2.1. Weighted blow-ups

Roughly speaking, a weighted blow-up replaces a closed substack $Y\subseteq X$ with a weighted projective bundle over Y. With respect to the ordinary case, the weighting offers an extra degree of freedom. Moreover, weighted blow-ups are not schematic. We recall some basic facts, referring the reader to the foundational work of Quek and Rydh [Reference Quek and Rydh31].

The Rees algebra $I_\bullet :=\bigoplus I_nt^n \subset \mathcal {O}_X[t]$ is $\mathbb Z$ -graded, corresponding to a $\mathbb {G}_{\mathrm {{m}}}$ -action on $\operatorname {Spec}_X(I_\bullet )$ , which is therefore a cone. Let $I_{+}$ be the ideal generated by t, corresponding to the vertex of the cone.

Definition 2.2. The weighted blow-up of X at $I_\bullet $ is

$$ \begin{align*}Bl_{I_{\bullet}}X:=\mathcal{P}\!\operatorname{roj}_X\left(I_{\bullet}\right):=[(\operatorname{Spec}_X(I_{\bullet}) \smallsetminus V(I_+))/{\mathbb{G}}_{\mathrm{{m}}}] \rightarrow X.\end{align*} $$

Here $\mathcal {P}\!\operatorname {roj}$ is taken in the sense of quotient stacks (as opposed to the usual $\operatorname {Proj}$ , that is, the schematic quotient). The weighted blow-up centre is $Y=V(I_1)\subseteq X$ .

Ordinary blow-ups correspond to taking $I_n=I^n$ . In this case, the Rees algebra is generated in degree $1$ , and the blow-up is schematic over the base.

Given a grading of the generators of an ideal I, we can define a filtration by defining $I_n\subset I$ to be the elements of degree $\geq n$ . When the filtration $I_\bullet $ comes from a grading, we say that the weighted blow-up is at $Y=V(I_1)$ rather than at $I_\bullet $ .

A fundamental example is the weighted blow-up of $\mathbb {A}^m$ at $0$ . Given weights $(a_1,\ldots ,a_m)$ , this defines a filtration of $I=(x_1,\ldots , x_m)$ . The weighted blow-up can then be described as

$$ \begin{align*}Bl_{0}\mathbb{A}^m \cong \mathcal{P}\!\operatorname{roj}\left(\frac{k[x_1,\dots x_m][s,x_1',\dots x_m']}{(x_1-s^{a_1}x_1',\dots, x_m-s^{a_m}x_m')}\right)\end{align*} $$

with $x_i$ in degree $0$ , $x_i'$ in degree $a_i$ , and s in degree $-1$ . The exceptional divisor in this case is the weighted projective stack

$$\begin{align*}\mathcal{P}(a_1,\ldots, a_m) := [\mathbb{A}^m \smallsetminus \{0\} / \mathbb{G}_{\mathrm{{m}}} ], \end{align*}$$

where $\mathbb {G}_{\mathrm {{m}}}$ acts on $\mathbb {A}^m$ with weights $(a_1,\ldots , a_m)$ . Given a subset $J=\{j_1,\ldots , j_r\} \subset I$ such that $\operatorname {gcd}(a_{j_1},\ldots , a_{j_r})=d$ , the vanishing locus of the $J^c$ -labelled coordinates has generic stabiliser $\mu _d$ . The projective stack $\mathcal {P}(a_1,\ldots , a_m)$ is always smooth, whereas its coarse moduli space is in general not smooth as a scheme.

Given a regular blow-up, the exceptional divisor is the weighted projective bundle over Y

$$\begin{align*}\mathcal{E} = [ \mathcal{N} \smallsetminus \mathcal{N}_0 / \mathbb{G}_{\mathrm{{m}}} ] \end{align*}$$

where $\mathcal {N}$ is the normal bundle of Y (and $\mathcal {N}_0$ its zero section). It inherits a $\mathbb {G}_{\mathrm {{m}}}$ -action that smooth-locally looks like the linear $\mathbb {G}_{\mathrm {{m}}}$ -action on $\mathbb {A}^m$ with weights $(a_1,\ldots , a_m)$ .

The following is a generalisation of Artin’s criterion for regular weighted blow-up.

