Hankel determinantal rings have rational singularities

TL;DR

This paper proves that Hankel determinantal rings possess rational singularities in characteristic zero and are F-pure in positive characteristic, also describing their divisor class groups and Cohen-Macaulay modules.

Contribution

It establishes the singularity type of Hankel determinantal rings across all characteristics and characterizes their divisor class groups and Cohen-Macaulay modules.

Findings

Hankel determinantal rings have rational singularities in characteristic zero.

They are F-pure in positive characteristic.

Complete description of divisor class groups and Cohen-Macaulay modules.

Abstract

Hankel determinantal rings, i.e., determinantal rings defined by minors of Hankel matrices of indeterminates, arise as homogeneous coordinate rings of higher order secant varieties of rational normal curves; they may also be viewed as linear specializations of generic determinantal rings. We prove that, over fields of characteristic zero, Hankel determinantal rings have rational singularities; in the case of positive prime characteristic, we prove that they are F-pure. Independent of the characteristic, we give a complete description of the divisor class groups of these rings, and show that each divisor class group element is the class of a maximal Cohen-Macaulay module.