A tropical calculation of the Welschinger invariants of real toric Del Pezzo surfaces

TL;DR

This paper introduces a tropical geometric formula to compute Welschinger invariants of real toric Del Pezzo surfaces, linking algebraic invariants with combinatorial tropical curve counts.

Contribution

It provides the first tropical formula for Welschinger invariants of real toric Del Pezzo surfaces applicable to any conjugation-invariant point configuration.

Findings

Derived a tropical formula expressing invariants via tropical curves and lattice paths.

Computed specific Welschinger invariants using the new formula.

Validated the formula through joint computational results.

Abstract

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 invariants of real toric Del Pezzo surfaces for any conjugation-invariant configuration of points. The formula expresses the Welschinger invariants via the total multiplicity of certain tropical curves (non-Archimedean amoebas) passing through generic configurations of points, and then via the total multiplicity of some lattice path in the convex lattice polygon associated with a given surface. We also present the results of computation of Welschinger invariants, obtained jointly with I. Itenberg and V. Kharlamov.