2.2. Integral Chow rings and blow-up formulae

The classical blow-up formulae of [Reference Fulton16, §6.7] have been generalised to weighted blow-ups by Arena and Obinna [Reference Arena, Obinna and Abramovich2]. We will only use a special case, which we recall for the reader’s convenience. Let $\widetilde {X}$ be a regular weighted blow-up of a smooth Deligne–Mumford stack X with centre Y, where $\iota \colon Y\hookrightarrow X$ denotes the regular embedding. Suppose moreover that the pullback homomorphism of Chow rings $\iota ^*\colon A^*(X) \to A^*(Y)$ is surjective. Then we have the following simple formula [Reference Arena, Obinna and Abramovich2, Corollary 6.5]:

(2.1) $$ \begin{align} A^*(\widetilde{X}) \simeq A^*(X)[t]/(t\cdot \ker(\iota^*), Q(t)) \end{align} $$

where $t=-[\mathcal {E}]$ is the opposite of the fundamental class of the exceptional divisor, and $Q(t)\in A^*(X)[t]$ is defined by replacing the t-constant term of $c_{\text {top}}^{\mathbb {G}_{\mathrm {{m}}}}(\mathcal {N})$ with the fundamental class of Y. We explain this in more detail.

By assumption, the normal bundle $\mathcal {N}$ of Y in X has a $\mathbb {G}_{\mathrm {{m}}}$ -action, which, smooth-locally on Y, is linear. We can therefore consider the $\mathbb {G}_{\mathrm {{m}}}$ -equivariant Chern classes of $\mathcal {N}$ , which are elements of the $\mathbb {G}_{\mathrm {{m}}}$ -equivariant Chow ring of Y. The latter being endowed with the trivial $\mathbb {G}_{\mathrm {{m}}}$ -action, we have $A^*_{\mathbb {G}_{\mathrm {{m}}}}(Y) \simeq A^*(Y)[t]$ . In particular, we can think of $c_{\text {top}}^{\mathbb {G}_{\mathrm {{m}}}}(\mathcal {N})$ as a polynomial in the equivariant variable t. The surjectivity of $\iota ^*$ implies that we can lift its coefficients to X. In particular, we can find an element $Q'(t)\in A^*(X)[t]$ such that $\iota ^*Q'(t) = c_{\text {top}}^{\mathbb {G}_{\mathrm {{m}}}}(\mathcal {N})(t)-c_{\text {top}}^{\mathbb {G}_{\mathrm {{m}}}}(\mathcal {N})(0)$ . Finally, the polynomial $Q(t)$ is defined by adding in the fundamental class of Y, that is, as

$$\begin{align*}Q(t)=Q'(t) + [Y]\in A^*(X)[t].\end{align*}$$

Putting everything together, we see that, given a presentation for $A^*(X)$ , a presentation for $A^*(\widetilde {X})$ can be computed from the following ingredients:

1. 1. generators of the ideal $\ker (\iota ^*\colon A^*(X) \to A^*(Y))$ ,

2. 2. the fundamental class of the centre $[Y]\in A^*(X)$ and

3. 3. the top equivariant Chern class $c_{\text {top}}^{\mathbb {G}_{\mathrm {{m}}}}(\mathcal {N}_{Y/X})$ .

2.3. Alternative compactifications of $\mathcal {M}_{1,n}$ , I

Smyth produced some alternative compactifications of $\mathcal {M}_{1,n}$ in [Reference Smyth34]. The idea is to trade a higher local/algebraic complexity (worse-than-nodal singularities, such as cusps and tacnodes) for a lower global/combinatorial complexity (subcurves of genus one are required to have many special points; in a smoothing family, those that don’t may be contracted to a singularity as above).

We start by recalling the relevant curve singularities. Smyth classified all the isolated Gorenstein singularities of genus oneFootnote ¹: there is exactly one analytic type for every number m of branches.

Definition 2.5. Let k be a field of characteristic different from $2,3$ . A k-point of a curve C is called an elliptic m-fold point if the analytic germ of C at p is one of the following:

where $I_m$ is the ideal generated by the binomials $x_ix_j-x_ix_h$ for all $i,j,h\in [m-1]$ .

