Blowup algebras of determinantal ideals in prime characteristic

TL;DR

This paper investigates the properties of blowup algebras associated with determinantal ideals in prime characteristic, focusing on their $F$-splitting, regularity, depth, and initial ideals, providing new bounds and structural insights.

Contribution

It introduces the concepts of $F$-split filtrations and symbolic $F$-split ideals, and establishes key properties of blowup algebras for various classes of determinantal ideals.

Findings

Blowup algebras are $F$-split or strongly $F$-regular under certain conditions.

The limit of normalized regularity of symbolic powers exists and depth stabilizes.

There exists a monomial order where initial ideals commute with symbolic powers for determinantal ideals.

Abstract

We study when blowup algebras are $F$-split or strongly $F$-regular. Our main focus is on algebras given by symbolic and ordinary powers of ideals of minors of a generic matrix, a symmetric matrix, and a Hankel matrix. We also study ideals of Pfaffians of a skew-symmetric matrix. We use these results to obtain bounds on the degrees of the defining equations for these algebras. We also prove that the limit of the normalized regularity of the symbolic powers of these ideals exists and that their depth stabilizes. Finally, we show that, for determinantal ideals, there exists a monomial order for which taking initial ideals commutes with taking symbolic powers. To obtain these results we develop the notion of $F$-split filtrations and symbolic $F$-split ideals.