2.1 Boundary gluing maps

A prestable graph $\Gamma $ of genus g with $n $ markings consists of the data

$$ \begin{align*}\Gamma=(V,\, H,\, \ell: L \to \{1, \ldots, n\} , \ g \colon V \to \mathbb{Z}_{\geq 0}\, , \ v\colon H \to V\, , \ \iota : H\to H)\end{align*} $$

satisfying the properties:

1. (i) V is a vertex set with a genus function $g\colon V\to \mathbb {Z}_{\geq 0}$ ,

2. (ii) H is a half-edge set equipped with a vertex assignment $v:H \to V$ and an involution $\iota $ ,

3. (iii) E, the edge set, is defined by the 2-cycles of $\iota $ in H (self-edges at vertices are permitted),

4. (iv) $L \subseteq H$ , the set of legs, is defined by the fixed points of $\iota $ and corresponds to n markings via the bijection $\ell : L \to \{1, \ldots , n\}$ ,

5. (v) the pair $(V,E)$ defines a connected graph satisfying the genus condition

   $$ \begin{align*}\sum_{v \in V} g(v) + h^1(\Gamma) = g,\end{align*} $$
   where $h^1(\Gamma ) = |E| - |V| + 1$ is the first Betti number of the graph $\Gamma $ .

A prestable graph $\Gamma $ is called stable if the following additional condition is satisfied:

1. (vi) for each vertex $w\in V$ , we have

   $$\begin{align*}2g(w)-2+n(w)>0,\end{align*}$$
   where $n(w) = |v^{-1}(w)|$ is the valence of w in $\Gamma $ : that is, the number of half-edges incident to w.

Given a second graph $\Gamma ' = (V',H', \ell ',g',v',\iota ')$ , an isomorphism $\varphi : \Gamma \to \Gamma '$ is the data of bijective maps

$$\begin{align*}\varphi_V : V \to V',\hspace{2mm} \varphi_H : H \to H'\end{align*}$$

that are compatible with the remaining data of the prestable graphs, in the sense that

$$\begin{align*}\ell' \circ \varphi_H|_{L} = \ell, \, g' \circ \varphi_V = g, \, v' \circ \varphi_H = \varphi_V \circ v, \, \iota' \circ \varphi_H = \varphi_H \circ \iota. \end{align*}$$

For every vertex $v \in V(\Gamma )$ , let $H(v)$ be the set of half-edges at v, with cardinality $n(v)$ . Then there exists a natural gluing morphism

$$ \begin{align*} \xi_{\Gamma} : \mathfrak{M}_{\Gamma}=\prod_{v \in V(\Gamma)} \mathfrak{M}_{g(v), n(v)} &\to \mathfrak{M}_{g,n}, \end{align*} $$

which assigns to a collection $((C_v, (p_h)_{h \in H(v)})$ the curve $(C,p_1, \ldots , p_n)$ obtained by identifying the markings $p_h, p_{h'}$ for each pair $(h,h')$ forming an edge of $\Gamma $ .Footnote ⁷ Restricting to the preimage of the open substack $\overline {\mathcal {M}}_{g,n} \subset \mathfrak {M}_{g,n}$ , we get the usual gluing maps

$$ \begin{align*} \xi_{\Gamma} : \overline{\mathcal{M}}_{\Gamma}=\prod_{v \in V(\Gamma)} \overline{\mathcal{M}}_{g(v), n(v)} &\to \overline{\mathcal{M}}_{g,n}. \end{align*} $$

Note that unless $\Gamma $ is stable, the left-hand side is empty.

On the other hand, given $m \geq 0$ , we have the forgetful morphism

$$ \begin{align*} F_m : \mathfrak{M}_{g,n+m} \to \mathfrak{M}_{g,n}, (C,p_1, \ldots, p_n, q_1, \ldots, q_m) \mapsto (C,p_1, \ldots, p_n). \end{align*} $$

Since the curve C remains prestable after forgetting a subset of the markings, there is no stabilization procedure in the morphism $F_m$ , and the underlying curve remains unchanged.

Lemma 2.1. The morphism $F_m$ is smooth and representable of relative dimension m, and the collection

$$ \begin{align*} \left(F_m|_{\overline{\mathcal{M}}_{g,n+m}} : \overline{\mathcal{M}}_{g,n+m} \to \mathfrak{M}_{g,n}\right)_{m \in \mathbb{Z}_{\geq 0}} \end{align*} $$

forms a smooth and representable cover of $\mathfrak {M}_{g,n}$ . The complement of the image of $\overline {\mathcal {M}}_{g,n+m}$ under $F_m$ in $\mathfrak {M}_{g,n}$ has codimension $\lfloor \frac {m}{2} \rfloor +1$ , except for finitely many m in the unstable setting $2g-2+n \leq 0$ .Footnote ⁸

Let $\mathrm {st}_m(\Gamma )$ be the set of stable graphs $\Gamma '$ in genus g with $n+m$ markings obtained from a prestable graph $\Gamma $ of genus g with n legs by adding m additional legs, labeled $n+1, \ldots , n+m$ , at vertices of $\Gamma $ . As explained above, for a fixed prestable graph $\Gamma $ , the set $\mathrm {st}_m(\Gamma )$ starts being nonempty for m sufficiently large.

Given $\Gamma ' \in \mathrm {st}_m(\Gamma )$ , there is a natural map

$$\begin{align*}F_{\Gamma' \to \Gamma} : \overline{\mathcal{M}}_{\Gamma'} \to \mathfrak{M}_{\Gamma}\end{align*}$$

that is just the product of forgetful maps $F_{m_v} : \overline {\mathcal {M}}_{g(v),n(v)+m_v} \to \mathfrak {M}_{g(v),n(v)}$ for each $v \in V(\Gamma )=V(\Gamma ')$ .

Lemma 2.2. For every prestable graph $\Gamma $ in genus g with n markings and every $m \geq 0$ , there is a fibre diagram

(5)

In particular, the map $\xi _{\Gamma }: \mathfrak {M}_{\Gamma } \to \mathfrak {M}_{g,n}$ is representable, proper and a local complete intersection.

Proof. An object of the fibre product of $\mathfrak {M}_{\Gamma }$ with $\overline {\mathcal {M}}_{g,n+m}$ over a (connected) scheme S is given by

• ○ a collection of families $(C_v, (p_h)_{h \in H(v)})$ of prestable curves over S for each $v \in V(\Gamma )$ ,

• ○ a family $(C',p^{\prime }_1, \ldots , p^{\prime }_n, q^{\prime }_1, \ldots , q^{\prime }_m)$ of stable curves over S,

• ○ an isomorphism (of families of prestable curves)

  $$\begin{align*}\varphi: C=\coprod_v C_v / (p_h \sim p_{h'}, (h,h') \in E(\Gamma)) \to C' \end{align*}$$
  satisfying $\varphi (p_i)=p_i'$ .

By the assumption that S is connected, for each $j=1, \ldots , m$ , there exists a unique $v=v(j) \in V(\Gamma )$ such that $q^{\prime }_j \in \varphi (C_v)$ at each point of S. This uses that via $\varphi $ , the smooth unmarked points of $C_v$ ( $v \in V(\Gamma )$ ) form a disjoint open cover of the smooth unmarked points of $(C',p_1, \ldots , p_n)$ in which $q^{\prime }_j$ is always contained.

