Rank-two stable sheaves with odd determinant on Fano threefolds of genus nine

According to Mukai and Iliev, a smooth prime Fano threefold of genus is associated with a surface , ruled over a smooth plane quartic , and the derived category of embeds into that of by a theorem of Kuznetsov. We use this setup to study the moduli spaces of rank- stable sheaves on with odd determinant. For each , we prove that a component of their moduli space is birational to a Brill-Noether locus of vector bundles with fixed rank and degree on , having enough sections when twisted by . For , we prove that is isomorphic to the blow-up of the Picard variety along the curve parametrizing lines contained in .