NOTES ON REGULARITY STABILIZATION

When M is a finitely generated graded module over a standard graded algebra S and I is an ideal of S, it is known from work of Cutkosky, Herzog, Kodiyalam, Romer, Trung and Wang that the Castelnuovo-Mumford regularity of (IM)-M-m has the form dm+e when m >> 0. We give an explicit bound on the m for which this is true, under the hypotheses that I is generated in a single degree and M/IM has finite length, and we explore the phenomena that occur when these hypotheses are not satisfied. Finally, we prove a regularity bound for a reduced, equidimensional projective scheme of codimension 2 that is similar to the bound in the Eisenbud-Goto conjecture, under the additional hypotheses that the scheme lies on a quadric and has nice singularities.