Hyperplane Arrangements in the Grassmannian

TL;DR

This paper investigates the Euler characteristic of the Grassmannian with hyperplane sections removed, providing combinatorial formulas and computational methods for both complex and real cases.

Contribution

It introduces a combinatorial approach to compute the Euler characteristic of hyperplane arrangements in the Grassmannian, focusing on generic and special Schubert divisors.

Findings

Derived a combinatorial formula for the Euler characteristic.

Developed practical methods for symbolic and numerical computation.

Analyzed both complex and real cases of hyperplane arrangements.

Abstract

The Euler characteristic of a very affine variety encodes the algebraic complexity of solving likelihood (or scattering) equations on this variety. We study this quantity for the Grassmannian with $d$ hyperplane sections removed. We provide a combinatorial formula, and explain how to compute this Euler characteristic in practice, both symbolically and numerically. Our particular focus is on generic hyperplane sections and on Schubert divisors. We also consider special Schubert arrangements relevant for physics. We study both the complex and the real case.