Related papers: Tropical moments of tropical Jacobians
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…
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…
By tropical Abel-Jacobi theorem, the Jacobian of a tropical curve is isomorphic to the Picard group. A tropical curve in $\mathbb{R}^2$ corresponds to an immersion from a tropical curve to $\mathbb{R}^2$. In this paper, we show that any…
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…
The classical Poincar\'e formula relates the rational homology classes of tautological cycles on a Jacobian to powers of the class of Riemann theta divisor. We prove a tropical analogue of this formula. Along the way, we prove several…
We introduce the notion of tropical curves of hyperelliptic type. These are tropical curves whose Jacobian is isomorphic to that of a hyperelliptic tropical curve, as polarized tropical abelian varieties. We show that this property depends…
If (Q,A) is a marked polygon with one interior point, then a general polynomial f in K[x,y] with support A defines an elliptic curve C on the toric surface X_A. If K has a non-archimedean valuation into the real numbers we can tropicalize C…
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…
We associate to an analytic subvariety of a torus a tropical variety. In the first part, we generalize the results from tropical algebraic geometry to this non-archimedean analytic situation. The periodic case is applied to a totally…
Let X be a tropical curve (or metric graph), and fix a base point p on X. We define the Jacobian group J(G) of a finite weighted graph G, and show that the Jacobian J(X) is canonically isomorphic to the direct limit of J(G) over all…
In this article we provide a stack-theoretic framework to study the universal tropical Jacobian over the moduli space of tropical curves. We develop two approaches to the process of tropicalization of the universal compactified Jacobian…
We study a notion of tropical linear series on metric graphs that combines two essential properties of tropicalizations of linear series on algebraic curves: the Baker-Norine rank and the independence rank. Our main results relate the local…
Algebraic curves have a discrete analogue in finite graphs. Pursuing this analogy we prove a Torelli theorem for graphs. Namely, we show that two graphs have the same Albanese torus if and only if the graphs obtained from them by…
We prove for a tropical rational map that if for any point the convex hull of Jacobian matrices at smooth points in a neighborhood of the point does not contain singular matrices then the map is an isomorphism. We also show that a tropical…
On tropical geometry in $\rr^2$, the divisor and the Jacobian variety are defined in analogy to algebraic geometry. For study of these objects, it is important to think of the `bunch' of a tropical curve. In this paper, we will show that if…
We propose a method to study the integrable cellular automata with periodic boundary conditions, via the tropical spectral curve and its Jacobian. We introduce the tropical version of eigenvector map from the isolevel set to a divisor class…
We study a rational matroid invariant, obtained as the tropicalization of the Feynman period integral. It equals the volume of the polar of the matroid polytope and we give efficient formulas for its computation. This invariant is proven to…
This paper is the second in a series of two papers which study the phenomenon of tropical split Jacobians. The first paper is a contemplative study, embedded in the broader context of exploring connections between the category of tropical…
Let $\mathrm{Out}(F_n)$ be the outer automorphism group of the free group $F_n$. It acts properly on the outer space $X_n$ of marked metric graphs, which is a finite-dimensional infinite simplicial complex with some simplicial faces…
This thesis delves into the geometry of abstract tropical curves, exploring their complete linear system and associated tropical submodules. We establish a lower bound on the dimension of tropical submodules in terms of the Baker-Norine…