Related papers: Tropical Weierstrass points and Weierstrass weight…

For a tropical curve and a finite subgroup of the isometry group of the tropical curve, we prove, extending the work by Haase, Musiker and Yu, that the invariant part of the complete linear system associated to a invariant effective divisor…

We prove new sharp asymptotic for counting the semistable elliptic curves with two marked Weierstrass points at $\infty$ and $0$ and also the cases where $0$ is a 2-torsion or a 3-torsion marked Weierstrass point over $\mathbb{F}_q(t)$ by…

We show that the non-Archimedean skeleton of the $d$-th symmetric power of a smooth projective algebraic curve $X$ is naturally isomorphic to the $d$-th symmetric power of the tropical curve that arises as the non-Archimedean skeleton of…

We show that the counting of rational curves on a complete toric variety that are in general position to the toric prime divisors coincides with the counting of certain tropical curves. The proof is algebraic-geometric and relies on…

We introduce tropical vector bundles, morphisms and rational sections of these bundles and define the pull-back of a tropical vector bundle and of a rational section along a morphism. Afterwards we use the bounded rational sections of a…

We construct a general framework for tropical differential equations based on idempotent semirings and an idempotent version of differential algebra. Over a differential ring equipped with a non-archimedean norm enhanced with additional…

Tropical geometry is a relatively recent field in mathematics created as a simplified model for certain problems in algebraic geometry. We introduce the definition of abstract and planar tropical curves as well as their properties,…

This paper is the first part in a series of three papers devoted to the study of enumerative invariants of abelian surfaces through the tropical approach. In this paper, we consider the enumeration of genus $g$ curves of fixed degree…

We consider the problem of determining Weierstrass gaps and pure Weierstrass gaps at several points. Using the notion of relative maximality in generalized Weierstrass semigroups due to Delgado \cite{D}, we present a description of these…

We show that the number of tropical curves of given genus and degree through some given general points in the plane does not depend on the position of the points. In the case when the degree of the curves contains only primitive integral…

The Weil reciprocity law asserts that given two meromorphic functions $f, g$ on a compact complex curve, the product of the values of $f$ over the roots and poles of $g$ is equal to the product of the values of $g$ over the roots and poles…

We construct moduli spaces of rational covers of an arbitrary smooth tropical curve in R^r as tropical varieties. They are contained in the balanced fan parametrizing tropical stable maps of the appropriate degree to R^r. The weights of the…

In this paper we study topology of moduli spaces of tropical curves of genus $g$ with $n$ marked points. We view the moduli spaces as being imbedded in a larger space, which we call the {\it moduli space of metric graphs with $n$ marked…

Hassett's moduli spaces of weighted stable curves form an important class of alternate modular compactifications of the moduli space of smooth curves with marked points. In this article we define a tropical analogue of these moduli spaces…

Let $X$ be an abstract tropical curve and let $G$ be a finite subgroup of the automorphism group of $X$. Let $D$ be a divisor on $X$ whose equivalence class is $G$-invariant. We address the following question: is there always a divisor $D'$…

In this lecture we give a brief introduction to Weierstrass points of curves and computational aspects of $q$-Weierstrass points on superelliptic curves.

We introduce and study the locus $\mathbb{M}_{g,d}^\textrm{nd}$ of genus $g$ tropical plane curves of gonality $d$ inside the moduli space $\mathbb{M}^{\textrm{nd}}_{g}$ of tropical plane curves of genus $g$. Each such tropical curve arises…

In this paper we construct a tropical moduli space parametrizing roots of divisors on tropical curves. We study the relation between this space and the skeleton of Jarvis moduli space of nets of limit roots on stable curves. We show that…

In complex algebraic geometry, the problem of enumerating plane elliptic curves of given degree with fixed complex structure has been solved by R.Pandharipande using Gromov-Witten theory. In this article we treat the tropical analogue of…

In this paper we study a construction of algebraic curves from combinatorial data. In the study of algebraic curves through degeneration, graphs usually appear as the dual intersection graph of the central fiber. Properties of such graphs…