• Corpus ID: 15059353

Floor decompositions of tropical curves : the planar case

@article{Brugall2008FloorDO,
title={Floor decompositions of tropical curves : the planar case},
author={Erwan Brugall{\'e} and Grigory Mikhalkin},
journal={arXiv: Algebraic Geometry},
year={2008},
pages={64-90}
}
• Published 17 December 2008
• Mathematics
• arXiv: Algebraic Geometry
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 proofs in the tropical geometry framework, in the case when the ambient variety is a complex surface, and give some examples of computations using floor diagrams. The focusing on dimension 2 is motivated by the special combinatoric of floor diagrams compared to arbitrary dimension. We treat a…

Figures and Tables from this paper

This paper is the third installment in a series of papers devoted to the computation of enumerative invariants of abelian surfaces through the tropical approach. We develop a pearl diagram algorithm
We use the tropical geometry approach to compute absolute and relative enumerative invariants of complex surfaces which are CP 1-bundles over an elliptic curve. We also show that the tropical
• Mathematics
L’Enseignement Mathématique
• 2019
We study the maximal values of Betti numbers of tropical subvarieties of a given dimension and degree in $\mathbb{TP}^n$. We provide a lower estimate for the maximal value of the top Betti number,
Tropical geometry is a piecewise linear "shadow" of algebraic geometry. It allows for the computation of several cohomological invariants of an algebraic variety. In particular, its application to
• Mathematics
• 2009
Floor diagrams are a class of weighted oriented graphs introduced by E. Brugalle and the second author. Tropical geometry arguments lead to combinatorial descriptions of (ordinary and relative)
• Mathematics
Combinatorial Theory
• 2022
Tropical refined invariants of toric surfaces constitute a fascinating interpolation between real and complex enumerative geometries via tropical geometry. They were originally introduced by Block
We enumerate rational curves in toric surfaces passing through points and satisfying cross-ratio constraints using tropical and combinatorial methods. Our starting point is (Tyomkin in Adv Math
We enumerate rational curves in toric surfaces passing through points and satisfying cross-ratio constraints using tropical and combinatorial methods. Our starting point is (Tyomkin in Adv Math
• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 2021
We study the stationary descendant Gromov–Witten theory of toric surfaces by combining and extending a range of techniques – tropical curves, floor diagrams and Fock spaces. A correspondence theorem
• Mathematics
• 2017
We study the stationary descendant Gromov–Witten theory of toric surfaces by combining and extending a range of techniques – tropical curves, floor diagrams, and Fock spaces. A correspondence theorem

References

SHOWING 1-10 OF 22 REFERENCES

The Welschinger invariants of real rational algebraic surfaces are natural analogues of the genus zero Gromov-Witten invariants. We establish a tropical formula to calculate the Welschinger
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.
• Mathematics
• 2005
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
• Mathematics
• 2003
Welschinger’s invariant bounds from below the number of real rational curves through a given generic collection of real points in the real projective plane. We estimate this invariant using
• Mathematics
• 1994
Within the broadly defined subject of topological field theory E. Witten suggested in [1] to study generalized “intersection numbers” on a compactified moduli space $${\bar M_{g,n}}$$ of Riemann
• Mathematics
• 2004
The Welschinger numbers, a kind of a real analogue of the Gromov-Witten numbers that count the complex rational curves through a given generic collection of points, bound from below the number of
• Mathematics
• 1994
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
• Mathematics
• 2008
Welschinger invariants of the real projective plane can be computed via the enumeration of enriched graphs, called marked floor diagrams. By a purely combinatorial study of these objects, we prove a
• Mathematics
• 2007
Denis Benois. Trivial zeros of p-adic L-functions and Iwasawa theory. We prove that the expected properties of Euler systems imply quite general MazurTate-Teitelbaum type formulas for derivatives of
Introduction Notation Part I. The Picard Group and the Riemann-Roch Theorem: Part II. Birational Maps: Part III. Ruled Surfaces: Part IV. Rational Surfaces: Part V. Castelnuovo's Theorem and