Geometric Shifts and Positroid Varieties

TL;DR

This paper investigates the geometric and combinatorial properties of positroid varieties within the Grassmannian, proposing methods to compute their cohomology classes and introducing the concept of expected codimension, with some limitations.

Contribution

It introduces a combinatorial method to compute the expected codimension of positroid varieties and explores degeneration techniques for certain matroids, highlighting their limitations.

Findings

Degeneration methods work for rank 3 positroids but fail beyond that.

Expected codimension matches actual codimension for positroid varieties.

Counterexamples show degeneration approaches are limited for general matroids.

Abstract

Matroid varieties are the closures in the Grassmannian of sets of points defined by specifying which Pl\"ucker coordinates vanish and which don't --- the set of nonvanishing Pl\"ucker coordinates forms a well-studied object called a matroid. Many problems in enumerative geometry could be solved if there were an efficient way to compute the cohomology class of a matroid variety in an efficient way just from the combinatorial data contained in the matroid itself. Unfortunately, in full generality, this problem is known to be completely intractable. In this thesis, we explore two attempts to get a handle on this problem in the special case of a better-behaved class of matroids called positroids. First we examine a method based on degenerations on the Grassmannian that's known to be successful for an even smaller class of matroids called interval rank matroids, showing that it can be made to work with some effort in rank 3 but that there is a counterexample demonstrating that it is hopeless beyond that. Finally, we give a way of producing a number for each matroid variety called its expected codimension that can be computed combinatorially solely from the matroid itself and show that it agrees with the actual codimension in the case of a positroid variety.