SYMBOLIC POWERS OF MONOMIAL IDEALS

We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal I subset of k[x(0), ..., x(n)] we show that I(t(m+e-1)-e+r) subset of m((t-1)(e-1)+r-1)(I-(m))(t) for all positive integers m, t and r, where e is the big-height of I and m = ( x(0), ..., x(n)). This captures two conjectures (r = 1 and r = e): one of Harbourne and Huneke, and one of Bocci et al. We also introduce the symbolic polyhedron of a monomial ideal and use this to explore symbolic powers of non-square-free monomial ideals.