Related papers: On Algebraic Theta Divisors and Rational Solutions…

The singularities of theta divisors have played an important role in the study of algebraic varieties. This paper surveys some of the recent progress in this subject, using as motivation some well known results, especially those for…

We prove the following converse of Riemann's Theorem: let (A,\Theta) be an indecomposable principally polarized abelian variety whose theta divisor can be written as a sum of a curve and a codimension two subvariety \Theta=C+Y. Then C is…

We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and establish tropical versions of the…

It is shown that it is possible to write down tau functions for the $n$-component KP hierarchy in terms of non-abelian theta functions. This is a generalization of the rank 1 situation; that is, the relation of theta functions of Jacobians…

The discriminant of a smooth plane cubic curve over the complex numbers can be written as a product of theta functions. This provides an important connection between algebraic and analytic objects. In this paper, we perform a new approach…

We study the intersection cohomology of the theta divisors on Jacobians of nonhyperelliptic curves.

We study the theta divisor of the compactified jacobian of a nodal, possibly reducible, curve. We compute its irreducible components and give it a geometric interpretation consistent with the classical Brill-Noether theory of smooth curves.…

We study the geometry and cohomology of algebraic super curves, using a new contour integral for holomorphic differentials. For a class of super curves (``generic SKP curves'') we define a period matrix. We show that the odd part of the…

We show how to calculate the Euler characteristic of an affine Jacobi variety of a spectral curve from its defining equations.

For the Jacobian of a curve, the Riemann singularity theorem gives a geometric interpretation of the singularities of the theta divisor in terms of special linear series on the curve. This paper proves an analogous theorem for Prym…

In this short survey we give a description of the theta functions of algebraic curves, half-integer theta-nulls, and the fundamental theta functions. We describe how to determine such fundamental theta functions and describe the components…

Using abelian differentials and periods of the universal Mumford curve, we study the universal expression and asymptotic behavior of tau functions defined for stably degenerating families of algebraic curves with additional data.…

Given a curve defined over an algebraically closed field which is complete with respect to a nontrivial valuation, we study its tropical Jacobian. This is done by first tropicalizing the curve, and then computing the Jacobian of the…

Let $X$ be an irreducible singular Riemann surface, with desingularisation $\widetilde X$. The generalised Jacobian $J(X)$ of $X$ fibers over the Jacobian $J(\widetilde{X})$ of $\widetilde X$, and there is an Abel map $A$ of $\widetilde X$…

We compare and contrast three different methods for the construction of the differential relations satisfied by the fundamental Abelian functions associated with an algebraic curve. We realize these Abelian functions as logarithmic…

We compute the class of arithmetic genus two Teichmueller curves in the Picard group of pseudo-Hilbert modular surfaces, distinguished according to their torsion order and spin invariant. As an application, we compute the number of genus…

The Jacobian varieties of smooth curves fit together to form a family, the universal Jacobian, over the moduli space of smooth marked curves, and the theta divisors of these curves form a divisor in the universal Jacobian. In this paper we…

Let $X$ be a compact Riemann surface of genus $g$. Jacobi's inversion theorem states that the Abel-Jacobi map $\varphi : X^{(g)} \longrightarrow J(X)$ is surjective, where $X^{(g)}$ is the symmetric product of $X$ of degree $g$ and $J(X)$…

We show that the only summands of theta divisors on Jacobians of curves and on intermediate Jacobians of cubic threefolds are the obvious powers of the curve and the Fano surface of lines on the threefold. The proof uses the decomposition…

In this papers, we study the geometric and arithmetic properties of the theta divisor associated to the sheaf of locally exact differential forms over a curve in positive characteristic. In this published version, we prove a stronger…