Some Results on Asymptotic Regularity of Ideal Sheaves

TL;DR

This paper investigates the asymptotic regularity of ideal sheaves on projective space, establishing conditions for linear growth and providing bounds in geometric settings involving nonsingular varieties and local complete intersections.

Contribution

It characterizes when the regularity of ideal sheaf powers grows linearly and offers regularity bounds in specific geometric contexts.

Findings

Linear asymptotic regularity when s=d

Regularity bounds for ideal sheaves on nonsingular varieties

Conditions for linear growth of regularity

Abstract

Let $\mathscr{I}$ be an ideal sheaf on $P^n$ defining a subscheme $X$. Associated to $\mathscr{I}$ there are two elementary invariants: the invariant $s$ which measures the positivity of $\mathscr{I}$, and the minimal number $d$ such that $\mathscr{I}(d)$ is generated by its global sections. It is now clear that the asymptotic behavior of $\reg \mathscr{I}^t$ 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 $\reg \mathscr{I}^t=dt+e$ for some positive constant $e$. We then turn to concrete geometric settings to study the asymptotic regularity of $\mathscr{I}$ in the case that $X$ is a nonsingular variety embedded by a very ample adjoint line bundle. Our approach also gives regularity bounds for $\mathscr{I}^t$ once we know $\reg \mathscr{I}$ and assume that $X$ is a local complete intersection.