Bioriented flags and resolutions of Schubert varieties

TL;DR

This paper introduces new geometric resolutions for Schubert varieties using incidence relations, offering alternatives to Bott-Samelson resolutions and leading to the development of W-flag varieties with applications in desingularization.

Contribution

It constructs novel Kempf-Laksov type and embedded resolutions for Schubert varieties, and introduces W-flag varieties as a new class of algebro-geometric objects.

Findings

Constructed Kempf-Laksov type resolutions for all Schubert varieties.

Developed embedded resolutions within Grassmannians.

Introduced W-flag varieties and achieved simple desingularization methods.

Abstract

We use incidence relations running in two directions in order to construct a Kempf-Laksov type resolution for any Schubert variety of the complete flag manifold but also an embedded resolution for any Schubert variety in the Grassmannian. These constructions are alternatives to the celebrated Bott-Samelson resolutions. The second process led to the introduction of W-flag varieties, algebro-geometric objects that interpolate between the standard flag manifolds and products of Grassmannians, but which are singular in general. The surprising simple desingularization of a particular such type of variety produces an embedded resolution of the Schubert variety within the Grassmannian.