Formats of 6 x 6 skew matrices of linear forms with vanishing Pfaffian

TL;DR

This paper classifies all 6x6 skew-symmetric matrices of linear forms with zero Pfaffian, revealing a finite set of types characterized by specific zero patterns, which aids in understanding moduli spaces of vector bundles.

Contribution

It provides a complete classification of 6x6 skew-symmetric linear form matrices with vanishing Pfaffian, identifying their types based on zero patterns and relations.

Findings

Finite classification of matrix types based on zero patterns

Characterization of matrices relevant for moduli space compactification

Identification of specific linear relations among matrix entries

Abstract

We show that every skew-symmetric 6 x 6 matrix of linear forms with vanishing Pfaffian is congruent to one of finitely many types of matrices, each of which is characterised by a specific pattern of zeroes (and some other linear relations) among its entries. Such matrices are for example important for compactifying moduli spaces of stable rank 2 vector bundles with Chern classes c_1=0, c_2=2 on cubic threefolds.