But for $j,v$ as above, we obtain a section $q_j=\varphi ^{-1} \circ q_j' : S \to C_v$ landing in the smooth unmarked locus of $C_v$ . Thus for every $v \in V(\Gamma )$ , this allows us to define a family

(6) $$ \begin{align} \widehat{C}_v=(C_v, (p_h)_{h \in H(v)}, (q_j)_{v(j)=v}) \to S \end{align} $$

of prestable curves over S. From the fact that $(C',p^{\prime }_1, \ldots , p^{\prime }_n,q^{\prime }_1, \ldots , q^{\prime }_m)$ is a family of stable curves, it follows that the family in equation (6) is actually a family of stable curves. Then one sees that the collection $({\widehat {C}}_v)_{v \in V(\Gamma ')}$ is exactly an S-point of one of the spaces $\overline {\mathcal {M}}_{\Gamma '}$ for the suitable $\Gamma ' \in \mathrm {st}_m(\Gamma )$ for which the marking $q_j$ is added at the vertex $v(j) \in V(\Gamma ')=V(\Gamma )$ .

The above operations define a map from $\mathfrak {M}_{\Gamma } \times _{\mathfrak {M}_{g,n}} \overline {\mathcal {M}}_{g,n+m}$ to the disjoint union of the $\overline {\mathcal {M}}_{\Gamma '}$ and clearly this disjoint union also maps to the fibre product using the maps $F_{\Gamma ' \to \Gamma }$ and $\xi _{\Gamma '}$ . One verifies that these are inverse isomorphisms.

Since being proper and being a local complete intersection is local on the target, and since the maps $F_m$ form a smooth cover of $\mathfrak {M}_{g,n}$ , these properties of $\xi _{\Gamma }$ follow from the corresponding properties of the maps $\xi _{\Gamma '}|_{\overline {\mathcal {M}}_{\Gamma '}}$ .

Later we will need some stronger statements about the locus of curves whose prestable graph is exactly a given graph $\Gamma $ . This locus is a locally closed substack $\mathfrak {M}^{\Gamma }$ of $\mathfrak {M}_{g,n}$ whose geometric points are precisely the curves $(C,p_1, \ldots , p_n)$ with prestable graph isomorphic to $\Gamma $ . However, since a family of prestable curves over an arbitrary base does not in general have a well-defined prestable graph, this definition is slightly tricky to write down in a functorial way. Thus we approach the definition from a different angle and then show that it defines the desired locus.

Definition 2.3. Let $\Gamma $ be a prestable graph in genus g with n markings, and let e be the number of edges of $\Gamma $ . Then we define

$$\begin{align*}\mathfrak{M}^{\Gamma} = \mathrm{im}(\xi_{\Gamma}) \setminus \bigcup_{\Gamma': |E(\Gamma')|=e+1} \mathrm{im}(\xi_{\Gamma'}),\end{align*}$$

where $\mathrm {im}$ denotes the stack theoretic image and the union goes over prestable graphs $\Gamma '$ with precisely $e+1$ edges.

By definition, we have that $\mathfrak {M}^{\Gamma }$ is a locally closed substack of $\mathfrak {M}_{g,n}$ . In the following lemma, we check that its geometric points are as desired.

We have the following neat description of $\mathfrak {M}^{\Gamma }$ , which is a generalization of [Reference FulghesuFul10b, Lemma 5.1]. For the statement, let

$$ \begin{align*} \mathfrak{M}_{g,n}^{\mathrm{sm}} \subset \mathfrak{M}_{g,n} \end{align*} $$

be the open substack where the curve C is smooth. For $g,n$ in the stable range, this is the usual stack $\mathcal {M}_{g,n}$ of smooth curves, but since the latter might be defined to be empty for $2g-2+n<0$ , we use the notation $\mathfrak {M}_{g,n}^{\mathrm {sm}}$ for clarity.

Proposition 2.5. For a prestable graph $\Gamma $ , consider the open substack

$$\begin{align*}\mathfrak{M}_{\Gamma}^{\mathrm{sm}} = \prod_{v \in V(\Gamma)} \mathfrak{M}_{g(v),n(v)}^{\mathrm{sm}} \subset \mathfrak{M}_{\Gamma}. \end{align*}$$

Then the restriction of the gluing map $\xi _{\Gamma }$ to $\mathfrak {M}_{\Gamma }^{\mathrm {sm}}$ factors through $\mathfrak {M}^{\Gamma }$ , and it is invariant under the natural action of $\mathrm {Aut}(\Gamma )$ . The induced map

(7) $$ \begin{align} \mathfrak{M}_{\Gamma}^{\mathrm{sm}} / \mathrm{Aut}(\Gamma) \xrightarrow{\xi_{\Gamma}} \mathfrak{M}^{\Gamma} \end{align} $$

from the quotient stackFootnote ⁹ of $\mathfrak {M}_{\Gamma }$ by $\mathrm {Aut}(\Gamma )$ is an isomorphism.

Proof. For each point $(C_v)_v = (C_v,(p_h)_{h\in H(v)})_{v \in V(\Gamma )} \in \mathfrak {M}_{\Gamma }^{\mathrm {sm}}$ , the stabilizer $\mathrm {Aut}(\Gamma )_{(C_v)_v}$ under the action of $\mathrm {Aut}(\Gamma )$ is the set of automorphisms of $\Gamma $ such that there exist compatible isomorphisms of the curves $(C_v,(p_h)_{h\in H(v)})$ . The stabilizer group of $[(C_v)_v] \in \mathfrak {M}_{\Gamma }^{\mathrm {sm}} / \mathrm {Aut}(\Gamma )$ is then an extension of the product of the automorphism groups of the $(C_v,(p_h)_{h\in H(v)})$ by the group $\mathrm {Aut}(\Gamma )_{(C_v)_v}$ .

On the other hand, for the curve $(C,p_1,\ldots ,p_n)$ obtained from $(C_v)_v$ by gluing and an element $\sigma \in \mathrm {Aut}(\Gamma )_{(C_v)_v}$ , the isomorphisms between the curves $C_v$ that are compatible with $\sigma $ can be glued to an automorphism of $(C,p_1,\ldots ,p_n)$ . From this it follows that there exists an exact sequence

$$\begin{align*}1\to\prod_{v\in V(\Gamma)} \mathrm{Aut}(C_v,(p_h)_{h\in H(v)}) \to\mathrm{Aut}(C,p_1,\ldots,p_n)\to \mathrm{Aut}(\Gamma)_{(C_v)_v} \to 1.\end{align*}$$

From this sequence we see that $\mathrm {Aut}(C,p_1,\ldots ,p_n)$ is precisely the group extension defining the stabilizer of $[(C_v)_v] \in \mathfrak {M}_{\Gamma }^{\mathrm {sm}} / \mathrm {Aut}(\Gamma )$ , and hence $\xi _{\Gamma }$ induces an isomorphism of each stabilizer. Thus the morphism $\xi _{\Gamma }$ in equation (7) is representable. It is easy to check that it is bijective on geometric points and it is separated by similar argument as in Lemma 2.2. So by [Sta20, 0DUD] it is enough to show that $\xi _{\Gamma }$ is an étale morphism to conclude that it is an isomorphism.

