Seminar in Real & Complex Geometry

Thursday, 01.11.2012, 16:15-17:30, Schreiber building, room 210

Franziska Schroether, Universitaet des Saarlandes

Broccoli curves of genus 0 and maybe 1


Welschinger numbers count (in a weighted manner) real rational curves on a toric Del Pezzo surface, e.g. $\mathbb{P}^2$, which pass through a generic configuration $\mathcal{P}$ of conjugation invariant points. Although it is possible to show the invariance of these numbers with methods from symplectic geometry, i.e. that they do not depend on the choice of $\mathcal{P}$, there was no proof in the tropical context yet when $\mathcal{P}$ also contains pairs of complex conjugate points. In this talk, I want to describe the tropical situation and I want to introduce certain tropical curves, namely broccoli curves, whose count is invariant. Under certain conditions, one then gets Welschinger invariants what yields a tropical proof of the invariance of these numbers. If time permits, I will also illustrate the case of broccoli curves of genus 1.