Residual Intersections and Schubert Varieties

TL;DR

This paper presents a uniform pattern for understanding residual intersections of certain Schubert varieties across ADE types, using representation theory to analyze their free resolutions and geometric linkages.

Contribution

It introduces a new uniform pattern for residual intersections of opposite Schubert varieties, supported by a representation-theoretic proof applicable across ADE types.

Findings

Identifies a pattern for residual intersections in Schubert varieties

Provides free resolutions relevant to the structure of these varieties

Uses representation theory for a uniform proof

Abstract

Inspired by the work of Ulrich and Huneke-Ulrich, we describe a pattern to show that the ideals of certain opposite embedded Schubert varieties defined by this pattern arise by taking residual intersections of two geometrically linked opposite Schubert varieties. This pattern is uniform for the ADE types. Some of the free resolutions of the Schubert varieties in question are important for the structure of finite free resolutions. Our proof is representation theoretical and uniform for our pattern, however it is possible to derive our results using case-by-case analysis and the aid of a computer.