Consider the atlas $F_m$ restricted to $\mathfrak {M}^{\Gamma }$ . Since being étale is local on the target, it is enough to show that $\xi _{\Gamma }$ is étale on each atlas. On each atlas, the dimension of the fibre is constantly zero. The domain of $\xi _{\Gamma }$ is smooth because it can be written as a quotient of a smooth algebraic space by a group scheme ([Sta20, 0DLS]). Following a slight variation of the proof of [Reference Arbarello, Cornalba and GriffithsACG11, Proposition 10.11], the stack $\mathfrak {M}^{\Gamma }$ is also smooth. Since the domain and the target of $\xi _{\Gamma }$ are smooth, the ‘miracle flatness’ ([Sta20, 00R3]) implies that $\xi _{\Gamma }$ is flat. Furthermore, the morphism is smooth because it is flat, and each geometric fibre is smooth. Smooth and quasi-finite morphisms are étale, and hence $\xi _{\Gamma }$ is an isomorphism.

2.2 $\mathcal {A}$ -valued prestable curves

For each $g,n$ , there exists the universal curve $\mathfrak {C}_{g,n} \to \mathfrak {M}_{g,n}$ . For later applications, it will be necessary to compute with tautological classes on $\mathfrak {C}_{g,n}$ (and tautological classes on the universal curve over $\mathfrak {C}_{g,n}$ , etc.). For the moduli spaces of stable curves, a separate theory is not necessary because the universal curve over $\overline {\mathcal {M}}_{g,n}$ is given by the forgetful map $\overline {\mathcal {M}}_{g,n+1} \to \overline {\mathcal {M}}_{g,n}$ . The same is not true for $\mathfrak {M}_{g,n}$ . Indeed, in Lemma 2.1, we saw that the forgetful morphism $\mathfrak {M}_{g,n+1} \to \mathfrak {M}_{g,n}$ is smooth, so it cannot be the universal curve over $\mathfrak {M}_{g,n}$ . In this section, we put an additional structure on prestable curves, called the $\mathcal {A}$ -value, which allows us to give a modular interpretation of the universal curve as a stack of $(n+1)$ -pointed curves with additional structure. This realization will be convenient to compute tautological classes on $\mathfrak {C}_{g,n}$ .

So let us start by recalling the notion of prestable curves with values in a semigroup $\mathcal {A}$ from [Reference CostelloCos06]. In what follows, let $\mathcal {A}$ be a commutative semigroup with unit

such that

• ○ $\mathcal {A}$ has indecomposable zero: that is, for $x,y \in \mathcal {A}$ , we have implies , ,

• ○ $\mathcal {A}$ has finite decomposition: that is, for $a \in \mathcal {A}$ the set

  $$ \begin{align*}\{(a_1,a_2) \in \mathcal{A} \times \mathcal{A} : a_1 + a_2 =a\}\end{align*} $$
  is finite.

Classical examples include

or $\mathcal {A}=\mathbb {N}$ , but later we are going to work with

Fixing $\mathcal {A}$ and an element $a \in \mathcal {A}$ , Behrend-Manin [Reference Behrend and ManinBM96] and Costello [Reference CostelloCos06] define an algebraic stack $\mathfrak {M}_{g,n,a}$ . A geometric point corresponds to a prestable curve $(C,p_1, \ldots , p_n)$ together with a map $C_v \mapsto a_{C_v}$ from the set of irreducible components $C_v$ of the normalization of C to $\mathcal {A}$ such that the sum of all $a_{C_v}$ equals a. The curve must satisfy the stability condition that for each $C_v$ either or that $C_v$ is stable, in the sense that for $g(C_v)=0$ it carries three special points and for $g(C_v)=1$ it carries at least one special point. Over an arbitrary base scheme, the definition of $\mathcal {A}$ -valued stable curves needs extra care; see [Reference CostelloCos06, p.569] for details. As an example, for any $\mathcal {A}$ as above and , we obtain .

Our main motivation for considering the moduli spaces $\mathfrak {M}_{g,n,a}$ is the fact that we have a forgetful morphism $\pi : \mathfrak {M}_{g,n+1,a} \to \mathfrak {M}_{g,n,a}$ making $\mathfrak {M}_{g,n+1,a}$ the universal curve over $\mathfrak {M}_{g,n,a}$ . The image of a point

$$\begin{align*}(C, p_1, \ldots, p_n, p_{n+1}, (a_{C_v})_v) \in \mathfrak{M}_{g,n+1,a}\end{align*}$$

under $\pi $ is formed by first forgetting the marked point $p_{n+1}$ . Then if the component $C_v$ of C containing $p_{n+1}$ becomes unstable,Footnote ¹⁰ the component $C_v$ of C is contracted. With this notation in place, we summarize the relevant properties of $\mathfrak {M}_{g,n,a}$ from [Reference CostelloCos06].

The fact that the universal curve is given by a moduli space of curves with an extra marked point turns out to be very convenient. Indeed, as discussed above, this is not the case for the forgetful morphism $\mathfrak {M}_{g,n+1} \to \mathfrak {M}_{g,n}$ . It is easy to identify $\mathfrak {M}_{g,n+1}$ as the open substack $\mathfrak {M}_{g,n+1} \subset \mathfrak {C}_{g,n}$ given as the complement of the set of markings and nodes.

Many other constructions we saw for prestable curves work in the $\mathcal {A}$ -valued setting. For instance, for $g_1+g_2=g$ , $n_1+n_2=n$ and $a_1, a_2 \in \mathcal {A}$ with $a_1 + a_2 = a$ , we have a gluing morphism

$$\begin{align*}\xi : \mathfrak{M}_{g_1,n_1+1,a_1} \times \mathfrak{M}_{g_2,n_2+1,a_2} \to \mathfrak{M}_{g,n,a}.\end{align*}$$

These gluing maps are again representable, proper and local complete intersections. Indeed, we have a fibre diagram

and the map at the bottom has all these properties by Lemma 2.2. More generally, one defines the notion of an $\mathcal {A}$ -valued stable graph, and the corresponding gluing map has all the desired properties.

The following result allows us to apply the machinery of Costello to the moduli spaces of prestable curves.

Proposition 2.7. Let

with

; then given $g,n$ , the subset

of $\mathcal {A}$ -valued curves $(C,p_1, \ldots , p_n; (a_{C_v})_v)$ such that one of the values $a_{C_v}$ equals

is closed. Let

be its complement. Then the composition

of the inclusion of $\mathfrak U_{g,n}$ with the morphism

forgetting the $\mathcal {A}$ -values defines an isomorphism $\mathfrak {U}_{g,n} \cong \mathfrak {M}_{g,n}$ .

Proof. The underlying reason why $\mathfrak {Z}_{g,n}$ is closed is that is indecomposable in $\mathcal {A}$ : given a curve $(C,p_1, \ldots , p_n; (a_{C_v})_v)$ such that some , any degeneration of this curve still has some component with value since in a degeneration of $C_v$ , $a_{C_v}$ must distribute to the components to which $C_v$ degenerates.

More concretely, we can write $\mathfrak {Z}_{g,n}$ as the union of images of gluing maps $\xi _{\Gamma }$ for suitable $\mathcal {A}$ -valued prestable graphs $\Gamma $ . Indeed, we exactly have to remove the images of $\xi _{\Gamma }$ for $\Gamma $ of the form

(8)

where $s \geq 1$ , $e_1, \ldots , e_s \in \mathbb {Z}_{>0}$ ,

$$\begin{align*}g_0+g_1+\ldots+g_s=g+\sum_{i=1}^s e_i-1\end{align*}$$

and $I_0 \coprod I_1 \coprod \ldots \coprod I_s = \{1, \ldots , n\}$ . Note that for this locus to be nonempty, we must require $g_0>0$ or $|I_0|+\sum _i e_i>2$ .

