Elliptic curves and rank-2 vector bundles on the prime Fano threefold of genus 7

Details Elliptic curves and rank-2 vector bundles on the prime Fano threefold of genus 7 Elliptic curves and rank-2 vector bundles on the prime Fano threefold of genus 7

2002-09-09

arxiv.org/abs/math/0209094v1

Mathematics

Algebraic Geometry

37 pages

Scientific paper

According to Mukai, any prime Fano threefold X of genus 7 is a linear section of the spinor tenfold in the projectivized half-spinor space of Spin(10). It is proven that the moduli space of stable rank-2 vector bundles with Chern classes c_1=1,c_2=5 on a generic X is isomorphic to the curve of genus 7 obtained by taking an orthogonal linear section of the spinor tenfold. This is an inverse of Mukai's result on the isomorphism of a non-abelian Brill--Noether locus on a curve of genus 7 to a Fano threefold of genus 7. An explicit geometric construction of both isomorphisms and a similar result for K3 surfaces of genus 7 are given.

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