Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Thomas Dedieu

TL;DR

This work presents a two-lecture framework for enumerating curves on toric surfaces through floor diagrams and tropical methods. The first lecture gives a purely algebro-geometric degeneration of $\mathbf{P}^2$ to a chain of Hirzebruch surfaces, producing floor diagrams that encode stable maps and yield a Correspondence Theorem equating plane-curve counts with combinatorial diagrams. The second lecture connects these diagrams to tropical geometry, introducing toric/tropical preliminaries, the tropical enumerative problem, and a detailed translation to floor diagrams via degenerations, with extensive examples from $K3$-surface degenerations and other toric cases. Together, the text demonstrates how complex algebro-geometric counts can be recast into tractable combinatorial and tropical problems, enabling explicit computations of plane curves with prescribed genus, degree, and boundary contact. The approach highlights the power of degeneration and tropicalization in bridging algebraic geometry, toric geometry, and combinatorics, with concrete multiplicities and degeneration models that recover classical counts and yield new tropical invariants. This framework has practical impact for computing enumerative invariants on toric surfaces and for understanding the tropical–algebraic correspondence in a broad geometric setting.

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.

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

TL;DR

This work presents a two-lecture framework for enumerating curves on toric surfaces through floor diagrams and tropical methods. The first lecture gives a purely algebro-geometric degeneration of to a chain of Hirzebruch surfaces, producing floor diagrams that encode stable maps and yield a Correspondence Theorem equating plane-curve counts with combinatorial diagrams. The second lecture connects these diagrams to tropical geometry, introducing toric/tropical preliminaries, the tropical enumerative problem, and a detailed translation to floor diagrams via degenerations, with extensive examples from -surface degenerations and other toric cases. Together, the text demonstrates how complex algebro-geometric counts can be recast into tractable combinatorial and tropical problems, enabling explicit computations of plane curves with prescribed genus, degree, and boundary contact. The approach highlights the power of degeneration and tropicalization in bridging algebraic geometry, toric geometry, and combinatorics, with concrete multiplicities and degeneration models that recover classical counts and yield new tropical invariants. This framework has practical impact for computing enumerative invariants on toric surfaces and for understanding the tropical–algebraic correspondence in a broad geometric setting.

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 surfaces, by degeneration to unions of rational surfaces with dual complex a tiling of the sphere.

Paper Structure

This paper contains 75 sections, 15 theorems, 128 equations, 40 figures.

Table of Contents

  1. Enumeration of curves by floor diagrams, via degenerations of the projective plane
  2. Notation
  3. Definitions and statement of the result
  4. Floor diagrams of plane curves relative to a line
  5. A special class of weighted oriented graphs
  6. Marked floor diagrams of degree $3$ and genus $0$ of type $(0,3)$
  7. Marked floor diagrams of degree $3$ and genus $0$ of type $(0,[1,1])$
  8. Marked floor diagrams of degree $3$ and genus $0$ of type $([0,1],1)$
  9. The enumerative problem
  10. The Correspondence Theorem
  11. Log-GW invariants of $\mathbf{P}^2$ and Hirzebruch surfaces
  12. Stable maps relative to a curve
  13. Log-GW invariants
  14. Enumeratively relevant components of dimension at most $1$
  15. Multiplicity for multiple fibres
  16. ...and 60 more sections

Key Result

Lemma (1.6)

Let $\mathcal{D}$ be a floor diagram of degree $d$ and genus $g$. The two following equalities hold:

Figures (40)

  • Figure 1: Floor diagrams of degree $3$ and genus $1$
  • Figure 2: Marked floor diagrams of degree $3$ and genus $0$ of type $(0,3)$.
  • Figure 3: Marked floor diagrams of degree $3$ and genus $0$ of type $(0,[1,1])$.
  • Figure 4: Marked floor diagrams of degree $3$ and genus $0$ of type $([0,1],1)$.
  • Figure 5: The degeneration $\pi_r : \mathcal{P}_r \to \mathbf{D}$, together with limit base points and tangency points along the boundary
  • ...and 35 more figures

Theorems & Definitions (53)

  • Definition (1.3): floor diagram
  • Example (1.4)
  • Lemma (1.6)
  • proof
  • Definition (1.7): marking
  • Definition (1.9)
  • Theorem (1.10): Brugallé--Mikhalkin BM, see also arroyoetal
  • Example (1.11)
  • Definition (2.12)
  • Theorem (2.15)
  • ...and 43 more