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

Abstract

Let X be a smooth prime Fano threefold of genus 7 and let Γ be its homologically projectively dual curve. We prove that, for d ≥ 6, an irreducible component of the moduli scheme MX(2, 1, d) of rank-2 stable sheaves on X with c1= 1, c2= d is birational to a generically smooth (2d - 9)-dimensional component of the Brill–Noether variety [Formula: see text] of stable vector bundles on Γ of rank d - 5 and degree 5d-24 with at least 2d - 10 independent global sections. The space MX(2, 1, 6) is proved to be isomorphic to [Formula: see text], and to be a smooth irreducible threefold if X is general enough.