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...

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...

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...

This survey discusses the geometry of the moduli space of Prym varieties. Several applications of Pryms in algebraic geometry are presented. The paper begins with with a historical discussion of the life and achievements of Friedrich Prym. Topics treated in subsequent sections include singularities and Kodaira dimension of the moduli space, syzygies of Prym-canonical embedding and the geometry ...

Abstract We prove an analogue of Kirchhoff’s matrix tree theorem for computing the volume tropical Prym variety double covers metric graphs. interpret formula in terms a semi-canonical decomposition variety, via careful study Abel–Prym map. In particular, we show that map is harmonic, determine its degree at every cell and global $2^{g-1}$ . Along way, use Ihara zeta function to provide new pro...

To any pair of coverings fi : X → Xi, i = 1, 2 of smooth projective curves one can associate an abelian subvariety of the Jacobian JX , the Prym variety P (f1, f2) of the pair (f1, f2). In some cases we can compute the type of the restriction of the canonical principal polarization of JX . We obtain 2 families of Prym-Tyurin varieties of exponent 6.

Let P ∪P ′ be the two component Prym variety associated to anétale double cover˜C → C of a non-hyperelliptic curve of genus g ≥ 6 and let |2Ξ 0 | and |2Ξ ′ 0 | be the linear systems of second order theta divisors on P and P ′ respectively. The component P ′ contains canonically the Prym curve˜C. We show that the base locus of the subseries of divisors containing˜C ⊂ P ′ is exactly the curve˜C. ...

Let P ∪P ′ be the two component Prym variety associated to an étale double cover C̃ → C of a non-hyperelliptic curve of genus g ≥ 6 and let |2Ξ0| and |2Ξ ′ 0| be the linear systems of second order theta divisors on P and P ′ respectively. The component P ′ contains canonically the Prym curve C̃. We show that the base locus of the subseries of divisors containing C̃ ⊂ P ′ is exactly the curve C̃. We...