Some results on asymptotic regularity of ideal sheaves

Let I be an ideal sheaf on Pn defining a subscheme X. Associated to I there are two elementary invariants: the invariant s which measures the positivity of I, and the minimal number d such that I(d) is generated by its global sections. It is now clear that the asymptotic behavior of is governed by s but usually not linear. In this paper, we first describe the linear behavior of the asymptotic regularity by showing that if s=d, i.e., s reaches its maximal value, then for t large enough for some positive constant e. We then turn to concrete geometric settings to study the asymptotic regularity of I in the case that X is a nonsingular variety embedded by a very ample adjoint line bundle. Our approach also gives regularity bounds for It once we know and assume that X is a local complete intersection.