While the image of each $\xi _{\Gamma }$ is closed, we use infinitely many of them. But in the open exhaustion of by the substacks of curves with at most $\ell $ nodes, each of these open substacks only intersects finitely many of the images of $\xi _{\Gamma }$ nontrivially, so the union of their images is still closed.

The fact that $\mathfrak U_{g,n} \to \mathfrak {M}_{g,n}$ is an isomorphism can be seen in different ways: its inverse is just given by the functor sending each prestable curve $(C,p_1, \ldots , p_n)$ to itself with value

on each component: that is,

Alternatively, one observes that $\mathfrak U_{g,n} \to \mathfrak {M}_{g,n}$ is étale, representable and a bijection on geometric points.

Corollary 2.8. The universal curve $\mathfrak {C}_{g,n} \to \mathfrak {M}_{g,n}$ is given by the morphism

forgetting the marking $n+1$ and contracting the component containing it if this component becomes unstable. The $\mathcal {A}$ -valued prestable graphs $\Gamma $ appearing in $\mathfrak {Z}_{g,n+1}$ but not contained in $\pi ^{-1}(\mathfrak Z_{g,n})$ are exactly of one of the three following forms:

• ○ for $i=1, \ldots , n$ , the graphs

  (9)

  
  corresponding to the n sections of the universal curve $\pi : \mathfrak {C}_{g,n} \to \mathfrak {M}_{g,n}$ ,

• ○ boundary divisors with edge subdivided, inserting a genus zero, value vertex carrying $n+1$

  
  where $g_1+g_2=g$ and
  
  corresponding to the locus of nodes inside the universal curve $\pi : \mathfrak {C}_{g,n} \to \mathfrak {M}_{g,n}$ .

Corollary 2.8 shows that in order to develop the intersection theory of $\mathfrak {M}_{g,n}$ and $\mathfrak {C}_{g,n}$ , it suffices to consider the general case of the intersection theory of (or even more generally, $\mathfrak {M}_{g,n,a}$ for any semigroup $\mathcal {A}$ and $a \in \mathcal {A}$ ).

3.1 Definitions

In this paper, we want to study the Chow groups (with $\mathbb {Q}$ -coefficients) of the stacks $\mathfrak {M}_{g,n}$ (and, more generally, the stacks $\mathfrak {M}_{g,n,a}$ for some element $a \in \mathcal {A}$ in a semigroup $\mathcal {A}$ ).

To define these Chow groups, recall that in [Reference KreschKre99], Kresch constructed Chow groups $\mathrm {CH}_*(\mathcal {X})$ for algebraic stacks $\mathcal {X}$ of finite type over a field k. Moreover, there is an intersection product on $\mathrm {CH}_*(\mathcal {X})$ when $\mathcal {X}$ is smooth and stratified by global quotient stacks Footnote ¹¹ ; see [Reference KreschKre99, Theorem 2.1.12]. This last condition can be checked point-wise: a reduced stack $\mathcal {X}$ is stratified by global quotient stacks if and only if the stabilizers of geometric points of $\mathcal {X}$ are affine ([Reference KreschKre99, Proposition 3.5.9]).

Now the spaces $\mathfrak {M}_{g,n,a}$ are in general not of finite type (only locally of finite type) and so we need to extend the definition of Chow groups above. Assume that $\mathfrak {M}$ is an algebraic stack, locally of finite type over a field k. Choose a directed systemFootnote ¹² $(\mathcal {U}_i)_{i \in I}$ of finite type open substacks of $\mathfrak {M}$ whose union is all of $\mathfrak {M}$ . Then we set

$$ \begin{align*} \mathrm{CH}_*(\mathfrak{M}) = \varprojlim_{i \in I} \mathrm{CH}_*(\mathcal U_i), \end{align*} $$

where for $\mathcal U_i \subseteq \mathcal U_j$ the transition map $\mathrm {CH}_*(\mathcal U_j) \to \mathrm {CH}_*(\mathcal U_i)$ is given by the restriction to $\mathcal U_i$ . In other words, we have

$$ \begin{align*} \mathrm{CH}_*(\mathfrak{M}) = \{(\alpha_i)_{i \in I} : \alpha_i \in \mathrm{CH}_*(\mathcal U_i), \alpha_j|_{\mathcal U_i} = \alpha_i\text{ for }\mathcal U_i \subseteq \mathcal U_j \}. \end{align*} $$

We give the details of this definition in Appendix A and show that the Chow groups of locally finite type stacks inherit all the usual properties (e.g., flat pullback, projective pushforward, Chern classes of vector bundles and Gysin pullbacks) of the Chow groups from [Reference KreschKre99]. Moreover, if $\mathfrak {M}$ is smooth and has affine stabilizer groups at geometric points, the intersection products on the groups $\mathrm {CH}_*(\mathcal U_i)$ give rise to an intersection product on $\mathrm {CH}_*(\mathfrak {M})$ . In this case, for $\mathfrak {M}$ equidimensional, we often use the cohomological degree convention

$$ \begin{align*} \mathrm{CH}^*(\mathfrak{M}) = \mathrm{CH}_{\dim \mathfrak{M} -*}(\mathfrak{M}). \end{align*} $$

Now recall from Definition 1.3 that the tautological rings $(\mathrm {R}^*(\mathfrak {M}_{g,n,a}))_{g,n,a}$ are defined as the smallest system of $\mathbb {Q}$ -subalgebras with unit of the Chow rings $(\mathrm {CH}^*(\mathfrak {M}_{g,n,a}))_{g,n,a}$ closed under taking pushforwards by the natural forgetful and gluing maps.

We recall the following particular examples of tautological classes:

Definition 3.2. Let $\pi : \mathfrak {M}_{g,n+1,a} \to \mathfrak {M}_{g,n,a}$ be the universal curve over $\mathfrak {M}_{g,n,a}$ , and for $i=1, \ldots , n$ , let $\sigma _i : \mathfrak {M}_{g,n,a} \to \mathfrak {M}_{g,n+1,a}$ be the section corresponding to the i-th marked points. Let $\omega _{\pi }$ be the relative canonical line bundle on $\mathfrak {M}_{g,n+1,a}$ . Then we define

(10) $$ \begin{align} \psi_i = \sigma_i^* c_1\left( \omega_{\pi} \right) \in \mathrm{CH}^1(\mathfrak{M}_{g,n,a}) \; \text{ for }i=1, \ldots, n \end{align} $$

and

(11) $$ \begin{align} \kappa_m &= \pi_* \big(\psi_{n+1}^{m+1}\big) \in \mathrm{CH}^m(\mathfrak{M}_{g,n,a}). \end{align} $$

Definition 3.3. Let $\Gamma $ be an $\mathcal {A}$ -valued prestable graph in genus g with n markings with total value $a \in \mathcal {A}$ . For $\mathfrak {M}_{\Gamma } = \prod _{v \in V(\Gamma )} \mathfrak {M}_{g(v),n(v),a(v)}$ , a decoration $\alpha $ on $\Gamma $ is an element of $\mathrm {CH}^*(\mathfrak {M}_{\Gamma })$ given by a product of $\kappa $ and $\psi $ -classes on the factors $\mathfrak {M}_{g(v),n(v),a(v)}$ of $\mathfrak {M}_{\Gamma }$ . Thus it has the form

(12) $$ \begin{align} \alpha = \prod_{v\in V} \left( \prod_{i \in H(v)} \psi_{v,i}^{a_i} \prod_{a=1}^{m_v} \kappa_{v,a}^{b_{v,a}} \right)\in \mathrm{CH}^*(\mathfrak{M}_{\Gamma}),\end{align} $$

where $a_i, b_{v,a}\geq 0$ and $m_v\geq 0$ are some integers. We define the decorated stratum class $[\Gamma ,\alpha ]$ as the pushforward

$$\begin{align*}[\Gamma,\alpha] = (\xi_{\Gamma})_* \alpha \in \mathrm{CH}^*(\mathfrak{M}_{g,n,a}).\end{align*}$$

One of the main goals of this section is to show that the set of tautological classes $\mathrm {R}^*(\mathfrak {M}_{g,n,a}) \subseteq \mathrm {CH}^*(\mathfrak {M}_{g,n,a})$ is the $\mathbb {Q}$ -linear span of all classes $[\Gamma ,\alpha ]$ .

