  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,
R\"omer, Trung and Wang that the Castelnuovo-Mumford regularity of I^mM 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 [1984], under the additional hypotheses that
the scheme lies on a quadric and has nice singularities.
