The symbolic defect of an ideal

Abstract

Let I be a homogeneous ideal of k[x0,…,xn]. To compare I(m), the m-th symbolic power of I, with Im, the regular m-th power, we introduce the m-th symbolic defect of I, denoted sdefect(I,m). Precisely, sdefect(I,m) is the minimal number of generators of the R-module I(m)/Im, or equivalently, the minimal number of generators one must add to Im to make I(m). In this paper, we take the first step towards understanding the symbolic defect by considering the case that I is either the defining ideal of a star configuration or the ideal associated to a finite set of points in P2. We are specifically interested in identifying ideals I with sdefect(I,2)=1.