The j-invariant of a plane tropical cubic

Several results in tropical geometry have related the j-invariant of an algebraic plane curve of genus one to the cycle length of a tropical curve of genus one. In this paper, we prove that for a plane cubic over the field of Puiseux series the negative of the generic valuation of the $j$-invariant is equal to the cycle length of the tropicalization of the curve, if there is a cycle at all.Comment: The proofs rely partly on computations done with polymake, topcom and Singula

An algorithm for lifting points in a tropical variety

The aim of this paper is to give a constructive proof of one of the basic theorems of tropical geometry: given a point on a tropical variety (defined using initial ideals), there exists a Puiseux-valued ``lift'' of this point in the algebraic variety. This theorem is so fundamental because it justifies why a tropical variety (defined combinatorially using initial ideals) carries information about algebraic varieties: it is the image of an algebraic variety over the Puiseux series under the valuation map. We have implemented the ``lifting algorithm'' using Singular and Gfan if the base field are the rational numbers. As a byproduct we get an algorithm to compute the Puiseux expansion of a space curve singularity in (K^{n+1},0).Comment: 33 page

Kontsevich's formula and the WDVV equations in tropical geometry

Using Gromov-Witten theory the numbers of complex plane rational curves of degree d through 3d-1 general given points can be computed recursively with Kontsevich's formula that follows from the so-called WDVV equations. In this paper we establish the same results entirely in the language of tropical geometry. In particular this shows how the concepts of moduli spaces of stable curves and maps, (evaluation and forgetful) morphisms, intersection multiplicities and their invariance under deformations can be carried over to the tropical world.Comment: 24 pages, minor changes to match the published versio

Triple-Point Defective Surfaces

In this paper we study the linear series $|L-3p|$ of hyperplane sections with a triple point $p$ on a surface $S$ embedded via a very ample line bundle $L$ for a \emph{general} point $p$. If this linear series does not have the expected dimension we call $(S,L)$ \emph{triple-point defective}. We show that on a triple-point defective surface through a general point every hyperplane section has either a triple component or the surface is rationally ruled and the hyperplane section contains twice a fibre of the ruling.Comment: The paper generalises the results in arXiv:0705.3912 using the same techniques. The assumptions both on the linear system and on the surface have been weakened. The interested reader should consult this new paper instead of the older on