Castelnuovo-Mumford regularity and Schubert geometry

TL;DR

This paper investigates the Castelnuovo-Mumford regularity of tangent cones of Schubert varieties, proposing conjectures and proving results for the covexillary case, advancing understanding in algebraic geometry and combinatorics.

Contribution

It introduces new conjectures on regularity of tangent cones and proves them for covexillary Schubert varieties, extending prior work on tableau rules and tangent cone analysis.

Findings

Conjectures on regularity are formulated.

Proved results for covexillary Schubert varieties.

Extended tableau rule methods for matrix Schubert varieties.

Abstract

We study the Castelnuovo-Mumford regularity of tangent cones of Schubert varieties. Conjectures about this statistic are presented; these are proved for the covexillary case. This builds on work of L. Li and the author on these tangent cones, as well as that of J. Rajchgot-Y. Ren-C. Robichaux-A. St. Dizier-A. Weigandt and J. Rajchgot-C. Robichaux-A. Weigandt on tableau rules for computing regularity of some matrix Schubert varieties.