A Castelnuovo-Mumford regularity bound for scrolls

Let X⊆Pr 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)≤d−e+1+g(e−1). This bound works over an algebraically closed field of arbitrary characteristic.