Related papers: Smooth tropical surfaces with infinitely many trop…
Given a tropical line $L$ and a smooth tropical surface $X$, we look at the position of $L$ on $X$. We introduce its primal and dual motif which are respectively a decorated graph and a subcomplex of the dual triangulation of $X$. They…
We study smooth tropical plane quartic curves and show that they satisfy certain properties analogous to (but also different from) smooth plane quartics in algebraic geometry. For example, we show that every such curve admits either…
We present results on the relative realizability of infinite families of lines on general smooth tropical cubic surfaces. Inspired by the problem of relative realizability of lines on surfaces, we investigate the information we can derive…
We study bitangents of non-smooth tropical plane quartics. Our main result is that with appropriate multiplicities, every such curve has 7 equivalence classes of bitangent lines. Moreover, the multiplicity of bitangent lines varies…
In this paper, we study tropicalisations of singular surfaces in toric threefolds. We completely classify singular tropical surfaces of maximal-dimensional type, show that they can generically have only finitely many singular points, and…
Complex algebraic varieties become easy piecewise-linear objects after passing to the so-called tropical limit. Geometry of these limiting objects is known as tropical geometry. In this short survey we take a look at motivation and…
We present tools and definitions to study abstract tropical manifolds in dimension 2, which we call simply tropical surfaces. This includes explicit descriptions of intersection numbers of 1-cycles, normal bundles to some curves and…
We provide some new local obstructions to approximating tropical curves in smooth tropical surfaces. These obstructions are based on the relation between tropical and complex intersection theories which is also established here. We give two…
We define transversal tropical triangles (affine and projective) and characterize them via six inequalities to be satisfied by the coordinates of the vertices. We prove that the vertices of a transversal tropical triangle are tropically…
We consider the enumeration of tropical curves in M\"obius strips for two different lattice structures and relate them to the enumeration of curves in two rational ruled surfaces over a complex elliptic curve. Using this correspondence, we…
We introduce the notion of families of n-marked smooth rational tropical curves over smooth tropical varieties and establish a one-to-one correspondence between (equivalence classes of) these families and morphisms from smooth tropical…
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…
We prove a correspondence theorem for singular tropical surfaces in real three space, which recovers singular algebraic surfaces in an appropriate toric three-fold that tropicalize to a given singular tropical surface. Furthermore, we…
We enumerate complex curves on toric surfaces of any given degree and genus, having a single cusp and nodes as their singularities, and matching appropriately many point constraints. The solution is obtained via tropical enumerative…
We study lifts of tropical bitangents to the tropicalization of a given complex algebraic curve together with their lifting multiplicities. Using this characterization, we show that generically all the seven bitangents of a smooth tropical…
The classical statement of Cayley-Salmon that there are 27 lines on every smooth cubic surface in P^3 fails to hold under tropicalization: a tropical cubic surface in TP^3 often contains infinitely many tropical lines. Under mild genericity…
This paper deals with surfaces with many lines. It is well-known that a cubic contains 27 of them and that the maximal number for a quartic is 64. In higher degree the question remains open. Here we study classical and new constructions of…
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…
Let k be a field of characteristic other than 2,3. We prove that there are no geometrically smooth quartic surfaces in IP^3 with more than 64 lines. As a key step, we derive the sharp bound that any line meets at most 20 other lines on a…
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…