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

Abstract

We discuss, following Mikhalkin, Brugallé, 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 Brugallé 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.