Two lectures on the enumeration of curves by means of floor diagrams

TL;DR

This paper explores the enumeration of algebraic curves on toric surfaces using floor diagrams, providing a combinatorial approach via degeneration techniques and connections to tropical geometry, with applications to K3 surfaces.

Contribution

It presents a proof of a correspondence theorem relating curve counting to floor diagrams without tropical geometry, and discusses the relationship with tropical geometry and toric varieties.

Findings

Proof of a correspondence theorem for plane curves using degeneration

Connection between floor diagrams and tropical geometry

Applications to enumerative geometry of K3 surfaces

Abstract

We discuss, following Mikhalkin, Brugall\'e, and many others, the counting of curves on toric surfaces with prescribed genus, Newton polygon, and intersection pattern with the toric boundary divisor, both at assigned and unassigned points. The first lecture is dedicated to the proof of a correspondence theorem (for plane curves) with the counting of floor diagrams, using a degeneration of the projective plane to a chain of rational ruled surfaces. This is due to Brugall\'e and does not involve any tropical geometry. The second lecture explores the relations with tropical geometry, and contains an introduction to toric varieties and tropical geometry. We discuss the correspondence theorem of Mikhalkin, and show how the corresponding tropical enumerative problem can be formulated in terms of the combinatorial problem of counting floor diagrams. We give many examples throughout, inspired by the study of the enumerative geometry of $K3$ surfaces, by degeneration to unions of rational surfaces with dual complex a tiling of the $\mathbf{S}^2$ sphere.