Maximal minors of $1$-generic matrices have rational singularities

TL;DR

This paper proves that the quotient rings formed by maximal minors of 1-generic matrices have rational singularities, confirming a longstanding conjecture and extending previous results on determinantal rings.

Contribution

It establishes that these quotient rings are normal and possess rational singularities, resolving Eisenbud's conjecture and generalizing earlier findings.

Findings

Quotient rings by maximal minors of 1-generic matrices have rational singularities.

These rings are shown to be normal.

The result extends to characteristic zero cases previously studied.

Abstract

We show that the quotient ring by the ideal of maximal minors of a $1$-generic matrix has rational singularities. This answers a conjecture of Eisenbud (1988) that such rings are normal, and generalizes a result of Conca, Mostafazadehfard, Singh and Varbaro (2018) that generic Hankel determinantal rings have rational singularities in characteristic zero.