A Recursive Formula for Osculating Curves

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.

Notation, small modification in Introduction, Prop. 4.1 and Lemma 5.2

Cited by in corpus (3)

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