Geometry of genus sixteen K3 surfaces

Polarized K3 surfaces of genus sixteen have a Mukai vector bundle of rank two. We study the geometry of the projectivization of this bundle. We prove that it has an embedding in ${\mathbb{P}}_9$ with an ideal generated by quadrics. We give an effective method to compute these quadrics from a general choice in Mukai’s unirationalization of the moduli space. This linear system gives a double cover of ${\mathbb{P}}_9$ ramified on a degree $10$ hypersurface. It gives relative Weddle/Kummer surfaces over a Peskine variety associated to an explicit trivector. This work is also motivated by hyperkähler geometry and Debarre-Voisin varieties. Oberdieck showed that the Hilbert square of a general K3-surface of genus $16$ is a Debarre-Voisin variety for some trivector. We start to investigate the relationship between these two trivectors.