Symbolic Powers of Monomial Ideals

We investigate symbolic and regular powers of monomial ideals. For a square-free monomial ideal \(I\) in \(k[x_0, \ldots, x_n]\) we show \(I^{t(m+e-1)-e+r)}\) is a 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 \(M = (x_0, \ldots, x_n)\). This captures two conjectures (\(r=1\) and \(r=e\)): one of Harbourne-Huneke and one of Bocci-Cooper-Harbourne. We also introduce the symbolic polyhedron of a monomial ideal and use this to explore symbolic powers of non-square-free monomial ideals.