Related papers: Tropical curves and covers and their moduli spaces
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,…
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to enumerative algebraic geometry led to…
Tropical algebraic geometry is an active new field of mathematics that establishes and studies some very general principles to translate algebro-geometric problems into purely combinatorial ones. This expository paper gives an introduction…
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…
We give an introduction to Tropical Geometry and prove some results in Tropical Intersection Theory. The first part of this paper is an introduction to tropical geometry aimed at researchers in Algebraic Geometry from the point of view of…
We describe a framework to construct tropical moduli spaces of rational stable maps to a smooth tropical hypersurface or curve. These moduli spaces will be tropical cycles of the expected dimension, corresponding to virtual fundamental…
These notes outline some basic notions of Tropical Geometry and survey some of its applications for problems in classical (real and complex) geometry. To appear in the Proceedings of the Madrid ICM.
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 define the tropical moduli space of covers of a tropical line in the plane as weighted abstract polyhedral complex, and the tropical branch map recording the images of the simple ramifications. Our main result is the invariance of the…
We construct the moduli space for equivalence classes of n-pointed tropical curves of genus g, together with its compactification given by weighted tropical curves, and establish some of its basic topological properties. We compare it to…
Tropical mathematics redefines the rules of arithmetic by replacing addition with taking a maximum, and by replacing multiplication with addition. After briefly discussing a tropical version of linear algebra, we study polynomials build…
In this paper, we study the deformation theory of degenerate algebraic curves on singular varieties which appear as the degenerate limit of families of varieties. For this purpose, we systematically develop a new method to calculate the…
We contribute to the foundations of tropical geometry with a view towards formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show…
We introduce tropical complexes, as an enrichment of the dual complex of a degeneration with additional data from non-transverse intersection numbers. We define cycles, divisors, and linear equivalence on tropical complexes, analogous both…
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…
We exploit three classical characterizations of smooth genus two curves to study their tropical and analytic counterparts. First, we provide a combinatorial rule to determine the dual graph of each algebraic curve and the metric structure…
Let $X$ be an algebraic variety and let $S$ be a tropical variety associated to $X$. We study the tropicalization map from the moduli space of stable maps into $X$ to the moduli space of tropical curves in $S$. We prove that it is a…
Tropical algebraic geometry is the geometry of the tropical semiring $(\mathbb{R},\min,+)$. Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on…
Abstractly, tropical hyperelliptic curves are metric graphs that admit a two-to-one harmonic morphism to a tree. They also appear as embedded tropical curves in the plane arising from triangulations of polygons with all interior lattice…
We give an overview of recently implemented polymake features for computations in tropical geometry. The main focus is on explicit examples rather than technical explanations. Our computations employ tropical hypersurfaces, moduli of…