Vector bundles on Fano varieties of genus ten

TL;DR

This paper describes a unique embedding of a Fano 4-fold of genus 10 into a Grassmannian and constructs moduli spaces of bundles on its linear sections, including a specific moduli space on K3 surfaces of genus 10.

Contribution

It introduces a unique linear embedding of a prime Fano 4-fold of genus 10 into G(3,6) and constructs related moduli spaces of bundles on linear sections.

Findings

Embedding of Fano 4-fold into Grassmannian G(3,6)

Construction of moduli space as a double cover of the plane

Description of moduli space on K3 surfaces of genus 10

Abstract

In this note we describe a unique linear embedding of a prime Fano 4-fold F of genus 10 into the Grassmannian G(3,6). We use this to construct some moduli spaces of bundles on linear sections of F. In particular the moduli space of bundles with Mukai vector (3,L,3) on a generic polarized K3 surface (S,L) of genus 10 is constructed as a double cover of the projective plane branched over a smooth sextic.