Nearly Toric Schubert Varieties of Type A

TL;DR

This paper introduces nearly toric varieties, explores their properties within Schubert varieties of type A, and establishes combinatorial characterizations, counting formulas, and connections to Dyck paths.

Contribution

It defines nearly toric varieties in the context of Schubert varieties and provides combinatorial criteria and enumerations for smooth and singular cases.

Findings

Characterization of smooth nearly toric Schubert varieties

Enumeration formulas for nearly toric Schubert varieties

Connection established between Dyck paths and nearly toric Schubert varieties

Abstract

A notion of a nearly toric variety is introduced. The examples of nearly toric varieties in the context of Schubert varieties are discussed. In particular, combinatorial characterizations of the smooth and singular nearly toric Schubert varieties are found. Furthermore, the counts and generating series are determined. Additionally, a connection between Dyck paths and a certain family of nearly toric Schubert varieties is established.