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Relative Gromov-Witten and maximal contact conics

Giosuè Muratore

TL;DR

The paper studies maximal-contact rational curves relative to a general smooth plane curve $Y$ of degree $d\ge3$ through relative Gromov--Witten theory, culminating in the exact count of sextactic conics: $n_d=3d(4d-9)$. The authors develop a framework of relative stable maps, derive explicit line-count formulas, and establish a polynomiality phenomenon in $d$ that determines many counts from finite data. A key technical achievement is isolating the double-cover contributions from inflectional lines, which allows the extraction of the true enumerative number of sextactic conics and clarifies the structure of the moduli space for degree $2$ maps. The results connect classical enumerative geometry with modern GW techniques, providing a clean GW-based derivation of Cayley’s sextactic count and suggesting natural generalizations to higher degrees and to higher-dimensional ambient spaces.

Abstract

We discuss some properties of the relative Gromov--Witten invariants counting rational curves with maximal contact order at one point. We compute the number of Cayley's sextactic conics to any smooth plane curve $Y$. In particular, we compute the contribution, from double covers of inflectional lines, to a certain degree $2$ relative Gromov--Witten invariant relative to $Y$.

Relative Gromov-Witten and maximal contact conics

TL;DR

The paper studies maximal-contact rational curves relative to a general smooth plane curve of degree through relative Gromov--Witten theory, culminating in the exact count of sextactic conics: . The authors develop a framework of relative stable maps, derive explicit line-count formulas, and establish a polynomiality phenomenon in that determines many counts from finite data. A key technical achievement is isolating the double-cover contributions from inflectional lines, which allows the extraction of the true enumerative number of sextactic conics and clarifies the structure of the moduli space for degree maps. The results connect classical enumerative geometry with modern GW techniques, providing a clean GW-based derivation of Cayley’s sextactic count and suggesting natural generalizations to higher degrees and to higher-dimensional ambient spaces.

Abstract

We discuss some properties of the relative Gromov--Witten invariants counting rational curves with maximal contact order at one point. We compute the number of Cayley's sextactic conics to any smooth plane curve . In particular, we compute the contribution, from double covers of inflectional lines, to a certain degree relative Gromov--Witten invariant relative to .
Paper Structure (7 sections, 10 theorems, 44 equations, 1 figure)

This paper contains 7 sections, 10 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and background
  3. Explicit computation of some relative GW invariants
  4. The case of lines
  5. The case of any degree
  6. Sextactic conics
  7. Curves of higher degree

Key Result

Theorem 1.1

Let $Y$ be a general smooth plane curve of degree $d\ge 3$. The number of conics of contact order $6$ with $Y$ is

Figures (1)

  • Figure 4.1: The space $M_l$.

Theorems & Definitions (26)

  • Theorem 1.1
  • Example 2.1
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Example 3.2
  • Proposition 3.3
  • proof
  • Proposition 3.4
  • proof
  • ...and 16 more