Parametrization of Singθ for a Fano 3-fold of Genus 7 by Moduli of Vector Bundles
نویسنده
چکیده
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). The orthogonal linear section of the spinor tenfold is a canonical genus-7 curve Γ, and the intermediate Jacobian J(X) is isomorphic to the Jacobian of Γ. It is proven that, for a generic X , the Abel-Jacobi map of the family of elliptic sextics on X factors through the moduli space of rank-2 vector bundles with c1 = −KX and deg c2 = 6 and that the latter is birational to the singular locus of the theta divisor of J(X).
منابع مشابه
Vector Bundles and Brill–Noether Theory
After a quick review of the Picard variety and Brill–Noether theory, we generalize them to holomorphic rank-two vector bundles of canonical determinant over a compact Riemann surface. We propose several problems of Brill–Noether type for such bundles and announce some of our results concerning the Brill–Noether loci and Fano threefolds. For example, the locus of rank-two bundles of canonical de...
متن کاملVector 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...
متن کاملVector Bundles on a K3 Surface
A K3 surface is a quaternionic analogue of an elliptic curve from a view point of moduli of vector bundles. We can prove the algebraicity of certain Hodge cycles and a rigidity of curve of genus eleven and gives two kind of descriptions of Fano threefolds as applications. In the final section we discuss a simplified construction of moduli spaces. 2000 Mathematics Subject Classification: 14J10, ...
متن کاملPfaffian Lines and Vector Bundles on Fano Threefolds of Genus 8 A. Iliev and L. Manivel
Let X be a general complex Fano threefold of genus 8. We prove that the moduli space of rank two semistable sheaves on X with Chern numbers c1 = 1, c2 = 6 and c3 = 0 is isomorphic to the Fano surface F (X) of conics on X . This surface is smooth and isomorphic to the Fano surface of lines in the orthogonal to X cubic threefold. Inside F (X), the non-locally free sheaves are parameterized by a s...
متن کاملMinimal Rational Curves in Moduli Spaces of Stable Bundles
Let C be a smooth projective curve of genus g ≥ 2 and L be a line bundle on C of degree d. Assume that r ≥ 2 is an integer coprime with d. Let M := UC(r,L) be the moduli space of stable vector bundles on C of rank r and with the fixed determinant L. It is well-known that M is a smooth projective Fano variety with Picard number 1. For any projective curve in M , we can define its degree with res...
متن کامل