A Prym Construction for the Cohomology of a Cubic Hypersurface
نویسنده
چکیده
Mumford defined a natural isomorphism between the intermediate jacobian of a conicbundle over P and the Prym variety of a naturally defined étale double cover of the discrminant curve of the conic-bundle. Clemens and Griffiths used this isomorphism to give a proof of the irrationality of a smooth cubic threefold and Beauville later generalized the isomorphism to intermediate jacobians of odd-dimensional quadric-bundles over P. We further generalize the isomorphism to the primitive cohomology of a smooth cubic hypersurface in Pn.
منابع مشابه
Prym-Tjurin Constructions on Cubic Hypersurfaces
In this paper, we give a Prym-Tjurin construction for the cohomology and the Chow groups of a cubic hypersurface. On the space of lines meeting a given rational curve, there is the incidence correspondence. This correspondence induces an action on the primitive cohomology and the primitive Chow groups. We first show that this action satisfies a quadratic equation. Then the Abel-Jacobi mapping i...
متن کامل1 1 M ay 2 00 4 Singularities of the Prym Theta Divisor
For the Jacobian of a curve, the Riemann singularity theorem gives a geometric interpretation of the singularities of the theta divisor in terms of special linear series on the curve. This paper proves an analogous theorem for Prym varieties. Applications of this theorem to cubic threefolds, and Prym varieties of dimension five, are also considered.
متن کاملS ep 2 00 8 Singularities of the Prym Theta Divisor Sebastian
For the Jacobian of a curve, the Riemann singularity theorem gives a geometric interpretation of the singularities of the theta divisor in terms of special linear series on the curve. This paper proves an analogous theorem for Prym varieties. Applications of this theorem to cubic threefolds, and Prym varieties of dimension five, are also considered.
متن کاملOn Relations among 1-cycles on Cubic Hypersurfaces
In this paper we give two explicit relations among 1-cycles modulo rational equivalence on a smooth cubic hypersurface X. Such a relation is given in terms of a (pair of) curve(s) and its secant lines. As the first application, we reprove Paranjape’s theorem that CH1(X) is always generated by lines and that it is isomorphic to Z if the dimension of X is at least 5. Another application is to the...
متن کاملThe Tangent Cones at Double points of Prym-Canonical Divisors of Curves of genus 7
Let η be a line bundle on a smooth curve X with η^2=0 such that π_η, the double covering induced by η is an etale morphism. Assume also that X_η be the Prym-canonical model of X associated to K_X.η and Q is a rank 4 quadric containing X_η. After stablishing the projective normality of the prym-canonical models of curves X with Clifford index 2, we obtain in this paper a sufficient condition for...
متن کامل