On the Chow Ring Of

We describe the Chow ring with rational coefficients of the moduli space of stable maps with marked points M0,m(P , d) as the subring of invariants of a ring B(M0,m(P , d);Q), relative to the action of the group of symmetries Sd. B (M0,m(P , d);Q) is computed by following a sequence of intermediate spaces for M0,m(P , d) and relating them to substrata of M0,1(P , d + m − 1). An additive basis for A(M0,m(P , d);Q) is given. Introduction The moduli spaces of stable maps introduced by Kontzevich in [KM2] have come to play a central role in the enumerative geometry of curves. In this paper we are concerned with M0,m(P n, d) when m > 0, which parametrizes stable maps from genus zero, at most nodal curves with m distinct, smooth marked points, into the projective space. M 0,m(P n, d) is a smooth DeligneMumford stack having a projective coarse model (see [BM], [FP]). Thus the fine and coarse moduli spaces share the same cohomology and Chow groups with rational coefficients ([V]). Intersection theory in codimension zero on M0,m(P n, d), known as part of Gromow-Witten theory, has been intensively studied starting with [KM2], [G]. The Picard group was determined by [P]. More recent results regard other aspects of the cohomology: the cohomology groups are generated by tautological classes ([O1], [O2]). An algorithm for computing the Betti numbers of all spaces M0,m(P n, d) is given in [GP]. Forays into the cohomology ring structure for specific cases of moduli spaces of stable maps start with the degree zero case M 0,m, whose Chow ring is calculated in [Ke]. The cohomology of the moduli space without marked points M0,0(P n, d) was computed in degree 2 and partially in degree 3 by [BO]. An additive basis for M0,2(P n, 2) was written down in [C], the ring structure of M0,2(P 1, 2) was determined in [C2]. [MM] contains a description of the Chow ring with rational coefficients of the moduli space M0,1(P n, d) as the subalgebra of invariants of a larger Q-algebra B(M0,1(P n, d);Q), relative to the action of the group of symmetries Sd. The study of the algebra B(M0,1(P n, d);Q) is both motivated and made possible by the existence of a sequence of intermediate spaces for M 0,1(P n, d), the birational morphisms between which can be understood in detail. In the present paper we extend Date: April 19, 2008. 1 2 ANDREI MUSTAŢǍ AND MAGDALENA ANCA MUSTAŢǍ this construction to M0,m(P n, d), whose intermediate moduli spaces turn out to be intimately related to substrata of M0,1(P n, d+m− 1). This leads to a natural construction of the ring B(M0,m(P n, d);Q), whose structure, shown in Theorem 5.1, follows mainly from work done in [MM]. An additive basis for A∗(M 0,m(P n, d);Q) is also easily discernible (Proposition 6.1). The elements of the basis are clearly tautological and can be counted as a sum indexed by rooted decorated trees with marked leaves. The paper is organized as follows: The first section is devoted to the construction of a system of intermediate spaces for M0,m(P n, d). As mentioned, these spaces have mostly already appeared as normal substrata of M0,1(P n, d+m− 1) in [MM]. However here they are introduced as smooth Deligne-Mumford stacks finely representing certain moduli problems. Via the rigidified moduli spaces of stable maps introduced in [FP], these moduli spaces turn out to be closely related to moduli spaces of rational weighted stable curves previously studied by [Has]. Indeed, rigidified moduli spaces form a cover of M0,m(P n, d) by finite morphisms and are at the same time torus bundles over Zariski open sets in M0,m+(n+1)d. This is clearly visible when taking into account, for each generic point (C, {pi}1≤i≤m → P n), (n+1) independent hyperplane sections on C. On the other hand, a system of weights on the marked points allows Hassett in [Has] to construct moduli spaces of curves where some marked points are allowed to coincide. Quotients by (Sd) n of torus bundles over Zariski open sets in these moduli spaces of curves glue together to smooth stacks birational to M0,m(P