 136

T the torus of Pk

d

or a toric variety 129, 131

Gm multiplicative group 129

G Ω0

F0 ··· Fd

homogeneous generalized fraction 129, 138

N, M dual lattices 130

σ∨

dual of a convex cone σ 130

σ(1) set of 1-dimensional faces of σ 130