I shall present recent work with Michal Kapustka,  describing a unique
linear embedding of a prime Fano 4-fold F of genus 10 into the
Grassmannian G(3,6).  Combining this with Mukai's description of a linear
embedding of F into G(2,7),  we construct some moduli spaces of bundles on
sections of F.  In particular the moduli space of bundles with Mukai
vector (3,L,3) on a generic polarized K3 surface (S,L) is constructed as a
     double cover of P^2 branched over a smooth sextic.