Combinatorics of Mumford-Morita-Miller Classes in Low Genus

Gilberto Bini

Dedicated to the Memory of Grigorios TSAGAS (1935-2003), President of Balkan Society of Geometers (1997-2003)

Abstract Here we use elementary combinatorial arguments to give explicit formulae and relations for some cohomology classes of moduli spaces of stable curves of low genus.

Mathematics Subject Classification: 14H10, 14H15. Key words: moduli of curves, cohomology, stable graphs.

1 Introduction

Let g and n be non-negative integers such that 2g − 2 + n > 0. We denote by \overline{M}_{g,n} the moduli space of stable n-pointed genus g curves. Its points are in one-to-one correspondence with isomorphism classes of pointed curves of arithmetic genus g curves with simple nodes and finitely many automorphisms. \overline{M}_{g,n} is a normal projective variety of complex dimension 3g − 3 + n. It can be viewed as a complex-analytic orbifold, in fact as a quotient of a smooth complete variety by a finite group (see [3], [12]). Further details on the properties of \overline{M}_{g,n} can be found, for instance, in [7], [13]. More generally, if P is a set with n elements, we will denote by \overline{M}_{g,P} the space whose elements are stable genus g curves with marked points indexed by P.

The fascinating geometry of \overline{M}_{g,n} has been only partially understood. In many instances, combinatorial arguments have been essential to prove various results in a natural way (cfr. [6], [11]). However, little emphasis has been given to the development of the combinatorics involved with moduli of curves, especially with its cohomology.

Here we obtain an explicit description for cohomology classes (in fact, algebraic) of fundamental importance in all genera. In Section 3.2, we also describe such classes in genus zero via the theory of hyperplane arrangements.

2 Preliminaries

For any g and P, |P | = n, in the range above, the collection of all moduli spaces is naturally equipped with some relevant maps. We briefly recall their definition since they will be used in what follows: for more details see, for example, [2].

∗ Balkan Journal of Geometry and Its Applications, Vol.8, No.2, 2003, pp. 11-19.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2003.

First of all, consider the projection

π : \overline{M}_{g,P ⋃\{q\}} → \overline{M}_{g,P} , (1)

which forgets the point q on any (n + 1)-pointed curve in the domain and contracts unstable components, i.e., without finitely many automorphisms. We denote by σ_{p}, p ∈ P, the canonical section of π and by D_{p} the corresponding divisor in \overline{M}_{g,P ∪\{q\}}. The relative dualizing sheaf ω_{π} of the map (1) yields the cohomology classes:

ψ_{p} = c_{1}(σ_{p}^{∗}(ω_{π})), p ∈ P,

K = c_{1}(ω_{π}(∑_{p∈P} D_{p})).

The ψ_{p}’s are usually called universal cotangent classes. Following [1], the Mumford classes are defined to be

κ_{m} = π_{∗}(K^{m+1}). (2)

Note that the push-forward in the last formula is well defined since the Poincaré duality with rational coefficients holds for orbifolds. For P = ∅, the analogue of (2) was first introduced in [13]. Another generalization to the case of n-pointed curves is given by

κ̃_{m} = π_{∗}(c_{1}(ω_{π})^{m+1}).

As shown in [2], the following relations hold:

κ_{m} = κ̃_{m} + ∑_{p∈P} ψ_{p}^{m}, (3)

π_{∗}(ψ_{q}^{m+1}) = κ_{m}. (4)

The set of the ψ_{p}’s, κ_{m}’s, and κ̃_{m}’s is called the set of Mumford-Morita-Miller classes.

