87 research outputs found

### 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

### Tropical covers of curves and their moduli spaces

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 degree of the branch map, which enables us to give a tropical
intersection-theoretic definition of tropical triple Hurwitz numbers. We show
that our intersection-theoretic definition coincides with the one given by
Bertrand, Brugall\'e and Mikhalkin in the article "Tropical Open Hurwitz
numbers" where a Correspondence Theorem for Hurwitz numbers is proved. Thus we
provide a tropical intersection-theoretic justification for the multiplicities
with which a tropical cover has to be counted. Our method of proof is to
establish a local duality between our tropical moduli spaces and certain moduli
spaces of relative stable maps to the projective line.Comment: 24 pages, 10 figure

### Counting tropical elliptic plane curves with fixed j-invariant

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 this problem, the determination of the number of tropical
elliptic plane curves of degree d and fixed ``tropical j-invariant''
interpolating an appropriate number of points in general position. We show that
this number is independent of the position of the points and the value of the
j-invariant and that it coincides with the number of complex elliptic curves.
The result can be used to simplify Mikhalkin's algorithm to count curves via
lattice paths in the case of rational plane curves.Comment: 34 pages; minor changes to match the published versio

### The space of tropically collinear points is shellable

The space T_{d,n} of n tropically collinear points in a fixed tropical
projective space TP^{d-1} is equivalent to the tropicalization of the
determinantal variety of matrices of rank at most 2, which consists of real d x
n matrices of tropical or Kapranov rank at most 2, modulo projective
equivalence of columns. We show that it is equal to the image of the moduli
space M_{0,n}(TP^{d-1},1) of n-marked tropical lines in TP^{d-1} under the
evaluation map. Thus we derive a natural simplicial fan structure for T_{d,n}
using a simplicial fan structure of M_{0,n}(TP^{d-1},1) which coincides with
that of the space of phylogenetic trees on d+n taxa. The space of phylogenetic
trees has been shown to be shellable by Trappmann and Ziegler. Using a similar
method, we show that T_{d,n} is shellable with our simplicial fan structure and
compute the homology of the link of the origin. The shellability of T_{d,n} has
been conjectured by Develin in 2005.Comment: final version, minor revision, 15 page

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