Maximal minors and linear powers

TL;DR

This paper proves that the ideal of maximal minors of a sufficiently general matrix with linear entries has linear powers, including all rational normal scrolls, based on conditions related to the heights of minors.

Contribution

It establishes conditions under which ideals of maximal minors have linear powers, extending to rational normal scrolls, based on genericity and height criteria.

Findings

Maximal minors of sufficiently general matrices have linear powers.

Rational normal scrolls possess linear powers.

Conditions involve heights of lower order minors.

Abstract

An ideal I in a polynomial ring S has linear powers if all the powers I^k of I have a linear free resolution. We show that the ideal of maximal minors of a sufficiently general matrix with linear entries has linear powers. The required genericity is expressed in terms of the heights of the ideals of lower order minors. In particular we prove that every rational normal scroll has linear powers.