  Let X = Proj R be a projective scheme over a field k, and let I be an ideal
in R generated by forms of the same degree d. Let Y --> X be the blowing up of
X along the subscheme defined by I, and let f: Y --> Z be the projection of Y
given by the divisor dH - E, where E is the exceptional divisor of the blowup
and H is the pullback of a general hyperplane in X. We investigate how the
asymptotic linearity of the regularity and a*-invariant of I^q (for q large) is
related to invariants of fibers of f.
