Smooth Points on Positroid Varieties

TL;DR

This paper provides a criterion to determine smoothness of positroid varieties at specific points in the Grassmannian, using affine pipe dreams, and explores minimal singular cases within a partially ordered set of positroid pairs.

Contribution

The paper introduces a quick test for smoothness of positroid varieties at fixed points, extending the understanding of their local geometry and singularities.

Findings

A criterion for smoothness at positroid points using affine pipe dreams.

Identification of minimal singular positroid pairs in the partial order.

Application of results to smoothness of Schubert varieties at 321-avoiding permutations.

Abstract

In the Grassmannian $Gr_{\mathbb{C}}(k,n)$ we have positroid varieties $\Pi_f$, each indexed by a bounded affine permutation $f$ and containing torus-fixed points $\lambda \in \Pi_f$. In this paper we consider the partially ordered set consisting of quadruples $(k,n,\Pi_f,\lambda)$ (or \textit{(positroid) pairs} $(\Pi_f,\lambda)$ for short). The partial order is the ordering given by the covering relation $\lessdot$ where $(\Pi_f',\lambda') \lessdot (\Pi_f,\lambda)$ if $\Pi_f'$ is obtained by $\Pi_f$ by \textit{deletion} or \textit{contraction.} Using the results of Snider [2010], we know that positroid varieties can be studied in a neighborhood of each of these points by \textit{affine pipe dreams.} Our main theorem provides a quick test of when a positroid variety is smooth at one of these given points. It is sufficient to test smoothness of a positroid variety by using the main result to test smoothness at each of these points. These results can also be applied to the question of whether Schubert varieties in flag manifolds are smooth at points given by 321-avoiding permutations, as studied in Graham/Kreimer [2020]. We have a secondary result, which describes the minimal singular positroid pairs in our ordering - these are the positroid pairs where any deletion or contraction causes it to become smooth.