Polarized K3 surfaces of genus thirteen and curves of genus three

TL;DR

This paper characterizes certain polarized K3 surfaces of genus thirteen as complete intersections within moduli spaces of vector bundles over genus three curves, revealing their geometric structure.

Contribution

It provides a new description of polarized K3 surfaces of genus thirteen as intersections in Grassmannians related to vector bundle moduli spaces, especially for hyperelliptic curves.

Findings

K3 surfaces of genus 13 are described as complete intersections.

For hyperelliptic curves, the moduli space is a subvariety of a Grassmannian.

General such K3 surfaces are intersections of contact homogeneous varieties.

Abstract

We describe a general (primitively) polarized K3 surface $(S,h)$ with $(h^2)=24$ as a complete intersection variety with respect to vector bundles on the $6$-dimensional moduli space $\mathcal{N}^-$ of the stable vector bundles of rank two with fixed odd determinant on a curve $C$ of genus $3$. If the curve $C$ is hyperelliptic, then $\mathcal{N}^-$ is a subvariety of the $12$-dimensional Grassmann variety $\operatorname{Gr}(\mathbf{C}^8,2)$ defined by a pencil of quadric forms. In this case, our description implies that a general $(S,h)$ is the intersection of two (7-dimensional) contact homogeneous varieties of $\operatorname{Spin}(7)$ in the Grassmann variety $\operatorname{Gr}(\mathbf{C}^8,2)$.