Free resolutions of some Schubert singularities

TL;DR

This paper constructs free resolutions for specific Schubert varieties, including determinantal varieties, using Schubert geometry, and explores their cohomological properties on partial flag varieties.

Contribution

It introduces a new geometric approach to resolving Schubert singularities, extending known results to broader classes of varieties.

Findings

Resolved determinantal varieties using Schubert geometry

Connected free resolutions to cohomology of non-reducible bundles

Provided explicit constructions for certain Schubert singularities

Abstract

In this paper we construct free resolutions of certain class of closed subvarieties of affine spaces (the so-called "opposite big cells" of Grassmannians). Our class covers the determinantal varieties, whose resolutions were first constructed by A. Lascoux (Adv. Math., 1978). Our approach uses the geometry of Schubert varieties. An interesting aspect of our work is its connection to the computation of the cohomology of homogeneous bundles (that are not necessarily completely reducible) on partial flag varieties.