Smyth proved that elliptic $\ell $ -fold points arise in one-parameter smoothing families exactly when we contract a genus one subcurve E of the nodal central fibre C, such that $\lvert E\cap \overline {C\setminus E}\rvert =\ell $ . Up to blowing up the markings, then, we can always replace genus one subcurves with $\ell <n$ special points with an elliptic $\ell $ -fold singularity. This motivates the following:

Definition 2.6. Let $0\leq m<n$ . An m-stable n-pointed genus one curve over S is

$$\begin{align*}(\pi\colon C\to S,p_1,\ldots,p_n)\end{align*}$$

where $C\to S$ is a proper curve (cf. Section 1.3) such that:

1. 1. the morphism $\pi \colon C\to S$ is a family of projective curves of arithmetic genus one, whose fibres have only nodes and elliptic $\ell $ -fold points as singularities, with $\ell \leq m$ ;

2. 2. the morphisms $p_i:S\to C$ are disjoint sections of $\pi $ whose images are contained in its smooth locus; let us denote their union by $\Sigma $ , then

3. 3. for every geometric point $s\in S$ , and for every connected subcurve of genus one $E\subset C_s$ , the latter contains more than m special points (Smyth calls this number the level of E):

   $$ \begin{align*}\operatorname{lev}(E):=|E\cap \overline{(C_s\smallsetminus E)}| + |\Sigma \cap E|>m;\end{align*} $$

4. 4. for every geometric point $s\in S$ , we have $H^0(C_s,\Omega _{C_s}^\vee (-\Sigma \cap C_s))=0$ .

Notice that $0$ -stability coincides with the classical Deligne–Mumford stability.

Every Gorenstein curve of arithmetic genus one admits a decomposition into a minimal elliptic subcurve (also called the core, or elliptic spine; this can be an elliptic curve, a nodal wheel of $\mathbb {P}^1$ , or an elliptic $\ell $ -fold point with exactly $\ell $ branches $\simeq \mathbb {P}^1$ ) and a union of rational trees, attached nodally. The dualising bundle of the core is trivial. Condition (3) above can be equivalently required of the core only. Condition (4) above implies the finiteness of the automorphism group. It can be phrased combinatorially: every branch of an elliptic $\ell $ -fold point must contain at least one marking or node, and at least one branch must contain at least two. Smyth proves the following:

Smoothness follows from the deformation theory of Gorenstein rings in codimension $3$ [Reference Smyth35, Cor. 2.2]. More generally, these stacks are normal, Gorenstein and smooth in codimension $\leq 6$ [Reference Lekili and Polishchuk24, §1.5]. So, only in the range $m\leq 5$ are the Chow rings of Smyth’s spaces defined.

2.4. Alternative compactifications of $\mathcal {M}_{1,n}$ , II

Motivated by homological mirror symmetry, Lekili and Polishchuk constructed alternative compactifications of $\mathcal M_{1,n}$ from $A_\infty $ -structures. They consider the moduli space $\mathcal {U}^{sns}_{1,m+1}$ of nonspecial curves $(C,p_1,\ldots ,p_{m+1})$ , which we will also call almost- m-stable and denote by $\widetilde {\mathcal {M}}_{1,m+1}$ , consistently with [Reference Di Lorenzo, Pernice and Vistoli11]. Lekili and Polishchuk compute an explicit normal form for these curves. When restricted to $\operatorname {Spec}(\mathbb {Z}[\frac {1}{6}])$ , this yields explicit coordinates on Smyth’s moduli spaces $\overline {\mathcal {M}}_{1,m+1}(m)$ .

The total space $\widetilde {\mathcal {U}}^{sns}_{1,m+1}$ of the $\mathbb {G}_{\mathrm {{m}}}$ -torsor associated to the Hodge line bundle $\widetilde {\mathcal {H}}=\pi _*\omega _\pi $ over $\widetilde {\mathcal {M}}_{1,m+1}$ is an affine scheme with natural coordinates, by which Lekili and Polishchuk prove:

Interestingly, it can be proved a posteriori that the singularities of an almost m-stable $m+1$ -marked curve are at worst elliptic $(m+1)$ -fold points. Indeed, the unique point with $\mathbb {G}_{\mathrm {{m}}}$ -stabiliser in each of the above corresponds to the $(m+1)$ -fold point with one marking for every branch, and the stack is nothing but (the quotient by the natural action of the multiplicative group of) the miniversal deformation of such a singularity (see also [Reference Stevens37]).

