Tropical descendant Gromov-Witten invariants (2008)
Markwig, Hannah, Rau, Johannes
We define tropical Psi-classes on the moduli space of rational tropical curves in R^2 and consider intersection products of Psi-classes and pull-backs of evaluations on this space. We show a certain...
Tropical enumerative invariants of F_0 and F_2 (2008)
Franz, Marina, Markwig, Hannah
There is an equation relating numbers of curves on F_0=P^1xP^1 satisfying incidence conditions and numbers of curves on F_2 satisfying incidence conditions. The purpose of this paper is to give a...
Tropical Hurwitz Numbers (2008)
Cavalieri, Renzo, Johnson, Paul, Markwig, Hannah
Hurwitz numbers count genus g, degree d covers of the projective line with fixed branch locus. This equals the degree of a natural branch map defined on the Hurwitz space. In tropical geometry,...
The tropical $j$-invariant (2008)
Katz, Eric, Markwig, Hannah, Markwig, Thomas
If (Q,A) is a marked polygon with one interior point, then a general polynomial f in K[x,y] with support A defines an elliptic curve C on the toric surface X_A. If K has a non-archimedean valuation...
The space of tropically collinear points is shellable (2007)
Markwig, Hannah, Yu, Josephine
The space $T_{d,n}$ of $n$ tropically collinear points in a fixed tropical projective space $\mathbb{TP}^{d-1}$ is the tropicalization of the determinantal variety of matrices of rank 2. It consists...
Intersecting Psi-classes on tropical M_{0,n} (2007)
Kerber, Michael, Markwig, Hannah
We apply the tropical intersection theory developed by L. Allermann and J. Rau to compute intersection products of tropical Psi-classes on the moduli space of rational tropical curves. We show that...
The j-invariant of a plane tropical cubic (2007)
Katz, Eric, Markwig, Hannah, Markwig, Thomas
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...
Tropical fans and the moduli spaces of tropical curves (2007)
Gathmann, Andreas, Kerber, Michael, Markwig, Hannah
We give a rigorous definition of tropical fans (the "local building blocks for tropical varieties") and their morphisms. For such a morphism of tropical fans of the same dimension we show that the...
An algorithm for lifting points in a tropical variety (2007)
Jensen, Anders Nedergaard, Markwig, Hannah, Markwig, Thomas
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...
Counting tropical elliptic plane curves with fixed j-invariant (2006)
Kerber, Michael, Markwig, Hannah
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...
Kontsevich's formula and the WDVV equations in tropical geometry (2005)
Gathmann, Andreas, Markwig, Hannah
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...
The numbers of tropical plane curves through points in general position (2005)
Gathmann, Andreas, Markwig, Hannah
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...
The Caporaso-Harris formula and plane relative Gromov-Witten invariants in tropical geometry (2005)
Gathmann, Andreas, Markwig, Hannah
Some years ago Caporaso and Harris have found a nice way to compute the numbers N(d,g) of complex plane curves of degree d and genus g through 3d+g-1 general points with the help of relative...