# Matroid connectivity and singularities of configuration hypersurfaces

## TL;DR

This paper explores the geometric and algebraic properties of configuration hypersurfaces derived from matroids, revealing how matroid connectivity influences their singularities and scheme structures.

## Contribution

It provides a detailed analysis of the singularities and degeneracy schemes of configuration hypersurfaces associated with matroids, highlighting the impact of matroid connectivity.

## Key findings

- Configuration hypersurfaces are integral for 2-connected matroids.
- Second degeneracy schemes are reduced Cohen-Macaulay of codimension 3 for 2-connected matroids.
- For 3-connected matroids, the second degeneracy scheme is also integral.

## Abstract

Consider a linear realization of a matroid over a field. One associates with it a configuration polynomial and a symmetric bilinear form with linear homogeneous coefficients. The corresponding configuration hypersurface and its non-smooth locus support the respective first and second degeneracy scheme of the bilinear form. We show that these schemes are reduced and describe the effect of matroid connectivity: for (2-)connected matroids, the configuration hypersurface is integral, and the second degeneracy scheme is reduced Cohen-Macaulay of codimension 3. If the matroid is 3-connected, then also the second degeneracy scheme is integral. In the process, we describe the behavior of configuration polynomials, forms and schemes with respect to various matroid constructions.