Symbolic powers of the generic linkage of maximal minors

TL;DR

This paper explicitly describes the generators of the generic linkage of maximal minors, establishes the equality of symbolic and ordinary powers, and analyzes the algebraic properties of related rings using Gr"obner degenerations.

Contribution

It provides a detailed Gr"obner degeneration approach to understand the generators and properties of the generic linkage of maximal minors, including symbolic power equality and ring regularity.

Findings

Equality of symbolic and ordinary powers of J

Gorenstein property of the associated graded ring

F-regularity of the blowup algebras in positive characteristic

Abstract

Let $I$ be the ideal generated by the maximal minors of a matrix of indeterminates over a field and let $J$ denote the generic link, i.e., the most general link, of $I$. The generators of the ideal $J$ are not known. We provide an explicit description of the lead terms of the generators of $J$ using Gr\"obner degeneration. Indeed, we construct a degeneration which preserves the entire graded Betti table of $J$ on passing to the initial ideal. We leverage this construction to establish the equality of the symbolic and ordinary powers of $J$. Our analysis of the initial ideal readily yields the Gorenstein property of the associated graded ring of $J$, and, in positive characteristic, the $F$-rationality of the Rees algebra of $J$. Using the technique of $F$-split filtrations, we further obtain the $F$-regularity of the blowup algebras of $J$.