Vector bundles on Fano varieties of genus ten

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 $$\mathbf P ^2$$ branched over a smooth sextic.