A Castelnuovo-Mumford regularity bound for scrolls

TL;DR

This paper establishes a linear bound on the Castelnuovo-Mumford regularity of scrolls over curves, relating it to degree, codimension, and genus, applicable in any characteristic.

Contribution

It provides the first characteristic-independent linear regularity bound for scrolls over smooth projective curves.

Findings

Regularity bound: reg(X) ≤ d - e + 1 + g(e - 1)

Applicable in arbitrary characteristic

Extends known bounds to a broader class of scrolls

Abstract

Let $X \subseteq \mathbb{P}^r$ be a scroll of codimension $e$ and degree $d$ over a smooth projective curve of genus $g$. The purpose of this paper is to prove a linear Castelnuovo-Mumford regularity bound that reg$(X) \leq d-e+1+g(e-1)$. This bound works over an algebraically closed field of arbitrary characteristic.