# The Betti Numbers of a Determinantal Variety

## TL;DR

This paper computes the Poincaré polynomial of the determinantal variety defined by matrices with zero determinant, revealing its topological structure in the context of algebraic geometry.

## Contribution

It provides an explicit calculation of the Betti numbers for the determinantal variety, a significant advancement in understanding its topological invariants.

## Key findings

- Explicit Poincaré polynomial derived for the determinantal variety
- Betti numbers characterized for the variety of singular matrices
- Topological structure of the determinantal variety clarified

## Abstract

We determine the Poincar\'e polynomial of the determinantal variety $\{\det = 0\}$ in the projective space associated with the monoid of $n\times n$ matrices.