Divisors in the moduli spaces of curves

Abstract

The calculation by Harer [12] of the second homology groups of the moduli spaces of smooth curves over C can be regarded as a major step towards the understanding of the enumerative geometry of the moduli spaces of curves [21, 17]. However, from the point of view of an algebraic geometer, Harer’s approach has the drawback of being entirely transcendental; in addition, his proof is anything but simple. It would be desirable to provide a proof of his result which is more elementary, and algebro-geometric in nature. While this cannot be done at the moment, as we shall explain in this note it is possible to reduce the transcendental part of the proof, at least for homology with rational coefficients, to a single result, also due to Harer [13], asserting that the homology of Mg,n, the moduli space of smooth n-pointed genus g curves, vanishes above a certain explicit degree. A sketch of the proof of Harer’s vanishing theorem, which is not at all difficult, will be presented in Section 5 of this survey. It must be observed that Harer’s vanishing result is an immediate consequence of an attractive algebro-geometric conjecture of Looijenga (Conjecture 1 in Section 5); an affirmative answer to the conjecture would thus give a completely algebro-geometric proof of Harer’s theorem on the second rational homology of moduli spaces of curves. In this note we describe how one can calculate the first and second rational (co)homology groups of Mg,n, and those of Mg,n, the moduli space of stable n-pointed curves of genus g, using only relatively simple algebraic geometry and Harer’s vanishing theorem. ForMg,n, this program was carried out in [5], where the third and fifth cohomology groups were also calculated and shown to always vanish; in Section 6, we give an outline of the argument, which uses in an essential way a simple Hodge-theoretic result due to Deligne [10]. In genus zero, we rely on Keel’s calculation of the Chow ring of M0,n; a simple proof of Keel’s result in the case of divisors is presented in Section 4. We finally give a new proof of Harer’s theorem for H(Mg,n;Q); we also