Related papers: The Abel Jacobi map for Mumford Curves via Integra…

We consider Abel maps for regular smoothing of nodal curves with values in the Esteves compactified Jacobian. In general, these maps are just rational, and an interesting question is to find an explicit resolution. We translate this problem…

Metric graphs are important models for capturing the structure of complex data across various domains. While much effort has been devoted to extracting geometric and topological features from graph data, computational aspects of metric…

Mumford showed that Schottky subgroups of $PGL(2,K)$ give rise to certain curves, now called Mumford curves, over a non-Archimedean field K. Such curves are foundational to subjects dealing with non-Archimedean varieties, including…

In tropical geometry, one studies algebraic curves using combinatorial techniques via the tropicalization procedure. The tropicalization depends on a map to an algebraic torus and the combinatorial methods are most useful when the…

In this paper we resolve the degree-2 Abel map for nodal curves. Our results are based on a previous work of the authors reducing the problem of the resolution of the Abel map to a combinatorial problem via tropical geometry. As an…

In this paper we give local conditions to the existence of Abel maps for nodal curves that are limits of Abel maps for smooth curves. We use this result to construct Abel maps for any degree for nodal curves with two components.

To a compact tropical variety of arbitrary dimension, we associate a collection of intermediate Jacobians defined in terms of tropical homology and tropical monodromy. We then develop an Abel-Jacobi theory in the tropical setting by…

Motivated by the recent surge of interest in the geometry of hybrid spaces, we prove an Abel-Jacobi theorem for a metrized complex of Riemann surfaces, generalizing both the classical Abel-Jacobi theorem and its tropical analogue.

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…

We calculate the tropical Dolbeault cohomology for the analytifications of the projective line and Mumford curves over non-archimedean fields. We show that the cohomology satisfies Poincar\'e duality and behaves analogously to the…

We construct an Abel-Jacobi type map on the homologically trivial part of Lawson homology groups. It generalizes the Abel-Jacobi map constructed by Griffiths. By using a result of H. Clemens, we give some examples of smooth projective…

In recent years a series of remarkable advances in tropical geometry and in non-archimedean geometry have brought new insights to the moduli theory of algebraic curves and their Jacobians. The goal of this survey, an expanded version of my…

We show how the Abel-Jacobi map provides all the principal properties of an ample family of integrable mechanical systems associated to hyperelliptic curves. We prove that derivative of the Abel-Jacobi map is just the St\"{a}ckel matrix,…

Given a smooth curve with weighted marked points, the Abel-Jacboi map produces a line bundle on the curve. This map fails to extend to the full boundary of the moduli space of stable pointed curves. Using logarithmic and tropical geometry,…

Many finite dimensional integrable systems qre expressed with the help of the Lax equation which highlights a spectral parameter and therefore a spectral curve. These spectral curves are the starting point of an algebro-geometric…

This is a survey article written for the Jahresberichte der DMV. Tropical geometry can be viewed as an efficient combinatorial tool to study degenerations in algebraic geometry. Abstract tropical curves are essentially metric graphs, and…

We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical multiplicities and a generalization of the Sturmfels-Tevelev…

This paper provides an overview of recent progress on the interplay between tropical geometry and non-archimedean analytic geometry in the sense of Berkovich. After briefly discussing results by Baker, Payne and Rabinoff in the case of…

We pose some questions about spaces parametrizing rational curves on rationally connected varieties. We give a partial answer for cubic threefolds. Many of our results were previously proved by Iliev, Markushevich and Tikhimirov by…

The tropicalization of an algebraic variety X is a combinatorial shadow of X, which is sensitive to a closed embedding of X into a toric variety. Given a good embedding, the tropicalization can provide a lot of information about X. We…