Tropical and algebraic curves with multiple points

TL;DR

This paper introduces a new patchworking theorem linking tropical curves with algebraic curves having multiple points, enabling computation of Welschinger invariants for certain surfaces in tropical enumerative geometry.

Contribution

A novel patchworking theorem that connects tropical and algebraic curves with multiple points, expanding computational tools in tropical enumerative geometry.

Findings

New patchworking theorem relating tropical and algebraic curves with multiple points

Application to computing Welschinger invariants of non-toric Del Pezzo surfaces

Enhanced methods for tropical enumerative geometry

Abstract

Patchworking theorems serve as a basic element of the correspondence between tropical and algebraic curves, which is a core of the tropical enumerative geometry. We present a new version of a patchworking theorem which relates plane tropical curves with complex and real algebraic curves having prescribed multiple points. It can be used to compute Welschinger invariants of non-toric Del Pezzo surfaces.