Vector bundles on Fano threefolds of genus 7 and Brill-Noether loci

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Given a smooth prime Fano threefold X of genus 7 we consider its homologically projectively dual curve Γ and the natural integral functor Φ!:Dᵇ(X) → Dᵇ(Γ). We prove that, for d≥ 6, Φ! gives a birational map from a component of the moduli scheme Mₓ(2,1,d) of rank 2 stable sheaves on X with c₁=1, c₂=d to a generically smooth 2d-9-dimensional component of the Brill-Noether variety W²ᵈ⁻¹¹d₋₅,₅d₋₂₄ of stable vector bundles on Γ of rank d-5 and degree 5d-24 with at least 2d-10 sections. This map turns out to be an isomorphism for d=6, and the moduli space Mₓ(2,1,6) is fine. For general X, this moduli space is a smooth irreducible threefold.