Generating functions in algebraic geometry and sums over trees.

*(English)*Zbl 0871.14022
Dijkgraaf, R. H. (ed.) et al., The moduli space of curves. Proceedings of the conference held on Texel Island, Netherlands, during the last week of April 1994. Basel: Birkhäuser. Prog. Math. 129, 401-417 (1995).

This paper solves three different and important enumerative problems in the setting of algebraic geometry, which are united by the fact that the relevant formulas which are used can be expressed as a sum of certain weights over isomorphism classes of marked trees. The technique used involves some tricks well known to physicists. The sum is written as a partition function that can be computed by evaluating a formal potential function at an appropriate critical point. Although these steps cannot be justified completely in the full generality of marked trees, the author is able to apply them, according to the mathematical rigour, in order to solve the following three problems.

The first problem is the computation of the Betti numbers of the moduli space \(\overline M_{0,n}\) of stable n-pointed curves of genus zero. These moduli spaces appear in the computation of Gromov-Witten invariants occurring in the theory of quantum cohomology. Some nice formulas are obtained for the generating function of the Poincaré polynomials (and of the Euler characteristic) which imply recursive formulas which are simpler than the formulas obtained by S. Keel [Trans. Am. Math. Soc. 330, No. 2, 545-574 (1992; Zbl 0768.14002)].

The second problem is the computation of the Betti numbers of the configuration space \(X[n]\), which is a natural compactification constructed by W. Fulton and R. MacPherson [Ann. Math., II. Ser. 139, No. 1, 183-225 (1994; Zbl 0820.14037)] of the space of n pairwise distinct labeled points on a smooth compact variety \(X\).

The third and last problem comes from the work of M. Kontsevich in the same volume as the paper under review [in: “The moduli space of curves”, Prog. Math. 129, 335-368 (1995)] and consists in the calculation of the number \(m_d\) which gives the contribution of maps of degree \(d\) from \({\mathbb{P}}^1\) to a quintic threefold \(X\) (which is Calabi-Yau) whose image has normal sheaf \({\mathcal O}(-1)\oplus {\mathcal O}(-1)\). The author shows that \(m_d=d^{-3}\) by managing a formula of Kontsevich. This last problem has been considered and generalized in a recent preprint of Graber and Pandharipande (alg-geom/9708001).

For the entire collection see [Zbl 0827.00037].

The first problem is the computation of the Betti numbers of the moduli space \(\overline M_{0,n}\) of stable n-pointed curves of genus zero. These moduli spaces appear in the computation of Gromov-Witten invariants occurring in the theory of quantum cohomology. Some nice formulas are obtained for the generating function of the Poincaré polynomials (and of the Euler characteristic) which imply recursive formulas which are simpler than the formulas obtained by S. Keel [Trans. Am. Math. Soc. 330, No. 2, 545-574 (1992; Zbl 0768.14002)].

The second problem is the computation of the Betti numbers of the configuration space \(X[n]\), which is a natural compactification constructed by W. Fulton and R. MacPherson [Ann. Math., II. Ser. 139, No. 1, 183-225 (1994; Zbl 0820.14037)] of the space of n pairwise distinct labeled points on a smooth compact variety \(X\).

The third and last problem comes from the work of M. Kontsevich in the same volume as the paper under review [in: “The moduli space of curves”, Prog. Math. 129, 335-368 (1995)] and consists in the calculation of the number \(m_d\) which gives the contribution of maps of degree \(d\) from \({\mathbb{P}}^1\) to a quintic threefold \(X\) (which is Calabi-Yau) whose image has normal sheaf \({\mathcal O}(-1)\oplus {\mathcal O}(-1)\). The author shows that \(m_d=d^{-3}\) by managing a formula of Kontsevich. This last problem has been considered and generalized in a recent preprint of Graber and Pandharipande (alg-geom/9708001).

For the entire collection see [Zbl 0827.00037].

Reviewer: G.Ottaviani (L’Aquila)

##### MSC:

14H10 | Families, moduli of curves (algebraic) |

14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |

14D20 | Algebraic moduli problems, moduli of vector bundles |