Embeddings of Projective Schemes Blown up at Subschemes

Suppose X is a nonsingular projective scheme, Z a nonsingular closed sub-scheme of X. Let˜X be the blowup of X centered at Z, E0 the pull-back of a general hyperplane in X, and E the exceptional divisor. In this paper, we study projective em-beddings of˜X given by divisors De,t = tE0 − eE. When X satisfies a necessary condition, 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 give a uniform bound for the regularities of the ideal sheaves of these embeddings, and study their asymptotic behaviour 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 possess property Np for all t ≫ e > 0.