# Hilbert schemes, commuting matrices, and hyperk\"ahler geometry

## TL;DR

This paper links algebraic curves to commuting matrices and demonstrates that certain Hilbert schemes are isomorphic to hyperk"ahler quotients, revealing new geometric structures in algebraic geometry.

## Contribution

It introduces a novel representation of algebraic curves using commuting matrix polynomials and establishes an isomorphism with hyperk"ahler quotients for specific Hilbert schemes.

## Key findings

- Hilbert scheme of stable twisted rational curves is isomorphic to a hyperk"ahler quotient.
- Representation of algebraic curves via commuting matrices.
- Connection between algebraic geometry and hyperk"ahler geometry.

## Abstract

We represent algebraic curves via commuting matrix polynomials. This allows us to show that the Hilbert scheme of cohomologically stable twisted rational curves of degree $d$ in ${\Bbb P}^3\backslash {\Bbb P}^1$ is isomorphic to a complexified hyperk\"ahler quotient of an open subset of a vector space by a non-reductive Lie group.