On Block-Goettsche multiplicities for planar tropical curves

@article{Itenberg2012OnBM,
  title={On Block-Goettsche multiplicities for planar tropical curves},
  author={I. V. Itenberg and Grigory Mikhalkin},
  journal={arXiv: Algebraic Geometry},
  year={2012}
}
We prove invariance for the number of planar tropical curves enhanced with polynomial multiplicities recently proposed by Florian Block and Lothar Goettsche. This invariance has a number of implications in tropical enumerative geometry. 

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References

SHOWING 1-10 OF 21 REFERENCES
Counting plane curves of any genus
We obtain a recursive formula answering the following question: How many irreducible, plane curves of degree d and (geometric) genus g pass through 3d-1+g general points in the plane? The formula is
Enumeration of curves via floor diagrams
Enumerative tropical algebraic geometry
The paper establishes a formula for enumeration of curves of arbitrary genus in toric surfaces. It turns out that such curves can be counted by means of certain lattice paths in the Newton polygon.
Tropical curves, their Jacobians and Theta functions
We study Jacobian varieties for tropical curves. These are real tori equipped with integral affine structure and symmetric bilinear form. We define tropical counterpart of the theta function and
Topological properties of real algebraic varieties: du coté de chez Rokhlin
The survey gives an overview of the achievements in topology of real algebraic varieties in the direction initiated in the early 70th by V. I. Arnold and V. A. Rokhlin. We make an attempt to
Floor decompositions of tropical curves : the planar case
In a previous paper, we announced a formula to compute Gromov-Witten and Welschinger invariants of some toric varieties, in terms of combinatorial objects called floor diagrams. We give here detailed
Tropical geometry and its applications
From a formal perspective tropical geometry can be viewed as a branch of geometry manipulating with certain piecewise-linear objects that take over the role of classical algebraic varieties. This
A Caporaso-Harris type formula for Welschinger invariants of real toric Del Pezzo surfaces
We define a series of relative tropical Welschinger-type invariants of real toric surfaces. In the Del Pezzo case, these invariants can be seen as real tropical analogs of relative Gromov�Witten
Gromov-Witten classes, quantum cohomology, and enumerative geometry
The paper is devoted to the mathematical aspects of topological quantum field theory and its applications to enumerative problems of algebraic geometry. In particular, it contains an axiomatic
The numbers of tropical plane curves through points in general position
Abstract 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
...
...