Vector Bundles on Fano Threefolds of Genus 7 and Brill-noether Loci

Given a smooth prime Fano threefold X of genus 7 we consider its homologically projectively dual curve Γ and the natural integral functor Φ : D(X) → D(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 (2 d − 9)-dimensional component of the BrillNoether variety W 2d−11 d−5,5 d−24 of stable vector bundles on Γ of rank d − 5 and degree 5 d − 24 with at least 2 d − 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.