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A Recursive Formula for Osculating Curves

Giosuè Muratore

Abstract

Let $X$ be a smooth complex projective variety. Using a construction devised to Gathmann, we present a recursive formula for some of the Gromov-Witten invariants of $X$. We prove that, when $X$ is homogeneous, this formula gives the number of osculating rational curves at a general point of a general hypersurface of $X$. This generalizes the classical well known pairs of inflexion (asymptotic) lines for surfaces in $\mathbb{P}^{3}$ of Salmon, as well as Darboux's $27$ osculating conics.

A Recursive Formula for Osculating Curves

Abstract

Let be a smooth complex projective variety. Using a construction devised to Gathmann, we present a recursive formula for some of the Gromov-Witten invariants of . We prove that, when is homogeneous, this formula gives the number of osculating rational curves at a general point of a general hypersurface of . This generalizes the classical well known pairs of inflexion (asymptotic) lines for surfaces in of Salmon, as well as Darboux's osculating conics.

Paper Structure

This paper contains 7 sections, 6 theorems, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Kontsevich Moduli Space of Stable Maps
  3. Gathmann Construction
  4. Osculating Curves
  5. Recursive Formula
  6. Applications
  7. Computational aspects

Key Result

Theorem 1.2

Let $X$ be a homogeneous variety, let $\beta$ be the homological class of a curve. There exists a recursive formula for the number of curves of class $\beta$ osculating a very ample hypersurface.

Figures (1)

  • Figure 6.1: Osculating curves of $X=\mathbb{P}^{s_{1}}\times...\times\mathbb{P}^{s_{t}}$.

Theorems & Definitions (20)

  • Definition 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 3.1: gathmann2002absolute
  • Theorem 3.2: gathmann2003relative
  • Definition 3.4: gathmann2003relative
  • Definition 3.5: gathmann2002absolute
  • Lemma 3.6: gathmann2003relative
  • ...and 10 more