A recursive formula for osculating curves

Let $X$ be a smooth complex projective variety. Using a construction devised by 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 inflection (asymptotic) lines for surfaces in $\mathbb{P}^3$ of Salmon, as well as Darboux’s 27 osculating conics.

Keywords

osculating, Gromov–Witten

2010 Mathematics Subject Classification

Primary 14N10. Secondary 14N15, 14N35.

Full Text (PDF format)

Received 6 March 2020

Received revised 2 September 2020

Accepted 11 September 2020

Published 4 May 2021