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

According to Mukai and Iliev, a smooth prime Fano threefold $X$ of genus $9$ is associated with a surface $mathbb{P }(mathcal{V })$ , ruled over a smooth plane quartic $varGamma $ , and the derived category of $varGamma $ embeds into that of $X$ by a theorem of Kuznetsov. We use this setup to study the moduli spaces of rank- $2$ stable sheaves on $X$ with odd determinant. For each $c_2 ge 7$ , we prove that a component of their moduli space $mathsf{M}_X(2,1,c_2)$ is birational to a Brill–Noether locus of vector bundles with fixed rank and degree on $varGamma $ , having enough sections when twisted by $mathcal{V }$ . For $c_2=7$ , we prove that $mathsf{M}_X(2,1,7)$ is isomorphic to the blow-up of the Picard variety $text{ Pic}^{2}({varGamma })$ along the curve parametrizing lines contained in $X$ .