The arc space of the Grassmannian

TL;DR

This paper investigates the arc space of the Grassmannian, linking its structure to singularities of Schubert varieties, and introduces combinatorial tools involving plane partitions to analyze their invariants and singularity measures.

Contribution

It introduces a decomposition of the arc space analogous to Schubert cell decomposition, connecting combinatorics of plane partitions to singularity invariants and solving related problems.

Findings

Reduced log canonical threshold computations to linear programming.

Proved the Nash map is bijective for Schubert varieties.

Established a combinatorial framework using plane partitions.

Abstract

We study the arc space of the Grassmannian from the point of view of the singularities of Schubert varieties. Our main tool is a decomposition of the arc space of the Grassmannian that resembles the Schubert cell decomposition of the Grassmannian itself. Just as the combinatorics of Schubert cells is controlled by partitions, the combinatorics in the arc space is controlled by plane partitions (sometimes also called 3d partitions). A combination of a geometric analysis of the pieces in the decomposition and a combinatorial analysis of plane partitions leads to invariants of the singularities. As an application we reduce the computation of log canonical thresholds of pairs involving Schubert varieties to an easy linear programming problem. We also study the Nash problem for Schubert varieties, showing that the Nash map is always bijective in this case.