Arkiv för Matematik

Volume 59 (2021)

Number 1

A recursive formula for osculating curves

Pages: 195 – 211



Giosuè Muratore (Department of Mathematics, Universidade Federal de Minas Gerais, Belo Horizonte, MG, Brazil)


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.


osculating, Gromov–Witten

2010 Mathematics Subject Classification

Primary 14N10. Secondary 14N15, 14N35.

Received 6 March 2020

Received revised 2 September 2020

Accepted 11 September 2020

Published 4 May 2021