Introduction 1 1. Kanev’s construction and Prym-Tyurin varieties of E6-type 7 2. The E6 lattice 11 3. Degenerations of Jacobians and Prym varieties 13 4. Degenerations of Prym-Tyurin-Kanev varieties 15 5. The global geometry of the Hurwitz space of E6-covers 20 6. The Prym-Tyurin map along boundary components of Hur 30 7. Ordinary Prym varieties regarded as Prym-Tyurin-Kanev varieties 39 8. The...

For an Abel-Prym curve contained in a Prym variety, we determine the cohomological support loci of its twisted ideal sheaves and the dimension of its theta-dual.

is a double covering, where C and C are nonsingular complete curves with Jacobians J and 3. The involution 1: C C interchanging sheets extends to t: I J, and up to some points of order two, 3 splits into an even part J and an odd part P, the Prym variety. The Prym P has a natural polarization on it, but only in two cases where 21 has zero or two branch points do we get a unique principal polari...

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...

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. ...

We give a local deformation theoretic proof of Farkas’ conjecture that a principally polarized complex abelian variety of dimension 4 whose theta divisor has an isolated double point of rank 3 at a point of order two is a Jacobian. The argument yields an explicit local normal form for the theta function near such a point. The proof depends only on the facts that the theta function is even, a ge...

Contents 1. Introduction 1 1.1. Notation and Conventions 2 1.2. Brill-Noether varieties and Abel maps 4 1.3. Stability and semistability 6 2. Technical groundwork 9 2.1. Basic estimates 9 2.2. Basic cases 12 2.3. Divisors imposing independent conditions 14 3. Irreducibility and dimension 19 3.1. Irreducible components of the Theta divisor 19 3.2. Dimension of the image of the Abel map 24 4. Com...