This has two fundamental consequences: first, by removing $0$ from $\widetilde {\mathcal {H}}$ , the $\mathbb {G}_{\mathrm {{m}}}$ -quotient is a weighted projective stack; and second, since $0$ represents the only m-unstable curve, the quotient can also be identified with Smyth’s $\overline {\mathcal {M}}_{1,m+1}(m)$ .

2.5. …and maps between them

Since the locus of smooth pointed elliptic curves is a dense open substack of every $\overline {\mathcal {M}}_{1,n}(m)$ , these spaces are all birational. The birational maps:

(2.2) $$ \begin{align} \overline{\mathcal{M}}_{1,n}(m-1)\dashrightarrow\overline{\mathcal{M}}_{1,n}(m) \end{align} $$

are regular if and only if $m=1$ (contraction of elliptic tails to ordinary cusps, cf. [Reference Schubert33, Reference Hassett and Hyeon17]) or $m=n-1$ [Reference Smyth35, Corollary 4.5].

A simultaneous resolution of indeterminacy, inspired by tropical geometry, has been devised by Ranganathan, Santos-Parker and Wise [Reference Ranganathan, Santos-Parker and Wise32].

For the purpose of our computations, it is more convenient to have a step-by-step resolution, performing only the necessary blow-ups every time. Such a resolution has been described by Smyth in [Reference Smyth36, §2.2]. The map (2.2) above is not defined precisely where the minimal subcurve of genus one has exactly level m. In order to distinguish the components of these loci, it is useful to refine the notion of level of a subcurve to take values in (set-)partitions of $\{1,\ldots , n\}$ . It was the intuition of [Reference Bozlee, Kuo and Neff8] that this is the way to obtain all the compactifications of $\mathcal {M}_{1,n}$ by Gorenstein curves with smooth, distinct markings. Set $[n]:=\{1,\ldots , n\}$ .

Let $S\in {[n]\choose m}$ be a partition of n into m parts $S_1,\ldots ,S_m$ . Up to reordering, suppose that $|S_i|=1$ if and only if $i>k=k(S)$ . We adopt the notation $S_1|\ldots |S_m$ to represent S.

In particular, we have an isomorphism:

$$\begin{align*}\mathbf{T}_S(m-1)=\overline{\mathcal{M}}_{1,[k]\sqcup S_{k+1}\sqcup\ldots\sqcup S_m}(m-1)\times\prod_{i=1}^k\overline{\mathcal{M}}_{0,\{\star\}\sqcup S_i}\simeq\overline{\mathcal{M}}_{1,m}(m-1)\times\prod_{i=1}^k\overline{\mathcal{M}}_{0,\lvert S_i\rvert+1}.\end{align*}$$

Since this is the worst allowed singularity in this moduli stack, the universal family of curves over $\mathbf {Ell}_S(m)$ is equisingular, in particular all of these curves admit a common normalisation, which is a (disconnected) pointed rational curve. We thus have a morphism:

$$\begin{align*}\mathbf{Ell}_S(m)\longrightarrow \overline{\mathcal{M}}_{0,S}=\prod_{i=1}^k\overline{\mathcal{M}}_{0,\{\star_i\}\sqcup S_i}.\end{align*}$$

