Floor decompositions of tropical curves : the planar case

TL;DR

This paper provides detailed proofs and computational methods for Gromov-Witten and Welschinger invariants of toric surfaces using floor diagrams within tropical geometry, focusing on the planar case.

Contribution

It introduces a detailed tropical geometry framework for computing invariants of toric surfaces using floor diagrams, extending previous combinatorial approaches.

Findings

Validated formulas for invariants in the planar case

Provided explicit examples of computations with floor diagrams

Extended methods to include Welschinger invariants with conjugate points

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.