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, Römer, Trung and Wang that the Castelnuovo-Mumford regularity of $I^mM$ has the form $dm+e$ when $m\gg 0$. 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.