Hilbert schemes, commuting matrices and hyperkähler geometry

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 P3∖P1${\mathbb {P}}^3\backslash {\mathbb {P}}^1$ is isomorphic to a complexified hyperkähler quotient of an open subset of a vector space by a non‐reductive Lie group.