Introduction: A good understanding of the geometry of a theta divisor Θ of a principally polarized abelian variety (A,Θ) requires a knowledge of properties of its canonical linear system, the Gauss linear system |OΘ(Θ)|. A striking feature of the theta divisor Θ(C) of the Jacobian of a curve C is that the dual of the branch divisor of the associated Gauss map γΘ on Θ, is not a hypersurface as e...

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.

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.

The primitive cohomology of the theta divisor of a principally polarized abelian variety of dimension g contains a Hodge structure of level g− 3 which we call the primal cohomology. The Hodge conjecture predicts that this is contained in the image, under the Abel-Jacobi map, of the cohomology of a family of curves in the theta divisor. In this paper we use the Prym map to show that this version...

A theorem of Mumford’s states that for a smooth cubic threefold X, the intermediate Jacobian JX is a principally polarized abelian variety of dimension 5 whose theta divisor has a unique singular point, which has multiplicity three. This talk describes joint work with R. Friedman, in which we prove a converse: if A is a principally polarized abelian variety of dimension 5 whose theta divisor ha...

We introduce two algebraic completely integrable analogues of the Mumford systems which we call hyperelliptic Prym systems, because every hyperelliptic Prym variety appears as a fiber of their momentum map. As an application we show that the general fiber of the momentum map of the periodic Volterra lattice ȧi = ai(ai−1 − ai+1), i = 1, . . . , n, an+1 = a1, is an affine part of a hyperelliptic ...

The Prym map of type (g, n, r) associates to every cyclic covering of degree n of a curve of genus g, ramified at a reduced divisor of degree r, the corresponding Prym variety. We show that the corresponding map of moduli spaces is generically finite in most cases. From this we deduce the dimension of the image of the Prym map.

Let A be a principally polarized abelian variety of dimension four and let Θ ⊂ A be a symmetric theta-divisor, which we assume to be smooth. Using the Hodge structure on H(Θ) we associate to A two abelian subvarieties J(K) ⊂ J(H) of the intermediate jacobian J(Θ) of Θ of dimensions five and nine respectively. We show that J(H) is generated by the image under the Abel-Jacobi map of the family F ...