Blowup algebras of determinantal modules

TL;DR

This paper investigates the algebraic properties of blowup algebras associated with modules formed from determinantal ideals, providing explicit Gröbner bases and revealing their favorable geometric and algebraic characteristics.

Contribution

It explicitly determines Gröbner bases for presentation ideals of multi-Rees algebras of determinantal modules and establishes their Cohen--Macaulay, Koszul, and singularity properties.

Findings

Multi-blowup algebras are Koszul Cohen--Macaulay normal domains.

They have rational singularities in characteristic zero.

They are F-rational in positive characteristic.

Abstract

We study the blowup algebras of the modules that are direct sums of ideals generated by either maximal minors of a ladder matrix or unit interval determinantal ideals. Specifically, we determine Gr\"{o}bner bases for the presentation ideals of multi-Rees algebras and their special fiber rings. Our analysis reveals that the multi-blowup algebras are Koszul Cohen--Macaulay normal domains, possess rational singularities in characteristic zero, and are F-rational in positive characteristic.