In addition to (1), further morphisms between moduli spaces of curves are defined via the collection of stable graphs: see, for example, [2] for their definition and properties. Here we just observe how to associate with them cohomology classes in H^{∗}(\overline{M}_{g,n}, Q). With the same notation adopted in [2], for any stable graph G, choose an ordering of the l(v) half-edges of G going out of each vertex v. Then consider the morphisms

ξ_{G} : ∏_{v∈V} \overline{M}_{g(v),l(v)} → \overline{M}_{g,P} , (5)

where the g(v)’s are non-negative integers which label the vertices of G. A point in the domain of ξ_{G} is the datum of an l(v)-pointed curve C_{v} for each v in the set of vertices V of G. The image point is the P -labelled genus g curve that is obtained by identifying the marked points of C_{v} which correspond to half-edges of G linked by an edge. By definition, the map ξ_{G} does not depend on the ordering chosen for the half-edges going out of each vertex. By properties of stable graphs, we have

g = ∑_{v∈V} g(v) + 1 − |V | + \frac{1}{2} ∑_{v∈V}(l(v) − |P |). (6)

For the purpose of what follows, we finally recall that for any stable graph G with at least one edge the corresponding boundary class is given by

\frac{1}{|Aut(G)|}ξ_{G,∗}(1),

where Aut(G) is the automorphism group of G. These classes are usually referred to as boundary classes since their Poincaré dual is supported on \overline{M}_{g,P} − M_{g,P}, i.e., the boundary of the moduli space of smooth P -labelled curves.

3 Relations among Mumford-Morita-Miller classes

In this section we show some explicit formulae for the classes ψ_{p}’s, κ_{m}’s in terms of boundary classes. We basically extend previous work in [8] and [9].

3.1 The genus zero case

The structure of the cohomology ring of \overline{M}_{0,n} has been determined in [10]. For each subset S⊆ {1, ..., n}, with |S| ≥ 2 and |S^{c}| ≥ 2, let δ_{0,S} be the cohomology class dual to the divisor of genus 0 curves with one node and two components with |S| and |S^{c}| marked points, respectively. The cohomology ring of \overline{M}_{0,n} is a quotient of the free Z-module

Z[δ_{0,S} : S ⊆ {1, ..., n}, |S|, |S^{c}| ≥ 2].

This means that any class can be written as a linear combination of monomials in the classes δ_{0,S}’s. Here we show how to use combinatorial arguments to obtain explicit expressions for Mumford-Morita-Miller classes in any codimension.

Take P to be a set with n-elements and consider the moduli space \overline{M}_{0,P}. For the ψ_{p}’s we recall that (see [2])

ψ_{p} = ∑_{{q_{1},q_{2}∉S, \atop p∈S}} δ_{0,S}, (7)

where p, q_{1}, q_{2} are arbitrary elements in P.

Let A_{m}(n) denote the collection of unordered m-ples (A_{1}, ..., A_{m}) of subsets A_{j} ⊂ P such that the following conditions are satisfied:

• for each A_{k} in (A_{1}, ..., A_{m}), |A_{k}| ≥ 2, |A_{k}^{c}| ≥ 2 and p, q ∉ A_{k} for any pair p, q ∈ P;

• for each k∈ {1, ..., m} and each choice of p, q ∈ P, A_{k} is not contained in any subset S of P \{p, q}, with |S| = m;

• for each pair A_{k}, A_{l} of an m-ple, one of the following conditions

A_{k} ⊂ A_{l}, A_{l} ⊂ A_{k}, A_{k} ⊂ A_{l}^{c}, A_{l}^{c} ⊂ A_{k}

is satisfied.

For each such m-ple we define the following coefficient

c(A_{1}, ..., A_{m}) = \{ \begin{matrix}|A_{1} ∩ ... ∩ A_{m}| + (−1)^{m}, & A_{k} ⊆ P \ S, \\ |A_{1} ∩ ... ∩ A_{m}| & otherwise. \end{matrix}

Theorem 1 In H^{∗}(\overline{M}_{0,n}; Q), for each m, 1 ≤ m ≤ n − 3, we have i)

ψ_{p}^{m} = ∑_{{p∈A_{i}, q_{1},q_{2}∉A_{i} \atop A_{1},...,Am}} δ_{0,A_{1}} … δ_{0,A_{m}},

ii)

κ_{m} = ∑_{(A_{1},...,A_{m})∈A_{m}(n)} c(A_{1}, ..., A_{m})δ_{0,A_{1}}...δ_{0,A_{m}}.

