Stabilization of the regularity of powers of an ideal

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, Her-zog, Kodiyalam, Römer, Trung and Wang that the Castelnuovo-Mumford regularity of ImM has the form dm + e when m 0. We give an ex-plicit 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 satis-fied. 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 [1984], under the additional hypotheses that the scheme lies on a quadric and has nice singularities