K3 surfaces of genus sixteen

Abstract

The generic polarized $K3$ surface $(S, h)$ of genus 16, that is, $(h^2)=30$, is described in a certain compactified moduli space $\mathcal T$ of twisted cubics in $\mathbb{P}^3$, as a complete intersection with respect to an almost homogeneous vector bundle of rank 10. As corollary we prove the unirationality of the moduli space $\mathcal{F}_{16}$ of such K3 surfaces.