Hilbert squares of genus 16 K3 surfaces

Abstract

We consider the geometry of a general polarized K3 surface $(S,h)$ of genus 16 and its Fourier-Mukai partner $(S^{\prime },h^{\prime })$. We prove that $S^{[2]}$ is isomorphic to the moduli space $M_{S^{\prime }}(2,h^{\prime },7)$ of stable sheaves with Mukai vector $(2,h^{\prime },7)$ and describe the embeddings of the projectivization of the stable vector bundle of Mukai vector $(2,-h^{\prime },8)$ over $S^{\prime }$ into these two isomorphic hyper-Kähler fourfolds. Following the work of Frédéric Han \cite{Frederic}, we explicitly construct a 3-form $t_1\in {\wedge }^3{V}_{10}^{\ast }$, such that $S^{[2]}$ is isomorphic to the Debarre-Voisin fourfold in $G(6,V_{10})$ associated to $t_1\in {\wedge }^3{V}_{10}^{\ast }$. Then the reconstruction program is also explained from two perspectives, i.e. in terms of families of stable representations and families of twisted cubics.