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} }
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…
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22 References
A tropical calculation of the Welschinger invariants of real toric Del Pezzo surfaces
- Mathematics
- 2004
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…
Enumerative tropical algebraic geometry in R^2
- Mathematics
- 2003
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.…
Enumerative tropical algebraic geometry
- Mathematics
- 2003
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.…
The numbers of tropical plane curves through points in general position
- 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…
Counting curves on rational surfaces
- Mathematics
- 2000
Abstract:In [CH3], Caporaso and Harris derive recursive formulas counting nodal plane curves of degree d and geometric genus g in the plane (through the appropriate number of fixed general points).…
Welschinger invariant and enumeration of real rational curves
- 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…
Quantum intersection rings
- 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…
Logarithmic equivalence of Welschinger and Gromov-Witten invariants
- 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…
Gromov-Witten classes, quantum cohomology, and enumerative geometry
- 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…
Recursive formulas for Welschinger invariants of the projective plane
- 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…