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

Abstract

Given a smooth prime Fano threefold $X$ of genus 7 we consider its homologically projectively dual curve $\Gamma$ and the natural integral functor ${\Phi }^{!}:D^b(X)\to D^b(\Gamma )$. We prove that, for $d\ge 6$, ${\Phi }^{!}$ gives a birational map from a component of the moduli scheme $M_X(2,1,d)$ of rank 2 stable sheaves on $X$ with $c_1=1$, $c_2=d$ to a generically smooth $2d-9$-dimensional component of the Brill-Noether variety $W_{d-5,5d-24}^{2d-11}$ of stable vector bundles on $\Gamma$ 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 $M_X(2,1,6)$ is fine. For general $X$, this moduli space is a smooth irreducible threefold.

37 pages; revised version

References in corpus (1)

• The Abel-Jacobi map of a moduli component of vector bundles on the cubic threefold

Cited by in corpus (1)

• On the period map for prime Fano threefolds of degree 10