Proof. i) The claim follows clearly from (7).

ii) Let π_{s} : \overline{M}_{0,P ∪\{s\}} → \overline{M}_{0,P} be the forgetful map. We prove the result by induction on |P | = n. The base of the induction follows from the fact that κ_{m} = 0 on \overline{M}_{0,m+2} by dimensional calculations. Thus, we can assume n ≥ m + 2. Since (see [2])

κ_{m} = π_{s}^{∗}(κ_{m}) + ψ_{q}^{m},

by induction hypothesis and the relation which expresses ψ_{q} in terms of boundary classes (see [1]),

κ_{m} = ∑_{(A_{1},...,A_{m})∈A_{m}(n)} c(A_{1}, ..., A_{m})π_{q}^{∗}(δ_{0,A_{1}}...δ_{0,A_{m}}) + ( ∑_{{s∈B, \atop p,q∉B}} δ_{0,B})^{m} =

∑_{(A_{1},...,A_{m})∈A_{m}(n)} c(A_{1}, ..., A_{m})π_{q}^{∗}(δ_{0,A_{1}}...δ_{0,A_{m}}) + ∑_{{A_{1},...,Am, \atop p,q∉A_{k},s∈A_{k}}} δ_{0,A_{1}}...δ_{0,A_{m}}. (8)

The claim is proved if we show that the sum in (8) can be rewritten as

∑_{(A′_{1},...,A′_{m})∈A_{m}(n+1)} c(A′_{1}, ..., A′_{m})δ_{0,A′_{1}}...δ_{0,A′_{m}}.

In fact, since (see [2])

π_{s}^{∗}(δ_{0,A_{l}}) = δ_{0,A_{l}} + δ_{0,A_{l}∪{s}},

we just distinguish two cases:

1. s ∈ A′_{k}, ∀k ∈ {1, ..., m};

2. s ∉ A′_{1} ∩ ... ∩ A′_{m}.

In case 1 we can assume that A′_{k}= A_{k} ∪ {s}. Therefore, by direct computations we have

c(A′_{1}, ..., A′_{m}) = c(A_{1}, ..., A_{m}) + 1 = |A_{1} ∩ ... ∩ A_{m}| + (−1)^{m} + 1
= |A′_{1} ∩ ... ∩ A′_{m}| + 1,

or

c(A′_{1}, ..., A′_{m}) = c(A_{1}, ..., A_{m}) + 1 = |A_{1} ∩ ... ∩ A_{m}| + 1 = |A′_{1} ∩ ... ∩ A′_{m}|.

In case 2), we have c(A′_{1}, ..., A′_{m}) = c(A_{1}, ..., A_{m}) since there is no contribution to the new coefficient coming from the expansion of ψ_{q}^{m} in terms of boundary classes; hence the result follows.

□

3.1.1 An alternative description via hyperplane arrangements

Moduli space of pointed genus 0 curves can be constructed in terms of the De Concini-Procesi models, i.e., via the theory of hyperplane arrangements. We omit their definition since it is rather technical: for a detailed presentation see [4] and references therein. Here we briefly recall the relationship between \overline{M}_{0,n} and these models so to express the classes ψ_{i} and κ_{i}, 0 ≤ i ≤ n, in terms of the combinatorics of the corresponding hyperplane arrangement.

The moduli space M_{0,n+1} can be viewed as the quotient of the set

{(p_{0}, ..., p_{n}) ∈ P^{1} × ... × P^{1} : p_{i} ≠ p_{j}, ∀i ≠ j}

modulo the group P GL(2, C) which acts componentwise - in the sequel, we consider the marked points with indices in the set {0, 1, ..., n}. Note that this action identifies the moduli space of (n + 1)-pointed rational curves with the set

{(q_{1}, ..., q_{n−2}) ∈ P^{1} × ... × P^{1} : q_{i} ≠ q_{j}, q_{i} ≠ 1, 0, ∞} . (9)

