Symbolic powers of edge ideals

Abstract

Let I⊂R=k[ X]=k[X 1,…,X n] be an ideal in a polynomial ring over the field k. We define the essential symbolic module of I to be the R/ I-module F(I)=⊕ r⩾2(I (r)/Σ r(I)), where Σ r(I)=∑ ı=1 r−1I (ı)I (r−ı) and I ( m) stands for the mth symbolic power of I. We will mainly focus on the case where I is generated by square-free monomials of degree two. Among our main results are optimal bounds for the degrees of the minimal generators of F(I) , several criteria for a monomial to be such a generator and an upper bound for the generation type of the symbolic Rees algebra of I. As a byproduct we recapture the result of Simis, Vasconcelos, and Villarreal on when such an ideal is normally torsionfree.