The theta divisor Θ of the Jacobian variety of a complex curve X is best viewed as a divisor inside the component Pic(X) consisting of (isomorphism classes of) line bundles of degree g−1. Then a line bundle L belongs to Θ if and only if it has non-zero sections. Riemann proved that the multiplicity of Θ at a point L is equal to dimH(X;L) − 1. Kempf ([Ke]) obtained a geometric proof of Riemann’s...

Let G be a finite group, Λ an absolutely irreducible Z[G]-module and w a weight of Λ. To any Galois covering with group G we associate two correspondences, the Schur and the Kanev correspondence. We work out their relation and compute their invariants. Using this, we give some new examples of Prym-Tyurin varieties.

In this paper, we study the bicanonical pencil of a Godeaux surface and of a determinantal Barlow surface. This study gives a simple proof for the unobstructedness of deformations of a determinantal Barlow surface. Then we compute the number of hyperelliptic curves in the bicanonical pencil of a determinantal Barlow surface via classical Prym theory.

We introduce endomorphisms of special jacobians and show that they satisfy polynomial equations with all integer roots which we compute. The eigen-abelian varieties for these endomorphisms are generalizations of Prym-Tyurin varieties and naturally contain special curves representing cohomology classes which are not expected to be represented by curves in generic abelian varieties.

We address issues related to (i) a proposal for resolving a long-standing tension between large volume cosmology and phenomenology as regards reconciliation of requirements of different gravitino masses within the same string-theoretic framework, as well as (ii) evaluation of soft supersymmetry breaking terms and open-string moduli masses in the context of type IIB large volume compactification...

The ideal class groups of hyperelliptic curves(HECs) can be used in cryptosystems based on the discrete loga-rithm problem. Recent developments of computational technolo-gies for scalar multiplications of divisor classes have shown thatthe performance of hyperelliptic curve cryptosystems (HECC) iscompatible to that of elliptic curve cryptosystems (ECC). Espe-cially, genu...

Let (A; ) be a principally polarized abelian variety (ppav) of dimension g over the eld C of complex numbers. This means that is an ample divisor on A, well-determined up to translation, with h(A; ) := dimH(A; ) = 1. Let [ ] 2 H(A;Z) be the cohomology class of the theta divisor . Then the cohomology class [ ] e is divisible by (g e)!. The class [ ] g e (g e)! is not divisible and it is called t...

In this paper, we study some operations which produce new divisor graphs from old ones. We prove that the contraction of a divisor graph along a bridge is a divisor graph. For two transmitters (receivers) u and v in some divisor orientation of a divisor graph G, it is shown that the merger G |u,v is also a divisor graph. Two special types of vertex splitting are introduced, one of which produce...

These notes discuss recent advances on syzygies on algebraic curves, especially concerning the Green, the Prym-Green and the Green-Lazarsfeld Secant Conjectures. The methods used are largely geometric and variational, with a special emphasis on examples and explicit calculations. The notes are based on series of lectures given in Daejeon (March 2013), Rome (November-December 2015) and Guanajuat...