# Who streamlined Kontsevich's count of rational curves?

Let $N_d$ denote the number of rational curves in $\mathbf P^2$ passing through $3d-1$ points in general position. Maxim Kontsevich discovered a famous recursion for these numbers: $$N_d = \sum_{k+l = d} N_k N_l k^2 l \left( l \binom{3d-4}{3k-2} - k \binom{3d-4}{3k-1}\right).$$ The proof of this recursion goes by interpreting $N_d$ as a Gromov-Witten invariant: one looks at the moduli space $\overline{M}_{0,3d-1}(\mathbf P^2, d)$ and pulls back the class of a point along each evaluation map. Taking the product of all these classes in the Chow ring produces a number. Using that $\mathbf P^2$ is homogeneous one can show that this number actually counts the number of stable maps where the markings are sent to the given points, and it is not hard to see that counting stable maps is the same thing as counting rational curves. Finally, the associativity of the quantum product can be translated to the WDVV differential equations for the Gromov-Witten potential, i.e. the generating function of all Gromov-Witten invariants. These differential equations translate into the above recursion.

At least, this is how the proof is stated in Kontsevich's "Enumeration of rational curves via torus actions" and Konstevich-Manin "Gromov-Witten classes, quantum cohomology and enumerative geometry", which seem to be the earliest published sources.

However, there is also a beautiful streamlined proof which avoids the use of the quantum product. Here one instead works with $\overline M_{0,3d}(\mathbf{P}^2,d)$ (one more marking) and takes the pullback of the classes of two lines in $\mathbf P^2$ along the first two markings and $3d-2$ classes of points for the remaining. The intersection of these are a curve in the moduli space and one then computes the intersection of this curve with two different linearly equivalent boundary divisors. These two intersection numbers can easily be computed by hand, producing the recursion. The proof that these two boundary divisors are linearly equivalent uses the forgetful map to $\overline{M}_{0,4}$ which is also used in the proof of the WDVV equations so in some sense it seems like this proof inlines the particular case of WDVV that is needed in a very clever way.

This latter version of the proof appears for instance in the book of Kock and Vainsencher, in Abramovich's "Lectures on Gromov-Witten invariants on orbifolds", and in the lecture notes I found here. But where is it from originally? All these sources just refer to it as "Kontsevich's proof" without attribution. Did Kontsevich also come up with this streamlined version but did not see it as worth publishing?

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the link is broken... –  Dmitri Mar 8 '11 at 16:29
Should $N_d$ appear on both sides of the given recursion? –  Daniel Litt Mar 8 '11 at 17:18
That's harmless, since all terms where $k$ or $l$ are equal to $d$ will vanish anyway. Perhaps it would have been clearer to write the sum over all $k+l = d$, $k \geq 1$, $l \geq 1$. –  Dan Petersen Mar 8 '11 at 17:33
@Dan: I think Daniel's referring to the typo $N_d$ on the RHS. –  ndkrempel Mar 8 '11 at 19:16
Oh. Thanks. Yes, that was a typo. –  Dan Petersen Mar 8 '11 at 19:49