2 00 2 Projective Embeddings of Projective Schemes Blown up at Subschemes Huy

Suppose X is a nonsingular arithmetically Cohen-Macaulay projective scheme, Z a nonsingular closed subscheme of X, and˜I the ideal sheaf of Z in X. Let˜X be the blowup of X along˜I, E0 the pull-back of a general hyperplane in X, and E the exceptional divisor. In this paper, we study projective embeddings of˜X given by the divisors De,t = tE0 − eE. We give explicit values of d and δ such that for all e > 0 and t > ed + δ, De,t embeds˜X as a projectively normal and arithmetically Cohen-Macaulay scheme. We also study asymptotic behaviour of regularities of the ideal sheaves of these embeddings as t gets large compared to e. When X is a surface and Z is a 0-dimensional subscheme, we further show that these embeddings possesses property Np for all t ≫ e > 0.