The Expected Codimension of a Matroid Variety

TL;DR

This paper introduces a combinatorial method to estimate the codimension of matroid varieties in the Grassmannian, proving its accuracy for positroid varieties, thus linking combinatorial data to geometric properties.

Contribution

It defines the expected codimension for matroid varieties based solely on vanishing Pl"ucker coordinates and proves its equality with actual codimension for positroid varieties.

Findings

Expected codimension can be computed combinatorially from vanishing Pl"ucker coordinates.

For positroid varieties, the expected codimension matches the actual codimension.

Provides a new tool for understanding the geometry of matroid varieties.

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. In general these varieties are very ill-behaved, but in many cases one can estimate their codimension by keeping careful track of the conditions imposed by the vanishing of each Pl\"ucker coordinates on the columns of the matrix representing a point of the Grassmannian. This paper presents a way to make this procedure precise, producing a number for each matroid variety called its expected codimension that can be computed combinatorially solely from the list of Pl\"ucker coordinates that are prescribed to vanish. We prove that for a special, well-studied class of matroid varieties called positroid varieties, the expected codimension coincides with the actual codimension.