n, d). These are the intermediate (weighted) moduli spaces of stable mapsM0,A(P n, d, a). They parametrize rational curves with weighted marked points mapping to Pn such that only tails of certain degrees are allowed. The second and third sections keep record of the canonical stratification of M0,m(P n, d) and its intermediate spaces by means of decorated trees and good monomials of stable two-partitions. The sources of this approach are [BM], [KM1], [LM]. Various decorations and extra structures are attached to the trees to account for the substrata of M0,A(P n, d, a), and the complex network of morphisms between them. In this context the language of 2-partitions fits perfectly into the construction of the Q-algebra B(M0,A(P n, d, a)). The ring B∗(M 0,A(P n, d, a)) introduced in section 4 holds inside the Chow groups of all substrata of M0,A(P n, d, a), and takes into account all morphisms between substrata indexed by contractions of trees. It is shown that the invariant subalgebra of B(M0,A(P n, d, a)) with respect to the natural action of Sd is A ∗(M 0,A(P n, d, a)). A formula for B∗(M 0,m(P n, d)) is given in Theorem 5.1, and an additive basis for A∗(M 0,m(P n, d)) is written in Proposition 6.1. The ensuing Poincairé polynomial computation, as a sum indexed by rooted decorated trees with marked leaves, becomes cumbersome in high degree, but is practical in the low-degree examples considered at the end of the paper. ON THE CHOW RING OF M0,m(P, d) 3 Acknowledgements. We are grateful to Kai Behrend and Daniel Krashen for helpful suggestions, to Jim Bryan for interest and hospitality at the University of British Columbia. 1. Intermediate moduli spaces of M0,m(P n, d) In this section we consider curves of any arithmetic genus g ≥ 0, with marked points and at most nodal singularities different from the markings, and morphisms to Pn. The following moduli spaces were introduced in [Has]: Definition 1.1. A collection of N weights is an N -tuple A = (a1, ..., aN ) ∈ QN such that 0 < aj ≤ 1 for all j=1,...,N and such that ∑N i=1 ai > 2 − g, where g ≥ 0. Given a collection of N weights A, a family of genus g nodal curves with marked points (C, s1, ..., sN ) → π S is (g,A)-stable if: (1) The map π : C → S is flat, projective and such that each geometric fiber is a nodal connected curve of arithmetic genus g. (2) The sections (s1, ..., sN ) of π lie in the smooth locus of π, and for any subset {si}i∈I with nonempty intersection, ∑ i∈I ai ≤ 1. (3) Kπ+ ∑N i=1 aisi is relatively ample, whereKπ is the relative canonical divisor. Theorem 1.2. (2.1 in [Has] ) The moduli problem of (g,A)-stable curves is finely represented by a smooth Deligne-Mumford stack Mg,A, proper over Z. The corresponding coarse moduli scheme is projective over Z. A natural collection of morphisms between these spaces can be constructed: Theorem 1.3. (4.1 in [Has] ) For two collections of weights A = (a1, ..., aN ) and (A′ = (a1, ..., a ′ N )) such that ai ≤ a ′ i for each i, there exists a natural birational reduction morphism ρ : Mg,A′ → M g,A. The image of an element (C, s1, ..., sN ) ∈ Mg,A′ is obtained by successively collapsing components along which the divisor KC + ∑N i=1 aisi fails to be ample. Consider in addition a number a > 0. Definition 1.4. Let t̄ = (t0 : ... : tn) denote a homogeneous coordinate system on Pn. Let d > 0 and m be natural numbers and let A = (a1, ..., am) denote a collection of m weights. A t̄-rigid, (A, a)-stable family of degree d nodal maps with m marked points to Pn consists of the following data: (π : C → B, {qi,j}0≤i≤n,1≤j≤d, {pi}1≤i≤m,L, e) such that (1) the family (π : C → S, {qi,j}0≤i≤n,1≤j≤d, {pi}1≤i≤m) is a (g,A ′)stable family of curves, where the system of weights A′ consists of: • ai,j = a (n+1) for the sections {qi,j}0≤i≤n,1≤j≤d; 4 ANDREI MUSTAŢǍ AND MAGDALENA ANCA MUSTAŢǍ • ai for pi, i = 1, ...,m. (2) L is a line bundle on C and e : O C → L is a morphism of sheaves such that, via the natural isomorphism H(P,OPn(1)) ∼= H0(C,O C ), there is an equality of Cartier divisors (e(t̄i) = 0) = d