Normally torsion-free edge ideals of weighted oriented graphs

Abstract

Let be the edge ideal of a weighted oriented graph D, let G be the underlying graph of D, and let be the n-th symbolic power of I defined using the minimal primes of I. We prove that if and only if the following conditions hold: (i) every vertex of D with weight greater than 1 is a sink and (ii) G has no triangles. Using a result of Mandal and Pradhan and the classification of normally torsion-free edge ideals of graphs, we prove that for all if and only if the following conditions hold: (a) every vertex of D with weight greater than 1 is a sink and (b) G is bipartite. If I has no embedded primes, conditions (a) and (b) classify when I is normally torsion-free. Using polyhedral geometry and integral closure, we give necessary conditions for the equality of ordinary and symbolic powers of monomial ideals with a minimal irreducible decomposition. Then, we classify when the dual of the edge ideal of a weighted oriented graph is normally torsion-free.