Symbolic powers of edge ideals

Abstract

Let I ⊂ R = k[X] = k[X1, . . . , Xn] 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)/Σr (I )), where Σr (I ) = r−1 ı=1 I (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.