Koszul determinantal rings and $2\times e$ matrices of linear forms

Abstract

Let $k$ be an algebraically closed field of characteristic $0$. Let $X$ be a $2\times e$ matrix of linear forms over a polynomial ring $k[{\mathsf{x}}_1,\dots ,{\mathsf{x}}_n]$ (where $e,n\ge 1$). We prove that the determinantal ring $R=k[{\mathsf{x}}_1,\dots ,{\mathsf{x}}_n]/I_2(X)$ is Koszul if and only if in the Kronecker-Weierstrass normal form of $X$, the largest length of a nilpotent block is at most twice the smallest length of a scroll block. As an application, we classify rational normal scrolls whose all section rings by natural coordinates are Koszul. This result settles a conjecture due to Conca.

Final version; 30 pages. To appear in Michigan Mathematical Journal

References in corpus (1)

• On the behavior of Castelnuovo-Mumford regularity with respect to some functors

Cited by in corpus (1)

• Eigenschemes and the Jordan canonical form