Hilbert schemes, commuting matrices, 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.