Symbolic powers of edge ideals

Let 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 where Σr(I)=∑ı=1r−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 , 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.