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.

中文翻译：

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曲线闭合的递归公式

令$ X $为平滑复数射影变型。使用加斯曼（Gathmann）设计的构造，我们为$ X $的一些Gromov–Witten不变量提供了递归公式。我们证明，当$ X $是齐次的时，该公式给出了在$ X $的一般超曲面的一般点处的有理曲线的数量。这概括了鲑鱼$ \ mathbb {P} ^ 3 $中的曲面的经典著名的拐点（渐近线）对，以及Darboux的27个密合圆锥曲线。