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Kontsevich's Formula for Rational Curves from Classical and Quantum Perspectives

Greg Weiler

TL;DR

This work establishes Kontsevich's recursion for counting rational curves of degree $d$ through $3d-1$ general points in $\mathbb{P}^2$ via two complementary approaches: a direct boundary-analysis on Kontsevich spaces and a cohomological route through Gromov–Witten invariants and quantum cohomology. It develops the requisite moduli spaces, analyzes their boundary strata with fundamental relations, and extends the framework to curves on $\mathbb{P}^1\times\mathbb{P}^1$ by working with bidegrees. The GW/quantum formalism yields generating functions, reconstruction results, and explicit small quantum rings for both targets, enabling recursive computation of all invariants from base cases like $N_1=1$, $N_{(1,0)}=N_{(0,1)}=1$. By leveraging associativity of the quantum product, the paper provides a cohesive and elegant derivation of Kontsevich's formula, connecting enumerative geometry and quantum cohomology in a unified framework with broad generalizations to other spaces and higher genus.

Abstract

Kontsevich's formula for rational plane curves is a recursive relation for the number $N_d$ of degree $d$ rational curves in $\mathbb{P}^2$ passing through $3d-1$ general points. We provide two proofs of this recursion: the first more direct and combinatoric, the second more abstract. In order to achieve this, we introduce several moduli spaces, such as the Deligne-Mumford-Knudsen spaces and the Kontsevich spaces, and exploit their properties. In particular, the boundary structure of these spaces gives rise to certain fundamental relations crucial to both proofs. For the second proof, we reconsider the objects in question from the cohomological viewpoint and generalize the numbers $N_d$ to Gromov-Witten invariants. We introduce quantum cohomology and deduce Kontsevich's formula from the associativity of the quantum product. We also adapt these steps to the case of curves in $\mathbb{P}^1\times\mathbb{P}^1$, whose bidegrees lead to slightly more complicated but analogous results.

Kontsevich's Formula for Rational Curves from Classical and Quantum Perspectives

TL;DR

This work establishes Kontsevich's recursion for counting rational curves of degree through general points in via two complementary approaches: a direct boundary-analysis on Kontsevich spaces and a cohomological route through Gromov–Witten invariants and quantum cohomology. It develops the requisite moduli spaces, analyzes their boundary strata with fundamental relations, and extends the framework to curves on by working with bidegrees. The GW/quantum formalism yields generating functions, reconstruction results, and explicit small quantum rings for both targets, enabling recursive computation of all invariants from base cases like , . By leveraging associativity of the quantum product, the paper provides a cohesive and elegant derivation of Kontsevich's formula, connecting enumerative geometry and quantum cohomology in a unified framework with broad generalizations to other spaces and higher genus.

Abstract

Kontsevich's formula for rational plane curves is a recursive relation for the number of degree rational curves in passing through general points. We provide two proofs of this recursion: the first more direct and combinatoric, the second more abstract. In order to achieve this, we introduce several moduli spaces, such as the Deligne-Mumford-Knudsen spaces and the Kontsevich spaces, and exploit their properties. In particular, the boundary structure of these spaces gives rise to certain fundamental relations crucial to both proofs. For the second proof, we reconsider the objects in question from the cohomological viewpoint and generalize the numbers to Gromov-Witten invariants. We introduce quantum cohomology and deduce Kontsevich's formula from the associativity of the quantum product. We also adapt these steps to the case of curves in , whose bidegrees lead to slightly more complicated but analogous results.
Paper Structure (28 sections, 60 theorems, 167 equations, 1 figure, 1 table)

This paper contains 28 sections, 60 theorems, 167 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Primer on Moduli Spaces
  3. Relevant Spaces and their Properties
  4. Moduli Spaces of Curves and Maps for $\mathbb{P}^r$
  5. Moduli Spaces of Curves and Maps for $\mathbb{P}^1\times\mathbb{P}^1$
  6. Kontsevich's Formula: Classical Approach
  7. Kontsevich's Formula for $\mathbb{P}^2$
  8. Kontsevich's Formula for $\mathbb{P}^1\times\mathbb{P}^1$
  9. Extending the Theory
  10. Sequences and Generating Functions
  11. Homology and Cohomology
  12. Quantum Formalism
  13. Gromov Witten Invariants and Quantum Cohomology for $\mathbb{P}^r$
  14. Gromov Witten Invariants and Quantum Cohomology for $\mathbb{P}^1\times\mathbb{P}^1$
  15. Computations
  16. ...and 13 more sections

Key Result

Proposition 3.2

For $n\ge3$, there exists a fine moduli space $M_{0,n}$ for the classification problem of $n$-pointed smooth rational curves up to isomorphism.

Figures (1)

  • Figure 1: Stratification of the boundary of $\overline{M}_{0,6}$

Theorems & Definitions (154)

  • Definition 2.2
  • Remark 2.3: Parameter Space
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Example 2.7: Universal Family
  • Remark 2.8: Intuition on the Existence of Moduli Spaces
  • Definition 3.1
  • Proposition 3.2
  • Definition 3.3
  • ...and 144 more