Let $p \in H^4(\mathbb{CP}^2)$ and $\ell \in H^2(\mathbb{CP}^2)$ be the cohomology classes Poincaré dual to a point and a line respectively.

Question. What is the Gromov-Witten invariant $\langle p, p, \ell\rangle_{0, 1}$ counting degree $1$, genus $0$ curves in $\mathbb{CP}^2$?

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    $\begingroup$ This is equivalent to asking for the number of straight lines passing through two general points in the plane, i.e. 1. $\endgroup$ Dec 6, 2016 at 6:02

1 Answer 1


As Dan says, you can use the divisor equation: $$\langle e_{\alpha_1}, \ldots, e_{\alpha_n}, \ell \rangle_{g,d} = d\ \langle e_{\alpha_1}, \ldots, e_{\alpha_n} \rangle_{g,d} $$ where the $e_{\alpha_i}$ are any cohomology classes of $\mathbb{CP}^2$, to reduce your invariant $\langle p,p,\ell \rangle_{0,1}$ to $\langle p,p\rangle_{0,1}$, the number of lines through two points.


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