3.2 Intersections and functoriality of tautological classes

In this section, we describe how the classes $[\Gamma ,\alpha ]$ behave under taking intersections as well as pullbacks and pushforwards under natural gluing, forgetful and stabilization maps.

Pushforwards by gluing maps

Pushing forward by gluing maps is by far the easiest operation: given an $\mathcal {A}$ -valued graph $\Gamma _0$ and classes $[\Gamma _v,\alpha _v] \in \mathrm {R}^*(\mathfrak {M}_{g(v),n(v),a(v)})$ for $v \in V(\Gamma _0)$ , the pushforward of the class

$$ \begin{align*}\prod_{v \in V(\Gamma)} [\Gamma_v,\alpha_v] \in \mathrm{CH}^*(\mathfrak{M}_{\Gamma})\end{align*} $$

is given by $[\Gamma ,\alpha ]$ , where $\Gamma $ is obtained by gluing the $\Gamma _i$ into the vertices of the outer graph $\Gamma _0$ and $\alpha $ is obtained by combining the decorations $\alpha _v$ using that $V(\Gamma ) = \coprod _{v \in V(\Gamma _0)} V(\Gamma _v)$ .

Pullbacks by gluing maps and intersection products

The next natural question is how a class $[B,\beta ]$ pulls back along a gluing morphism $\xi _A$ for an $\mathcal {A}$ -valued graph A. This operation allows a purely combinatorial description, generalizing the description in $\overline {\mathcal {M}}_{g,n}$ from [Reference Graber and PandharipandeGP03] (and already discussed for graphs A with exactly one edge in [Reference CostelloCos06, Section 4]). As combinatorial preparation, we recall the notion of morphisms of $\mathcal {A}$ -valued stable graphs.

Definition 3.5. An A-structure on an $\mathcal {A}$ -valued prestable graph $\Gamma $ (write $\Gamma \to A$ ) is a choice of subgraphs $\Gamma _v$ of $\Gamma $ such that $\Gamma $ can be constructed by replacing each vertex v of A by the corresponding $\mathcal {A}$ -valued graph $\Gamma _v$ . More precisely, the data of $\Gamma \to A$ is given by maps

$$\begin{align*}V(\Gamma) \to V(A)\text{ and }H(A) \to H(\Gamma). \end{align*}$$

They must satisfy that $V(\Gamma ) \to V(A)$ is surjective, such that the preimage of $v \in V(A)$ are the vertices of a subgraph $\Gamma _v$ of $\Gamma $ with total $\mathcal {A}$ -value $a_v$ . The map $H(A) \to H(\Gamma )$ of half-edges in the opposite direction is required to be injective and allows one to see half-edges $h \in H(v)$ of A with legs of the graph $\Gamma _v$ . These maps must respect the incidence relation of half-edges and vertices and the pairs of half-edges forming edges. In particular, the injection of half-edges allows us to see the set of edges $E(A)$ of A as a subset of the set of edges $E(\Gamma )$ of $\Gamma $ (see, e.g., [Reference Schmitt and van ZelmSvZ20, Definition 2.5] for more details in the case of stable graphs).

Given an A-structure $\Gamma \to A$ , there exists a gluing morphism

$$\begin{align*}\xi_{\Gamma\to A}\colon \mathfrak{M}_{\Gamma} \to \mathfrak{M}_A.\end{align*}$$

For a decoration $\alpha $ on $\mathfrak {M}_A$ as in equation (12), it is easy to describe $\xi _{\Gamma \to A}^* \alpha $ using that

• ○ $\xi _{\Gamma \to A}^* \psi _{v,i} = \psi _{w,j}$ if $\Gamma \to A$ maps half-edge i in A to half-edge j in $\Gamma $ ,

• ○ $\xi _{\Gamma \to A}^* \kappa _{v,\ell } = \sum _{w \mapsto v} \kappa _{w,\ell }$ , where the sum goes over vertices w of $\Gamma $ mapping to the vertex v of A on which $\kappa _{v,\ell }$ lives.

Both these properties follow immediately from the definitionsFootnote ¹³ of $\kappa $ and $\psi $ -classes.

Let $f_A : \Gamma \to A$ , $f_B : \Gamma \to B$ be A and B-structures on the prestable graph $\Gamma $ . The pair $f = (f_A, f_B)$ is called a generic $(A,B)$ -structure $f=(f_A,f_B)$ on $\Gamma $ if every half-edge of $\Gamma $ corresponds to a half-edge of A or a half-edge of B. Given a second $(A,B)$ -structure $f'=(f_A': \Gamma ' \to A, f_B': \Gamma ' \to B)$ , an isomorphism from f to $f'$ is an isomorphism $\Gamma \to \Gamma '$ commuting with the maps to $A,B$ . Let $\mathcal {G}_{A,B}$ be the set of isomorphism classes of prestable graphs $\Gamma $ together with a generic $(A,B)$ -structures on $\Gamma $ .

Proposition 3.6. Let $A,B$ be $\mathcal {A}$ -valued prestable graphs for $\mathfrak {M}_{g,n,a}$ ; then the fibre product of the gluing maps $\xi _A : \mathfrak {M}_A \to \mathfrak {M}_{g,n,a}$ and $\xi _B : \mathfrak {M}_B \to \mathfrak {M}_{g,n,a}$ is given by a disjoint union

(13)

of spaces $\mathfrak {M}_{\Gamma }$ for the set of isomorphism classes of generic $(A,B)$ -structures on prestable graphs $\Gamma $ . The top Chern class of the excess bundle

(14) $$ \begin{align} E_{\Gamma}=\xi_{\Gamma \to A}^* \mathcal{N}_{\xi_A} / \mathcal{N}_{\xi_{\Gamma \to B}} \\[-15pt]\nonumber\end{align} $$

is given by

(15) $$ \begin{align} c_{\mathrm{top}}(E_{\Gamma}) = \prod_{e=(h,h') \in E(A) \cap E(B) \subset E(\Gamma)} - \psi_h - \psi_{h'},\\[-15pt]\nonumber \end{align} $$

where the product is over the edges of $\Gamma $ coming both from edges of A and edges of B in the generic $(A,B)$ -structure.

Proof. The proof from [Reference Graber and PandharipandeGP03, Proposition 9] of the analogous result for the moduli spaces of stable curves goes through verbatim (see also [Reference Schmitt and van ZelmSvZ20, Section 2] for a more detailed version of the argument).

Using the projection formula, we can then also intersect tautological classes.

