On the Gorenstein property of the diagonals of the Rees algebra. (Dedicated to the memory of Fernando Serrano.)

Abstract

Let Y be a closed subscheme of Pn−1 k defined by a homogeneous ideal I⊂ A=k[X1,...,Xn], and X obtained by blowing up Pn−1 k along Y. Denote by Ic the degree c part of I and assume that I is generated by forms of degree ≤ d. Then the rings k[(Ie)c] are coordinate rings of projective embeddings of X in PN−1 k , where N=dimk(Ie)c for c ≥ de+1. The aim of this paper is to study the Gorenstein property of the rings k[(Ie)c] . Under mild hypothesis we prove that there exist at most a finite number of diagonals (c, e) such that k[(Ie)c] is Gorenstein, and we determine them for several families of ideals.