$F$-purity and the $F$-pure threshold as invariants of linkage

TL;DR

This paper investigates how $F$-purity and the $F$-pure threshold behave under generic linkage, identifying cases where these invariants are preserved and establishing $F$-regularity of certain generic links.

Contribution

It demonstrates the preservation of $F$-purity and the $F$-pure threshold in specific classes of ideals under generic linkage and establishes $F$-regularity for these cases.

Findings

$F$-purity and the $F$-pure threshold are preserved in certain classes of ideals.

The $F$-regularity of generic links of these ideals is established.

The $F$-pure threshold of generic residual intersections of a complete intersection is studied.

Abstract

The generic link of an unmixed radical ideal is radical (in fact, prime). We show that the squarefreeness of the initial ideal and $F$-purity are, however, not preserved along generic links. On the flip side, for several important cases in liaison theory, including generic height three Gorenstein ideals and the maximal minors of a generic matrix, we show that the squarefreeness of the initial ideal, $F$-purity, and the $F$-pure threshold are each preserved along generic links by identifying a property of such ideals which propagates along generic links. We use this property to establish the $F$-regularity of the generic links of such ideals. Finally, we study the $F$-pure threshold of the generic residual intersections of a complete intersection ideal and answer a related question of Kim--Miller--Niu.