The symbolic defect of an ideal

Let I be a homogeneous ideal of k [ x 0 , … , x n ] . To compare I ( m ) , the m-th symbolic power of I, with I m , 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 ) / I m , or equivalently, the minimal number of generators one must add to I m 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 P 2 . We are specifically interested in identifying ideals I with sdefect ( I , 2 ) = 1 .