Hilbert schemes, commuting matrices and hyperkähler geometry

Abstract

We represent algebraic curves via commuting matrix polynomials. This allows us to show that the Hilbert scheme of cohomologically stable non-planar curves of genus 0 and degree d $d$ in P 3 ∖ P 1 ${\mathbb {P}}^3\backslash {\mathbb {P}}^1$ is isomorphic to a complexified hyperkähler quotient of an open subset of a vector space by a non-reductive Lie group.