Background

Abstract

Throughout this book, ℙⁿ will be the n-dimensional projective space over an algebraically closed field K of characteristic zero, $R = K\left[ {x_0 ,x_1 , \ldots ,x_n } \right]$ and $m = \left[ {x_0 ,x_1 , \ldots ,x_n } \right]$ its homogeneous maximal ideal. If M is a graded R-module, we distinguish two types of duals of M: the R-dual M^*:= Hom _R (M, R) and the K-dual M^V:= Hom_K (M, K). A scheme V ⊂ ℙⁿ will mean an equidimensional locally Cohen-Macaulay closed subscheme of ℙⁿ. For a subscheme V of ℙⁿ we denote by I _V its ideal sheaf, $I\left( V \right) = H_ * ^0 \left( {\mathcal{I}_V } \right): = \oplus _{t \in \mathbb{Z}} H^0 \left( {\mathbb{P}^n ,\mathcal{I}_V \left( t \right)} \right)$ its saturated homogeneous ideal (unless $V = \not 0$, in which case we let I(V) = m), A(V) = R/I(V) the homogeneous coordinate ring, and N _v = Hom(I _V , O _V )the normal sheaf of V.