The symbolic defect of an ideal

2016/10/01 by Galetto, Federico, Geramita, Anthony V., Shin, Yong-Su +1 · 1 citation
#13A15 #14M05 #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1610.00176

Abstract

Let I be a homogeneous ideal of \Bbbk[x₀,…,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 ℙ². We are specifically interested in identifying ideals I with sdefect(I,2) = 1.

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