Let us now consider C^{n} with the standard scalar product denoted by (., .) and the hyperplane arrangement given by the hyperplanes z_{ij} : x_{i} − x_{j} = 0, where x_{i} ∈ (C^{n})^{∗} are the coordinate functions. Moreover, be N the intersection of all the hyperplanes and π : C^{n} → C^{n}/N := V the projection onto the quotient. In [4], and with the same notation adopted there, the set in (3.1.1) is identified with the complement of the projective arrangement \overline{A}_{n−1} := ∪_{h,k=1,...,n}H_{hk}, where H_{hk} = Ψ(π(z_{hk})), with Ψ the projectivization map from V to P(V).

This description allows constructing a De Concini-Procesi model which is denoted by \overline{Y}_{F_{A_{n−1}}} in the literature. With the same notation of [5], the cohomology of such a model is generated over the integers by cohomology classes c_{A}, where A ranges over the subsets of \{0, 1, …, n+1\}. Furthermore, as proved in [4], \overline{Y}_{F_{A_{n−1}}} is isomorphic to \overline{M}_{0,n+1}. Let Φ be this isomorphism and denote by Φ^{∗} the map induced in cohomology. Then

Φ^{∗}(δ_{0,A}) = c_{A},

for A⊂ {1, ..., n}, and

−Φ^{∗}( ∑_{{i,j}⊂A⊂{1,...,n}} δ_{0,A}) = c_{{1,...,n}}

for every {i, j} ⊂ {1, ..., n}, i ≠ j. Although this definiton may seem rather strange, it is the natural correspondence between the second cohomology groups of \overline{M}_{0,n+1} and \overline{Y}_{F_{A_{n−1}}} .

We can now describe the image of the classes ψ_{i}’s and κ_{i}’s under the map Φ^{∗}. To this end, let us denote by τ_{j} the transposition of S_{n+1} given by the exchange of the two marked points with indices 0 and j. We recall that there is a natural action of S_{n+1} on \overline{M}_{0,n+1} given as follows:

τ_{j} · [C; p_{0}, p_{1}, ..., p_{j}, ..., p_{n}] = [C; p_{j}, p_{1}, ..., p_{0}, …, p_{n}].

Proposition 2 Let z ∈ {0, 1, ..., n} and j ∈ {1, ..., n}. Then

Φ^{∗}(ψ_{z}) = τ_{z} ( ∑_{{j,z}⊂B⊂{1,...,n}} c_{{1,...,n}\B}) ,

with 1 ≤ |B| ≤ n − 2.

Proof. Since

τ_{z} (  ∑_{{j,z}⊂B⊂{1,...,n}} c_{{1,...,n}\B}) = ∑_{j∉A⊂{1,...,n},z∈A} c_{A},

the result follows from Proposition 1.6 in [2].

□

By Theorem 1, and with the same notation, we immediately get the following

Proposition 3 For each m ≥ 0,

Φ^{∗}(κ_{m}) = ∑_{(A_{1},...,A_{m})∈A_{m}(n)} c(A_{1}, ..., A_{m})c_{A_{1}} … c_{A_{2}}.

3.2 The genus one case

In this section we give expressions for the Mumford classes and the powers of the universal cotangent classes in genus one. To this end, let us consider the stable graphs G_{1} and G_{S} (S ⊂ P, |S| ≥ 2)

As defined in (5), the morphisms associated with the graphs above will be denoted by

ξ_{G_{1}} : \overline{M}_{0,P ∪\{q_{1},q_{2}\}} → \overline{M}_{1,P}

and

ξ_{G_{S}} : \overline{M}_{0,S∪\{r_{1}\}} × \overline{M}_{1,S^{c}∪\{r_{2}\}} → \overline{M}_{1,P} .

As shown in [2], we have

ψ_{q} = \frac{1}{24}ξ_{irr,∗}(1) + ∑_{q∈S,|S|≥2} ξ_{S,∗}(1), ∀p ∈ P.

