Broccoli curves and the tropical invariance of Welschinger numbers

TL;DR

This paper introduces broccoli curves, a new class of tropical curves, and demonstrates their invariance properties, establishing a tropical proof of the invariance of Welschinger numbers and providing a method to compute them.

Contribution

It defines broccoli curves and proves their invariance, linking them to Welschinger invariants and offering a tropical approach to compute these invariants.

Findings

Broccoli invariants equal Welschinger invariants in toric Del Pezzo cases.

Broccoli invariants are independent of point and end direction choices.

Provides a tropical Caporaso-Harris formula for computing Welschinger invariants.

Abstract

In this paper we introduce broccoli curves, certain plane tropical curves of genus zero related to real algebraic curves. The numbers of these broccoli curves through given points are independent of the chosen points - for arbitrary choices of the directions of the ends of the curves, possibly with higher weights, and also if some of the ends are fixed. In the toric Del Pezzo case we show that these broccoli invariants are equal to the Welschinger invariants (with real and complex conjugate point conditions), thus providing a proof of the independence of Welschinger invariants of the point conditions within tropical geometry. The general case gives rise to a tropical Caporaso-Harris formula for broccoli curves which suffices to compute all Welschinger invariants of the plane.