Floor decompositions of tropical curves : the planar case
arXiv:0812.3354
Abstract
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 general toric surface case in this dimension: the curve is given by an arbitrary lattice polygon and include computation of Welschinger invariants with pairs of conjugate points. See also \cite{FM} for combinatorial treatment of floor diagrams in the projective case.
20 pages, 17 figures. V2: remove an ambiguity in the proof of theorem 6.2
References in corpus (5)
Cited by in corpus (19)
- On the tropical Torelli map
- Refined curve counting with tropical geometry
- Brief introduction to tropical geometry
- The open Gromov-Witten-Welschinger theory of blowups of the projective plane
- Psi-floor diagrams and a Caporaso-Harris type recursion
- Point-like bounding chains in open Gromov-Witten theory
- A Fock space approach to Severi degrees
- A reverse isoperimetric inequality for J-holomorphic curves
- Tropical approach to Nagata's conjecture in positive characteristic
- Refined floor diagrams from higher genera and lambda classes
- Tropical Mirror Symmetry in Dimension One
- A combinatorial analysis of Severi degrees
- Enumeration of plane unicuspidal curves of any genus via tropical geometry
- Computing Severi Degrees with Long-edge Graphs
- Real Log Curves in Toric Varieties, Tropical Curves, and Log Welschinger Invariants
- Welschinger invariants of Blow-ups of symplectic 4-manifolds
- Labeled floor diagrams for plane curves
- Tropical classes
- Polynomiality properties of tropical refined invariants