Powers of Ideals and Fibers of Morphisms
  Let X\subset PP^n be a projective scheme over a field, and let phi:X --> Y be a finite morphism. Our main result is a formula in terms of global data for the maximum of the Castelnuovo-Mumford regularity of the fibers of \phi, considered as subschemes of \PP^n.   From an algebraic point of view, our formula is related to the theorem of Cutkosky-Herzog-Trung and Kodiyalam showing that for any homogeneous ideal I in a standard graded algebra S, the regularity of I^t can be written as dt+\epsilon for some non-negative integers d, \epsilon, and all large t. In the special case where I contains a power of S_+ and is generated by forms of a single degree, our formula gives an interpretation of \epsilon: it is one less than the maximum regularity of a fiber of the morphism associated to I.   These formulas have strong consequences for ideals generated by generic forms. 
David Eisenbud and Joe Harris