Corollary 3.7. Given decorated stratum classes $[A,\alpha ]$ , $[B,\beta ]$ on $\mathfrak {M}_{g,n,a}$ , their product is given by

(16) $$ \begin{align} [A,\alpha] \cdot [B,\beta] = \sum_{\Gamma \in \mathcal{G}_{A,B}} (\xi_{\Gamma})_* \left(\xi_{\Gamma \to A}^* \alpha \cdot \xi_{\Gamma \to B}^* \beta \cdot c_{\mathrm{top}}(E_{\Gamma}) \right).\\[-15pt]\nonumber \end{align} $$

Pushforwards and pullbacks by forgetful maps of points

In this section, we look at the behaviour of tautological classes under the forgetful map $\pi : \mathfrak {M}_{g,n+1,a} \to \mathfrak {M}_{g,n,a}$ , which is the universal curve over $\mathfrak {M}_{g,n,a}$ . As such, it is both flat and proper, so we can compute pullbacks as well as pushforwards. We will start with pullbacks.

Proposition 3.8. Given an $\mathcal {A}$ -valued prestable graph $\Gamma $ for $\mathfrak {M}_{g,n,a}$ , we have a commutative diagram

(17)

where the graph $\widehat \Gamma _v$ is obtained from $\Gamma $ by adding the marking $n+1$ at vertex v and the map $\pi _v$ is the identity on the factors of $\mathfrak {M}_{\widehat \Gamma _v}$ for vertices $w \neq v$ and the forgetful map of marking $n+1$ at the vertex v. The induced map

(18) $$ \begin{align} \coprod_{v \in V(\Gamma)} \mathfrak{M}_{\widehat \Gamma_v} \to \mathfrak{M}_{\Gamma} \times_{\mathfrak{M}_{g,n,a}} \mathfrak{M}_{g,n+1,a}\\[-15pt]\nonumber\end{align} $$

satisfies that the fundamental class on the left pushes forward to the fundamental class on the right.

Proof. This follows from the definition of the gluing map $\xi _{\Gamma }$ : giving the map $\xi _{\Gamma }$ is the same as giving the universal curve over $\mathfrak {M}_{\Gamma }$ , and this curve is obtained by gluing the universal curves $\mathfrak {M}_{\widehat \Gamma _v}$ over the various factors along the half-edges connected in $\Gamma $ . The map in equation (18) is obtained by taking, for each edge $\{h_1,h_2\} \in E(\Gamma )$ the loci inside $\mathfrak {M}_{\widehat \Gamma _{v_i}}$ , where marking $p_{n+1}$ and marking $q_{h_i}$ are on a contracted component and identifying them. Thus, if $p_{n+1}$ is not on a contracted component, the map is an isomorphism in a neighborhood. Therefore the map in equation (18) is an isomorphism at the general point of each component of the right-hand side, and the fundamental class pushes forward to the fundamental class.

Corollary 3.9. Given a tautological class $[\Gamma ,\alpha ]$ , write $\alpha = \prod _{v \in V(\Gamma )} \alpha _v$ with $\alpha _v$ the factors of $\alpha $ located at vertex v of $\Gamma $ . Then we have

$$\begin{align*}\pi^* [\Gamma,\alpha] = \sum_{v \in V(\Gamma)} [\widehat \Gamma_v, (\pi_v^* \alpha_v) \cdot \prod_{w \neq v} \alpha_w].\end{align*}$$

Proof. The class $[\Gamma ,\alpha ]$ is represented by $\xi _{\Gamma *}\left (\alpha \cap \,[\mathfrak {M}_{\Gamma }]\right )$ , where $\alpha $ is an operational Chow class in $\mathrm {CH}_{\mathrm {OP}}^*(\mathfrak {M}_{\Gamma })$ . We refer the reader to Appendix C for definitions and properties of these operational classes. By Proposition 3.8, the diagram in equation (17) together with the map in equation (18) satisfies assumptions in Lemma C.8. Therefore the equality follows from Lemma C.8.

The above corollary shows that to finish our understanding of pullbacks of tautological classes, it suffices to understand how $\kappa $ and $\psi $ -classes pull back.

Proposition 3.10. For the universal curve morphism $\pi : \mathfrak {M}_{g,n+1,a} \to \mathfrak {M}_{g,n,a}$ , we have

(19) $$ \begin{align} \pi^* \psi_i &= \psi_i - D_{i,n+1}, \end{align} $$

(20) $$ \begin{align} \pi^* \kappa_a &= \kappa_a - \psi_{n+1}^a, \end{align} $$

where $D_{i,n+1} \subset \mathfrak {M}_{g,n+1,a}$ is the image of the section $\sigma _i$ of $\pi $ corresponding to the ith marked point. It can be seen as the tautological class corresponding to the (undecorated) graph in equation (9) above.

Proof. The statement is a generalization of the classical pullback formulas for $\overline {\mathcal {M}}_{g,n}$ (which are the case

). A convenient way to prove it is to use that

(21) $$ \begin{align} \psi_i = -\pi_*(D_{i,n+1}^2). \end{align} $$

To show this, we note that $\sigma _i$ can be identified with the gluing map

where we glue the ith marking on $\mathfrak {M}_{g,n,a}$ with the marking $\bullet $ on

. Then indeed the locus $D_{i,n+1}$ is the image of the above gluing map (similar to the usual case of stable maps), and equation (21) follows from Corollary 3.7. On the other hand, it can also be seen directly from the fact that $\sigma _i$ is a closed embedding with normal bundle $\sigma _i^*(\omega _{\pi }^{\vee })$ .

Now we have a commutative diagram

(22)

and the space in the upper left maps birationally to the fibre product of the two forgetful maps. Then

Similarly, using the same diagram, the definition of $\kappa _a$ and the pullback formula for $\psi $ , one concludes the pullback formula for $\kappa _a$ .

We now turn to the question how to push forward tautological classes $[\Gamma , \alpha ] \in \mathrm {R}^*(\mathfrak {M}_{g,n+1,a})$ under the map $\pi $ .

Proposition 3.11. Let $[\Gamma , \alpha ] \in \mathrm {R}^*(\mathfrak {M}_{g,n+1,a})$ with $\alpha = \prod _{v \in V(\Gamma )} \alpha _v$ . Let $v \in V(\Gamma )$ be the vertex incident to $n+1$ , and let $\Gamma '$ be the graph obtained from $\Gamma $ by forgetting the marking $n+1$ and stabilizing if the vertex v becomes unstable. There are two cases:

• ○ If the vertex v remains stable, then

  $$\begin{align*}\pi_* [\Gamma,\alpha] = (\xi_{\Gamma'})_*\left((\pi_v)_* \alpha_v \cdot \prod_{w \neq v} \alpha_w \right), \end{align*}$$
  where $\pi _v$ is the forgetful map of marking $n+1$ of vertex v.

• ○ If the vertex v becomes unstable, then $g(v)=0$ , $n(v)=3$ and . If $\alpha _v \neq 1$ , then $[\Gamma , \alpha ]=0$ . Otherwise, we have

  $$\begin{align*}\pi_* [\Gamma,\alpha] = [\Gamma', \prod_{w \neq v} \alpha_w].\end{align*}$$

Proof. The result follows from the fact that the composition of the gluing map $\xi _{\Gamma }$ and the forgetful map $\pi $ factors through the gluing map $\xi _{\Gamma '}$ downstairs. In the second part, we use that , so any nontrivial decoration by $\kappa $ and $\psi $ -classes on this space vanishes.