What is missing in order to reconstruct the elliptic m-fold point is the datum of its local ring as a subalgebra of the semilocal ring of the normalisation. The parameter space for these data is called the crimping space in [Reference van der Wyck39], and the moduli space of attaching data in [Reference Smyth35, §2.2]. Smyth [Reference Smyth34, Lemma 2.2] showed that, for an elliptic m-fold singularity, the embedding of $\hat {\mathcal {O}}_{C,q}$ in $\oplus _{i=1}^m\hat {\mathcal {O}}_{C^\nu ,\star _i}$ is determined by the choice of a hyperplane in $\oplus _{i=1}^m \mathfrak {m}_{\star _i}/\mathfrak {m}_{\star _i}^2$ not containing any of the lines $\mathfrak {m}_{\star _{i}}/\mathfrak {m}^2_{\star _i}$ ; this hyperplane is identified with the pullback of the cotangent space to the elliptic singularity under the normalisation map. Dually, the hyperplane corresponds to a line in $\oplus _{i=1}^m (\mathfrak {m}_{\star _i} /\mathfrak {m}_{\star _i}^2)^\vee $ not contained in any of the hyperplanes $\operatorname {Ann}(\mathfrak {m}_{\star _i} /\mathfrak {m}_{\star _i}^2)$ . The elliptic singularity can be obtained from its seminormalisation $C^{sn}$ (an ordinary, rational m-fold point) by pinching along this line. Smyth [Reference Smyth35, §2.2] explains how to modularly compactify this algebraic torus bundle into a projective bundle by sprouting the universal curve along the boundary.

The rational morphism $\overline {\mathcal {M}}_{1,n}(m-1)\dashrightarrow \overline {\mathcal {M}}_{1,n}(m)$ flips the locus $\mathbf {T}_S(m-1)$ to the locus $\mathbf {Ell}_S(m)$ .

Over $\overline {\mathcal {M}}_{1,n}(m-\frac {1}{2})$ , Smyth [Reference Smyth36, Proposition 2.3] constructs a diagram of curves:

where $\sigma $ is the blow-up of the sections $p_{S_{k+1}},\ldots ,p_{S_m}$ over $\mathbf {Exc}(m-\frac {1}{2})$ , and $\tau $ is the contraction of the strict transform of the core, that is, the minimal subcurve of arithmetic genus one, over $\mathbf {Exc}(m-\frac {1}{2})$ . The resulting curve $\mathcal {C}(m)^\prime $ is m-stable, as $(m-1)$ -stability implies that at least one part $S_i$ has cardinality at least two, hence it induces a morphism:

$$\begin{align*}\rho_{m}\colon \overline{\mathcal{M}}_{1,n}\left(m-\frac{1}{2}\right)\to \overline{\mathcal{M}}_{1,n}(m).\end{align*}$$

We state our first result. We denote by $\mathbf {Ell}_m(m)$ the locus of curves in $\overline {\mathcal {M}}_{1,n}(m)$ with the worst possible singularity, that is an elliptic m-fold point; $\mathbf {Ell}_m(m)$ is the union of all the irreducible substacks $\mathbf {Ell}_{S}(m)\subset \overline {\mathcal {M}}_{1,n}(m)$ with $|S|=m$ .

We postpone the proof of this theorem to §2.7. Next, we describe the first three special cases. The result below recovers Inchiostro’s description of $\overline {\mathcal {M}}_{1,2}$ [Reference Inchiostro18].

Corollary 2.18. There is a diagram:

where $\rho _2$ is a blow-up with weights $(2,3,4)$ in the three points $\mathbf {Ell}_{i,j|k}(2)=:\mathbf {Tac}_k$ parametrising tacnodal curves, and $\rho _1$ is the blow-up with weights $(4,6)$ in the curve $\mathbf {Ell}_{[3]}(1)=:\mathbf {Cu}_3\simeq \overline {\mathcal {M}}_{0,4}$ parametrising cuspidal curves.

Corollary 2.19. There is a diagram:

where

• ○ $\rho _3$ is a blow-up with weights $(1,2,2,3)$ in the six points $\mathbf {Ell}_{h,i|j|k}(3)=:\mathbf {Tri}_{h,i}$ parametrising curves with planar triple points;

• ○ $\rho _2$ is the blow-up with weights $(2,3,4)$ in the four curves $\mathbf {Ell}_{h|i,j,k}(2)=:\mathbf {Tac}_h\simeq \overline {\mathcal {M}}_{0,4}$ parametrising tacnodal curves with a component marked only by $p_h$ , and in the three curves $\mathbf {Ell}_{h,i|j,k}(2)=:\mathbf {Tac}_{h,i}\simeq \mathbb {P}^1$ parametrising tacnodal curves with a component marked only by $p_h$ and $p_i$ ;

• ○ $\rho _{\frac {3}{2}}$ is the ordinary blow-up in the three surfaces $\mathbf {T}_{h,i|j,k}(1)\simeq \overline {\mathcal {M}}_{1,2}(1)$ of elliptic bridges;

• ○ $\rho _1$ is the blow-up with weights $(4,6)$ in the surface $\mathbf {Ell}_{[4]}(1)=:\mathbf {Cu}_4\simeq \overline {\mathcal {M}}_{0,5}$ of cuspidal curves.

