We construct a three-parameter family of nonhyperelliptic and bielliptic plane genus-three curves whose associated Prym variety is two-isogenous to the Jacobian general hyperelliptic genus-two curve. Our construction based on existence special elliptic fibrations with section Kummer surfaces that provide simple geometric interpretation for rational double cover induced by two-isogeny between Ab...

Given a tame Galois branched cover of curves π : X → Y with any finite Galois group G whose representations are rational, we compute the dimension of the (generalized) Prym variety Prymρ(X) corresponding to any irreducible representation ρ of G. This formula can be applied to the study of algebraic integrable systems using Lax pairs, in particular systems associated with Seiberg-Witten theory. ...

For a class of non-hyperelliptic genus 3 curves C which are 2-fold coverings of elliptic curves E, we give an explicit algebraic description of all birationally nonequivalent genus 2 curves whose Jacobians are degree 2 isogeneous to the Prym varieties associated to such coverings. Our description is based on previous studies of Prym varieties with polarization (1,2) in connection with separatio...

Introduction. Let G be a finite group acting on a smooth projective curve X. This induces an action of G on the Jacobian JX of X and thus a decomposition of G of JX up to isogeny. The most prominent example of such a situation is the case of the group G ≃ Z/2Z of two elements. Let π : X → Y = X/G denote the canonical quotient map. The first to notice that JX is isogenous to the product JY × P (...

Using lattice theory on specialK3 surfaces, calculations on moduli stacks of pointed curves and Voisin’s proof of Green’s Conjecture on syzygies of canonical curves, we prove the Prym-Green Conjecture on the naturality of the resolution of a general Prym-canonical curve of odd genus, as well as (many cases of) the Green-Lazarsfeld Secant Conjecture on syzygies of non-special line bundles on gen...

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...

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...

where (Pη,Ξη) is the principally polarised Prym variety corresponding to a 2-torsion point η ∈ J2(C), or is the Jacobian (J(C),Θ) in case η = 0; and where H ± denotes even/odd sections. The interest of (0.1) is that it is one step in the direction, initiated by Beauville and others ([B1], [B2], [BNR]), of relating the Verlinde spaces of C—i.e. the vector spaces H(M,L) where M is some moduli spa...

Prym varieties provide a correspondence between the moduli spaces of curves and abelian varieties Mg and Ag−1, via the Prym map Pg : Rg → Ag−1 from the moduli space Rg parameterizing pairs [C, η], where [C] ∈ Mg is a smooth curve and η ∈ Pic(C)[2] is a torsion point of order 2. When g ≤ 6 the Prym map is dominant and Rg can be used directly to determine the birational type ofAg−1. It is known t...

The periods of Prym differentials can be used to prove the invariance of Picard bundles on Jacobian varieties. Let S be a compact Riemann surface of genus g with universal covering surface 77 with the deck transformation group D. For any homomorphism X: D -> C* = C {0} into the group of complex units, a (meromorphic) Prym differential on S1 with multipliers A1 is a meromorphic differential w on...