The proposition allows us to reduce to computing forgetful pushforwards of products of $\kappa $ and $\psi $ -classes. As in the case of $\overline {\mathcal {M}}_{g,n}$ , these can be computed using the projection formula. Indeed, given a product

$$\begin{align*}\alpha = \prod_{a} \kappa_a^{e_a} \cdot \prod_{i=1}^n \psi_i^{\ell_i} \cdot \psi_{n+1}^{\ell_{n+1}} \in \mathrm{R}^*(\mathfrak{M}_{g,n+1,a}),\end{align*}$$

we can use Proposition 3.10 and the known intersection formulas on $\mathfrak {M}_{g,n+1,a}$ to write it as

$$\begin{align*}\alpha = \pi^* \left( \prod_{a} \kappa_a^{e_a} \cdot \prod_{i=1}^n \psi_i^{\ell_i}\right)\cdot \psi_{n+1}^{\ell_{n+1}} + \text{boundary terms}. \end{align*}$$

Using the projection formula, we conclude

$$\begin{align*}\pi_*(\alpha) = \left( \prod_{a} \kappa_a^{e_a} \cdot \prod_{i=1}^n \psi_i^{\ell_i}\right)\cdot \kappa_{\ell_{n+1}-1} + \pi_*(\text{boundary terms}), \end{align*}$$

where $\kappa _0=2g-2+n$ and $\kappa _{-1}=0$ . The boundary terms are handled by induction on the degree together with Proposition 3.11.

Together with the previous results of this section, this shows that the $\mathbb {Q}$ -linear span of the decorated strata classes $[\Gamma , \alpha ]$ in $\mathrm {CH}^*(\mathfrak {M}_{g,n,a})$ is closed under intersections as well as pushforwards under gluing and forgetful maps. Thus, by definition, it equals the tautological ring of $\mathfrak {M}_{g,n,a}$ so that we have finished the proof of Theorem 1.4.

Pullbacks by forgetful maps of $\mathcal {A}$ -values

Proposition 3.12. For the map $F_{\mathcal {A}}: \mathfrak {M}_{g,n,a} \to \mathfrak {M}_{g,n}$ forgetting the $\mathcal {A}$ -values on all components of the curve, without stabilizing, we have

$$\begin{align*}F_{\mathcal{A}}^* [\Gamma,\alpha] = \sum_{\substack{(a_v)_{v \in V(\Gamma)}\\\sum_v a_v=a}} [\Gamma_{(a_v)_v}, \alpha],\end{align*}$$

where the sum is over tuples $(a_v)_v$ of elements of $\mathcal {A}$ summing to a, such that the $\mathcal {A}$ -valuation $v \mapsto a_v$ on the vertices v of $\Gamma $ gives a well-defined $\mathcal {A}$ -valued graph $\Gamma _{(a_v)_v}$ .

Proof. The fibre product of $F_{\mathcal {A}}$ and a gluing map $\xi _{\Gamma }$ is the disjoint union of the gluing maps $\xi _{\Gamma _{(a_v)_v}}$ . A short computation shows that $F_{\mathcal {A}}^* \psi _i = \psi _i$ and $F_{\mathcal {A}}^* \kappa _a = \kappa _a$ .

Pullback by stabilization map

We saw before that the stabilization morphism $\mathrm {st}: \mathfrak {M}_{g,n} \to \overline {\mathcal {M}}_{g,n}$ is flat, so we can ask how to pull back tautological classes along this morphism. We start by computing the pullback of gluing maps under $\mathrm {st}$ .

Proposition 3.13. Given a stable graph $\Gamma $ in genus g with n marked points (for $2g-2+n>0$ ), we have a commutative diagram

(23)

where $\mathrm {st}_v : \mathfrak {M}_{g(v),n(v)} \to \overline {\mathcal {M}}_{g(v),n(v)}$ is the stabilization morphism at vertex v. Moreover, the induced map

(24) $$ \begin{align} \mathfrak{M}_{\Gamma} \to \mathfrak{M}_{g,n} \times_{\overline{\mathcal{M}}_{g,n}} \overline{\mathcal{M}}_{\Gamma} \end{align} $$

is proper and birational. In particular

(25) $$ \begin{align} \mathrm{st}^* \left[\Gamma, \prod_v \alpha_v\right] = (\xi_{\Gamma})_* \left( \prod_v {\mathrm{st}_v^*} \alpha_v \right). \end{align} $$

Proof. The commutativity of the diagram in equation (23) follows from the definition of the stabilization. The map in equation (24) is easily seen to be birational, and its properness follows from the diagram

and the cancellation property of proper morphisms (in the diagram, the map $\xi _{\Gamma }$ and $\mathrm {pr}_1$ are proper). Equation (25) again follows by an application of Lemma C.8.

The proposition above reduces the pullback of tautological classes under $\mathrm {st}$ to computing the pullback of $\kappa $ and $\psi $ -classes.

Proposition 3.14. Let $g,n$ with $2g-2+n>0$ , and let . Then for $1 \leq i \leq n$ and the stabilization map , we have

(26)

Proof. Consider the commutative diagram

(27)

where the right square is Cartesian and the map c is the map contracting the unstable components of the universal curve . By the cancellation property of proper morphisms, the map c is proper and easily seen to be birational.

For computing the pullback of $\psi _i$ under $\mathrm {st}_{\mathcal {A}}$ , we use that $\psi _i = -\pi _*( D_{i,n+1}^2)$ on $\overline {\mathcal {M}}_{g,n}$ , where $D_{i,n+1} \subset \overline {\mathcal {M}}_{g,n+1}$ is the image of the ith section. By Lemma C.8, we have

$$\begin{align*}\mathrm{st}_{\mathcal{A}}^* \psi_i = - \mathrm{st}_{\mathcal{A}}^* \pi_*( D_{i,n+1}^2) = (\pi_{\mathcal{A}})_* \left((c \circ \widehat{\mathrm{st}}_{\mathcal{A}})^* D_{i,n+1} \right)^2.\end{align*}$$

The composition $c \circ \widehat {\mathrm {st}}_{\mathcal {A}}$ is just the usual stabilization map, and the pullback of $D_{i,n+1}$ under this map is the sum of three boundary divisors of : their underlying graph is the same as for $D_{i,n+1}$ and the $\mathcal {A}$ -values correspond to the three different ways to distribute the value to the two vertices. A short computation using the rules for intersection and pushforward presented earlier gives the formula in equation (26).

The formula for the pullback of $\kappa $ -classes is more involved, and we need to introduce a bit of notation to state it. Fix $g,n$ with $2g-2+n>0$ ; then for $k \geq 0$ , let $\widehat G_k, G_k$ be the following $(n+1)$ and n-pointed prestable graphs in genus g with k edges

Here $v_0$ is the leftmost vertex and, for $k \geq 1$ , $h_0$ is the unique half-edge incident to this vertex. For $k=0$ , the graphs $\widehat {G}_k, G_k$ are the trivial graphs, respectively.

