Asymptotic behaviour of the Castelnuovo-Mumford regularity

In this paper the asymptotic behavior of the Castelnuovo-Mumford regularity of powers of a homogeneous ideal I is studied. It is shown that there is a linear bound for the regularity of the powers I-n whose slope is the maximum degree of a homogeneous generator of I, and that the regularity of I-n is a linear function for large n. Similar results hold for the integral closures of the powers of I. On the other hand we give examples of ideal for which the regularity of the saturated powers is asymptotically not a linear function, not even a linear function with periodic coefficients.