Floor diagrams relative to a conic, and GW-W invariants of Del Pezzo surfaces

TL;DR

This paper uses floor diagrams and degeneration formulas to count complex and real curves in blown-up projective planes, deriving Gromov-Witten and Welschinger invariants of Del Pezzo surfaces.

Contribution

It introduces a novel enumeration method via floor diagrams for curves in blown-up planes and computes related invariants using degeneration techniques.

Findings

Enumeration of curves via floor diagrams in blown-up planes

Calculation of Gromov-Witten invariants for Del Pezzo surfaces

Determination of Welschinger invariants for real curves

Abstract

We enumerate, via floor diagrams, complex and real curves in the projective plane blown up in $n$ points on a conic. As an application, we deduce Gromov-Witten and Welschinger invariants of Del Pezzo surfaces. These results are mainly obtained using Li's degeneration formula and its real counterpart.