Also, in the proposition below, we consider the power series

$$\begin{align*}\Phi(t)=\frac{\exp(t)-1}{t} = 1 + \frac{t}{2} + \frac{t^2}{6}+ \ldots .\end{align*}$$

We use the notation $[\Phi (t)]_{t^a \mapsto \kappa _a}$ to indicate that in the power series $\Phi $ , the term $t^a$ is substituted with the class $\kappa _a$ , getting the mixed-degree Chow class

$$\begin{align*}[\Phi(t)]_{t^a \mapsto \kappa_a}= 1 + \frac{\kappa_1}{2} + \frac{\kappa_2}{6}+ \ldots .\end{align*}$$

Proposition 3.15. For $g,n$ with $2g-2+n>0$ , the stabilization morphism $\mathrm {st}: \mathfrak {M}_{g,n} \to \overline {\mathcal {M}}_{g,n}$ satisfies the following equality of mixed-degree Chow classes on $\mathfrak {M}_{g,n}$ :

(28) $$ \begin{align} &\mathrm{st}^*\left[\Phi(t)\right]_{t^a \mapsto \kappa_a} \nonumber \\=& \left[\Phi(t)\right]_{t^a \mapsto \kappa_a} + \sum_{k \geq 1} (\xi_{G_k})_* \left( \left(\left[\Phi(t) \right]_{t^a \mapsto \kappa_{v_0,a}} + \psi_{h_0}^{-1} \right) \cdot \mathrm{Cont}_{E(G_k)} \right). \end{align} $$

Here $\mathrm {Cont}_{E(G_k)}$ is the mixed-degree class

$$\begin{align*}\mathrm{Cont}_{E(G_k)} = \prod_{(h,h') \in E(G_k)} - \Phi(\psi_h + \psi_{h'}) \end{align*}$$

on $\mathfrak {M}_{G_k}$ . In the formula above, the term $\psi _{h_0}^{-1}$ is understood to vanish unless it pairs with a term of $\mathrm {Cont}_{E(G_k)}$ containing a positive power of $\psi _{h_0}$ , and we have $\kappa _{v_0,0}=2\cdot 0 -2+1=-1$ .

To obtain the pullback of an individual class $\kappa _a$ under $\mathrm {st}$ , we take the degree a part of equation (28) and obtain a formula of the form

$$\begin{align*}\mathrm{st}^* \kappa_a = \kappa_a + \text{ boundary corrections}.\end{align*}$$

As an example, for $a=1,2$ , we obtain

$$ \begin{align*} \mathrm{st}^*\kappa_1=\kappa_1+[G_1] \end{align*} $$

and

$$ \begin{align*} \mathrm{st}^*\kappa_2=\kappa_2 -3[G_1,\kappa_{v_0,1}]+2[G_1,\psi_{h_0}]+[G_1,\psi_{h_1}] -3[G_2] \end{align*} $$

where $e=(h_0,h_1)$ is the unique edge of the graph $G_1$ .

Proof of Proposition 3.15

Consider the following commutative diagram:

(29)

Then as $\mathfrak {C}_{g,n}$ maps proper and birationally to the fibre product in the right diagram, we have

(30) $$ \begin{align} \mathrm{st}^* \kappa_a = \mathrm{st}^* \pi_* \psi_{n+1}^{a+1} = \pi^{\prime}_* \widehat{\mathrm{st}}^* \psi_{n+1}^{a+1} = \pi^{\prime}_* \left(\psi_{n+1} - [\widehat{G}_1] \right)^{a+1}. \end{align} $$

Here we use that computations in $\mathfrak {C}_{g,n}$ can be performed in together with the pullback formula from Proposition 3.14 (noting that the third term in in equation (26) vanishes since it lies in the complement of the open substack

From now on, it will be more convenient working with mixed-degree classes and exponentials. In this language, equation (30) translates to

(31) $$ \begin{align} \mathrm{st}^*[\Phi(t)]_{t^a \mapsto \kappa_a} = \pi^{\prime}_* \exp(\psi_{n+1} - [\widehat{G}_1]) = \pi^{\prime}_* \left(\exp(\psi_{n+1}) \cdot \exp(- [\widehat{G}_1]) \right). \end{align} $$

The occurrence of the power series $\Phi $ is due to the discrepancy between the degree a of $\kappa _a$ on the left of equation (30) and the degree $a+1$ of the term on the right. Using the rules for intersections of tautological classes, one shows

(32) $$ \begin{align} \exp(- [\widehat{G}_1]) = \sum_{k\geq 1} (\xi_{\widehat{G}_k})_* \left( \prod_{(h,h') \in E(\widehat{G}_k)} - \Phi(\psi_h + \psi_{h'}) \right). \end{align} $$

Now, in the pushforward in equation (31), the only terms supported on the trivial graph are those from

$$\begin{align*}\pi^{\prime}_*(\exp(\psi_{n+1}))=[\Phi(t)]_{t^a \mapsto \kappa_a},\end{align*}$$

explaining the first term of the answer. All other terms of the product of the exponentials are supported on some $\widehat {G}_k$ for $k \geq 1$ , where marking $n+1$ is on a rational component with just one other half-edge $h_0$ . Using the formulas for the pushforward by the forgetful map $\pi '$ from Proposition 3.11, the only nontrivial pushforward we have to compute is the one by the universal curve $\pi _{0,1}: \mathfrak {C}_{0,1} \to \mathfrak {M}_{0,1}$ , corresponding to forgetting $n+1$ on the $2$ -marked genus $0$ component $v_0$ of $\widehat {G}_k$ . Here, a short computation shows

(33) $$ \begin{align} (\pi_{0,1})_* \psi_1^a \psi_2^b = \psi_1^a \kappa_{b-1} + \delta_{a,0} \psi_1^{a-1}, \end{align} $$

where $\delta _{a,0}$ is the Kronecker delta, and we have the convention $\kappa _{-1}=\psi _1^{-1}=0$ . Applying this formula for the pushforward, the first term in equation (33) gives rise to the term of the result involving $\left [\Phi (t) \right ]_{t^a \mapsto \kappa _{v_0,a}}$ , where again $\Phi $ appears due to the shift of degree from b to $b-1$ in equation (33). The second term of equation (33) gives rise to the term involving $\psi _{h_0}^{-1}$ , where due to the Kronecker delta $\delta _{a,0}$ only the constant term of $\exp (\psi _{n+1})$ survives in the pushforward.

Remark 3.16. The following is a nontrivial check and application of the computations from the last sections: for $g,n,m$ with $2g-2+n>0$ , consider the diagram

(34)

where $F_m$ is the map forgetting the last m markings (without stabilizing the curve), the map $\mathrm {st}$ is the stabilization map and their composition $\pi $ is the ‘usual’ forgetful map between moduli spaces of stable curves. The pullback of tautological classes under $\pi $ is known classically, and the pullback by the two other maps has been computed in the previous sections. Since the pullbacks must be compatible, this gives rise to tautological relations, which we can verify in examples.

For instance, for the class $\kappa _1 \in \mathrm {CH}^1(\overline {\mathcal {M}}_{g,n})$ , we have

$$\begin{align*}\pi^* \kappa_1 = \kappa_1 - \sum_{i=n+1}^{n+m} \psi_i + \sum_{\substack{I \subset \{n+1, \ldots, n+m\}\\|I| \geq 2}} D_{0,I},\end{align*}$$

where $D_{0,I} \subset \overline {\mathcal {M}}_{g,n+m}$ is the boundary divisor of curves with a rational component carrying markings I. On the other hand we have

$$ \begin{align*} \mathrm{st}^* \kappa_1 &= \kappa_1 + [G_1] = \kappa_1 + D_{0,\emptyset}\\ \pi^* \mathrm{st}^* \kappa_1 &= \kappa_1 - \sum_{i=n+1}^{n+m} \psi_i + \sum_{\substack{I \subset \{n+1, \ldots, n+m\}\\|I| \geq 2}} D_{0,I}. \end{align*} $$

So indeed we get the same answer.