BCM-regularity of diagonal hypersurfaces and plus-pure thresholds in mixed characteristic

TL;DR

This paper introduces a new method for computing plus-pure thresholds in mixed characteristic, providing bounds, conditions for BCM-regularity, and classifying certain hypersurfaces.

Contribution

It develops a novel approach for plus-pure thresholds, establishes bounds, and classifies BCM-regular diagonal hypersurfaces in mixed characteristic.

Findings

Established lower bounds for plus-pure thresholds of diagonal hypersurfaces.

Provided necessary and sufficient conditions for BCM-regularity.

Classified BCM-regular diagonal hypersurfaces in mixed characteristic (0,2).

Abstract

We introduce a new method for computing plus-pure thresholds, a mixed-characteristic analogue of both log canonical thresholds and $F$-pure thresholds. We obtain some necessary conditions and some sufficient conditions for BCM-regularity of Fermat-type hypersurfaces. We also establish lower bounds for plus-pure thresholds of diagonal hypersurfaces in mixed characteristic. Furthermore, we give bounds for plus-pure thresholds of hypersurfaces in mixed characteristic $(0,2)$ using splitting-order sequences, introduced by Yoshikawa. As an application, we classify BCM-regular diagonal hypersurfaces in mixed characteristic $(0,2)$.