Geometric regularity of powers of two-dimensional squarefree monomial ideals

Abstract

Let I be a two-dimensional squarefree monomial ideal of a polynomial ring S. We evaluate the geometric regularity, \(a_i\)-invariants of \(S/I^n\) for \(i\ge 2\). It turns out that they are all linear functions in n from \(n=2\). Also, it is shown that \(\text{ g-reg }(S/I^n)={\text {reg}}(S/I^{(n)})\) for all \(n\ge 1\).