Geometry of genus sixteen K3 surfaces

TL;DR

This paper explores the geometry of genus sixteen K3 surfaces, revealing their embeddings, associated quadrics, and connections to hyperk"ahler and Debarre-Voisin varieties, advancing understanding of their moduli and related structures.

Contribution

It introduces an explicit geometric construction of genus sixteen K3 surfaces via quadrics and investigates their relation to hyperk"ahler and Debarre-Voisin varieties.

Findings

Embedding of the projectivization in bP_9 with quadrics generating the ideal

Effective method to compute these quadrics from Mukai's unirationalization

Double cover of bP_9 ramified on a degree 10 hypersurface

Abstract

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 hyperk\"ahler 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.