Our main theorem is an improvement of the Criterion of Kanev about Prym–Tyurin varieties induced by correspondences, which includes correspondences with fixed points. We give some examples of Prym–Tyurin varieties using this criterion.

For every genus g, we construct a smooth, complete, rational polarized algebraic variety DMg together with a normal crossing divisor D = ⋃ Di, such that for every moduli space MΣ(2, 0) of semistable topologically trivial vector bundles of rank 2 on an algebraic curve Σ of genus g there exists a holomorphic isomorphism f : MΣ(2, 0) \ K2 → DMg \ D, where K2 is the Kummer variety of the Jacobian o...

We give upper and lower bounds for the number of rational points on Prym varieties over finite fields. Moreover, we determine the exact maximum and minimum number of rational points on Prym varieties of dimension 2.

It is shown that the determinants of the correlation functions of the generalized bc-system introduced recently are given as pullbacks of the non-abelian theta divisor. The usual bc-system appearing in bosonic string theory 1;2 is very well understood 1?4 and has also been considered in a rigorous algebro-geometric way by Raina 5;6. Assuming some natural physical axioms, Raina showed the existe...

We prove a formula, which, given principally polarized abelian variety $(A,\lambda)$ over the field of algebraic numbers, relates stable Faltings height $A$ with N\'eron--Tate symmetric theta divisor on $A$. Our formula completes earlier results due to Bost, Hindry, Autissier and Wagener. The local non-archimedean terms in our can be expressed as tropical moments tropicalizations $(A,\lambda)$.

The object of this paper is the theta divisor of the compactified Jacobian of a nodal curve of genus g. We determine its irreducible components and give it a geometric interpretation. A characterization of hyperelliptic irreducible curves in M g is appended as an application.

We prove that k-th singular cohomology group of the complement of the theta divisor in a hyperelliptic Jacobian is isomorphic to the k-th fundamental representation of the symplectic group Sp(2g,C). This is one of the conjectures in the paper [11].