A section K on a genus g canonical curve C is identified as the key tool to prove new results on the geometry of the singular locus Θ s of the theta divisor. The K divisor is characterized by the condition of linear dependence of a set of quadrics containing C and naturally associated to a degree g effective divisor on C. K counts the number of intersections of special varieties on the Jacobian...

In this paper we solve an arbitrary matrix Riemann-Hilbert (inverse monodromy) problem with quasi-permutation monodromy representations outside of a divisor in the space of monodromy data. This divisor is characterized in terms of the theta-divisor on the Jacobi manifold of an auxiliary compact Riemann surface realized as an appropriate branched covering of CP 1 . The solution is given in terms...

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 ...

in a family of even hypersurfaces Roy Smith and Robert Varley Abstract Assume given a family of even local analytic hypersurfaces, whose central fiber has an isolated singularity at x = 0 which is not an ordinary double point. We prove that if the family is sufficiently general, for instance if the general fiber is smooth and the general singular fiber has only ordinary double points, then the ...

makes a good alternative to the well-known Sine-Gordon equation as one more model of the two-dimensional integrable relativistic field theory with the self-action. In [3] the authors had suggested the construction of the finite-gap solutions for this equation. These are periodic and/or almost periodic function in x, t associated with special classes of double sheet ramified coverings of Riemann...

We establish a Lefschetz hyperplane theorem for the Berkovich analytifications of Jacobians curves over an algebraically closed non-Archimedean field. Let J be Jacobian curve X, and let $$W_{d} \subset J$$ locus effective divisor classes degree d. show that pair $$(J^{an},W_d^{an})$$ is d-connected, thus in particular inclusion analytification theta $$\Theta ^{an}$$ into $$J^{an}$$ satisfies $$...

In response to a question of Beauville, we give a new class of examples of base points for the linear system |Θ| on the moduli space SUX(r) of semistable rank r vector bundles of trivial determinant on a curve X and we prove that for sufficiently large r the base locus is positive dimensional.

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...