Linear spaces of matrices of constant rank and instanton bundles

TL;DR

This paper introduces a novel approach using vector bundles and derived categories to study linear spaces of skew-symmetric matrices with constant rank, leading to new examples and explicit constructions.

Contribution

It develops a new method for analyzing matrices of constant rank via instanton bundles, producing new examples and explaining previous unique cases.

Findings

Existence of new 10x10 and 14x14 matrix examples

Explicit algorithm for constructing a 14x14 matrix

Enhanced understanding of matrices with constant rank through vector bundle techniques

Abstract

We present a new method to study 4-dimensional linear spaces of skew-symmetric matrices of constant co-rank 2, based on rank 2 vector bundles on P^3 and derived category tools. The method allows one to prove the existence of new examples of size 10x10 and 14x14 via instanton bundles of charge 2 and 4 respectively, and provides an explanation for what used to be the only known example (Westwick 1996). We also give an algorithm to construct explicitly a matrix of size 14 of this type.