Intersection theory on moduli spaces of curves ... - User Web Pages

1.2. <strong>Intersection</strong> <strong>theory</strong> on moduli spaces 21bers. Generating functions are commonly used to encode information about the intersection<strong>theory</strong> on M g,n . That this result can be expressed so succinctly using them highlights the combinatorialnature of the tautological ring. This fact is reinforced by the presence of the Bernoullinumbers, which will make another appearance in Chapter 2.The following statement was first conjectured by Hain and Looijenga [20].Conjecture 1.13. The tautological ring R ∗ (M g,n ) is a Poincaré duality ring of dimension 3g − 3 + n.To understand the content of this conjecture, we break it into three parts.Vanishing conjecture. For k > 3g − 3 + n, R k (M g,n ) ∼ = 0.Socle conjecture. R 3g−3+n (M g,n ) ∼ = Q.Perfect pairing conjecture. For 0 ≤ k ≤ 3g − 3 + n, the following natural product is a perfectpairing.R k (M g,n ) × R 3g−3+n−k (M g,n ) → R 3g−3+n (M g,n ) ∼ = QIn order to appreciate the depth of these statements, one must consider the tautological rings assubsets of the corresponding Chow rings. For instance, if we were to consider the tautologicalring as a subset of the cohomology ring, then the socle conjecture would be trivially true. On theother hand, the socle conjecture is neither obvious in the tautological Chow ring nor even truein the full Chow ring. At present, the conjecture remains unresolved, although there is a fairamount of low genus evidence. And if the conjecture is true, then one important consequencewould be that it is possible to recover the structure of the entire tautological ring from the topintersections of tautological classes alone. Furthermore, from Definition 1.8, any top intersectionsin the tautological ring can be determined from the top intersections of psi-classes alone.Therefore, we are motivated to study intersection numbers of the form∫ψ α 11M ψα 22 . . . ψα nng,n∈ Qwhere |α| = 3g − 3 + n or equivalently, g =3 1 (|α| − n + 3). It will be convenient to adopt Witten’snotation for these psi-class intersection numbers, which suppresses the genus and encodesthe symmetry between the psi-classes.∫〈τ α1 τ α2 . . . τ αn 〉 = ψ α 11M ψα 22 . . . ψα nng,nWe treat the τ variables as commuting, so that we can write intersection numbers in the form〈τ d 00 τd 11 τd 22 . . .〉 and we set 〈τ α 1τ α2 . . . τ αn 〉 = 0 if n = 0 or if the genus g = 1 3(|α| − n + 3) isnon-integral or negative. In this way, we have defined a linear functional〈·〉 : Q[τ 0 , τ 1 , τ 2 , . . .] → Q.