In order to give relations in genus one we need some additional notation. If t is a stable graph such that g(v) = 0 for each vertex v and the half-edges are labelled by P ∪ {a, b}, then we denote by L(t) the stable graph obtained from t by identifying the half-edges of t labelled with a and b. Moreover, for a stable graph such that g(v) = 0 for each vertex v and the half-edges are labelled by P ∪ {q_{1}}, we consider the stable graph S(t) given as follows. Fix a subset S of P such that |S| ≥ 2. For such an S, substitute in t the half-edge with label q_{1} with an edge that ends with a vertex v (g(v) = 1 ) and with half-edges labelled by the elements of the set S^{c}, the complement of S in P. Then

Theorem 4 i)

ψ_{q}^{m+1} = ∑_{{S_{1},...,Sm, \atop |S_{i}|≥2}} ξ_{S_{1},∗}(1) … ξ_{S_{m},∗}(1) + \frac{m}{24}ξ_{irr,∗} ∑_{{S_{1},...,S_{m−1}, \atop |S_{i}|≥2}} ξ_{S_{1},∗}(1) … ξ_{S_{m−1},∗}(1);

ii)

\begin{aligned}κ_{m} & = & \frac{1}{24} ∑_{t∈G_{m−1,n+2}} C(m − 1; a_{1}, …, a_{m}, n + 2)ξ_{L(t),∗}(1) \\ & + & ∑_{S,|S|≥2} ∑_{t∈G_{m−1,|S|+1}} C(m − 1; a_{1}, …, a_{m}, |S| + 1)ξ_{S(t),∗}(1), \end{aligned}

where

C(m; a_{1}, …, a_{m+1}, n) = \frac{(m + 1)!(a_{1} − 1) … (a_{m+1} − 1)a_{1} … a_{m+1}}{n(n − 1)a_{1}(a_{1} + a_{2}) … (a_{1} + … + a_{m})} .

Proof. i) By dimension computation, the Poincaré dual of the class ξ_{irr,∗}(1) is a point in \overline{M}_{1,1}. Therefore, by pull-back under the map π : \overline{M}_{1,n} → \overline{M}_{1,1} which forgets all marked points but the last one, we have

(ξ_{irr,∗}(1))^{2} = 0

on \overline{M}_{1,n} for any n. This completes the proof.

ii) As proved in [8], the following recursive relation holds:

κ_{m} = \frac{1}{24}ξ_{irr,∗}(κ_{m−1}) + ∑_{S,|S|≥2} ξ_{S,∗}(κ_{m−1} ⊗ 1). (10)

Let us denote by G_{m,n} the collection of P -labelled stable graphs t (|P | = n) such that g(v) = 0 for each vertex v and with an ordering of the set of vertices. Additionally, every graph in G_{m,n} has m edges - consequently m + 1 vertices - and each vertex has at most two incident edges. As proved in [9],

κ_{m} = ∑_{t∈G_{m,n}} C(m; a_{1}, …, a_{m+1}, n)ξ_{t,∗}(1), (11)

where a_{i} denotes the number of half-edges going out of the i-th vertex of t (1 ≤ i ≤ m + 1), and ξt is the map corresponding to t as defined in (5).

By combining (10) and (11), we get

(12) κ_{m} = \frac{1}{24} ∑_{t∈G_{m−1,n+2}} C(m − 1; a_{1}, …, a_{m}, n + 2)ξ_{irr,∗} (ξ_{t,∗}(1))

+ ∑_{S,|S|≥2} ∑_{t∈G_{m−1,|S|+1}} C(m − 1; a_{1}, …, a_{m}, |S| + 1)ξ_{S(t),∗}(ξ_{t,∗} ⊗ 1).

By definition of ξ_{irr} and ξ_{S}, the claim follows.

□

Acknowledgments. I wish to express my gratitude to Enrico Arbarello, Giovanni Gaiffi and Marzia Polito for stimulating conversations and useful remarks.

The author is member of Eager, GNSAGA-INdAM.

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Korteweg-de Vries Instituut voor Wiskunde Plantage Muidergracht 24 1018 TV Amsterdam (NL) email:gbini@science.uva.nl