2.6. Universal line bundles and tautological classes

From a flat family of projective, Gorenstein curves with smooth, distinct markings, we can define the following bundles over the base:

$$\begin{align*}\mathbb L_i(\pi\colon C\to S, p_1,\ldots,p_n) := p_i^*\omega_{\pi},\quad \mathcal{H}(\pi\colon C\to S, p_1,\ldots,p_n):=\pi_*\omega_{\pi}.\end{align*}$$

The first one is the cotangent line bundle at the i-th marking, and the second one is called the Hodge bundle. Their first Chern classes will be denoted by $\psi _{i,\pi }$ , respectively by $\lambda _\pi $ . A morphism between moduli spaces is often induced by a morphism between their universal curves, in which case we will be able to compare the bundles above; to avoid confusion we will decorate them as appropriate: for instance, we shall use $\mathcal {H}(m)$ and $\lambda _{(m)}$ for the universal m-stable curve over $\overline {\mathcal {M}}_{1,n}(m)$ (unless the number of markings plays a role).

Let us now specialise to the case of arithmetic genus one. In this case, the Hodge bundle has rank one, and the evaluation map $\mathcal {H}=\pi _*\omega _\pi \to p_i^*\omega _\pi =\mathbb L_i$ is generically an isomorphism. Indeed, we can be more precise: by pushing down the short exact sequence

$$\begin{align*}0\to \omega_\pi(-p_i)\to\omega_\pi\to\omega_{\pi|p_i}\to 0,\end{align*}$$

and noticing that $\operatorname {R}^1\pi _*\omega _\pi (-p_i)\to \operatorname {R}^1\pi _*\omega _\pi $ satisfies cohomology and base-change, we conclude that the evaluation map fails to be surjective precisely when $h^0(\mathcal {O}(p_i))=h^1(\omega (-p_i))>h^1(\omega )=1$ , that is, when $p_i$ does not lie on the minimal subcurve of genus one (it is special, in the terminology of Lekili and Polishchuk).

Proposition 2.20 [Reference Lekili and Polishchuk24, Lemma 1.1.1].

Let $(\pi \colon C\to S, p_1,\ldots ,p_n)$ be a pointed Gorenstein curve of genus one. Let $\Delta _{0,i}$ denote the locus in S where the i-th marking $p_i$ lies on a rational tail, and let us assume that $\Delta _{0,i}$ is a Cartier divisor in S, whose class we denote by $\delta _{0,i}$ . Then:

(2.3) $$ \begin{align} \psi_{i,\pi}=\lambda_{i,\pi}+\delta_{0,i}. \end{align} $$

In particular, over $\overline {\mathcal {M}}_{1,m+1}(m)$ , every $\mathbb L_i$ is identified with $\mathcal {H}(m)$ .Footnote ²

Let us now assume that we have two families $\pi _i\colon C_i \to S$ of pointed Gorenstein curves of genus one over a base S, related by a birational contraction $\rho \colon C_1\to C_2$ – an isomorphism generically on $C_2$ , and with connected fibres. Assume that the exceptional locus E of $\rho $ is a vertical Cartier divisor, that is, a subcurve of $C_1$ over a Cartier divisor $\Delta $ in the base (whose class we denote by $\delta $ ) such that $C_1$ is regular around $\Sigma :=E\cap \overline {C_{1|\Delta }\setminus E}$ . We shall compare the Hodge line bundles associated to $C_1$ and $C_2$ ; in order to do this, it is enough to assume that S is the spectrum of a discrete valuation ring. Denote by $\omega _i$ the relative dualising sheaf of $\pi _i$ , and let $\mathcal {H}_i=\pi _{i*}\omega _i$ be the Hodge line bundle associated to $C_i$ , whose class we denote by $\lambda _i$ . We recall Smyth’s definition of a balanced subcurve of genus one, borrowing the notation E and $\Sigma $ from above: the condition is that the chains of rational curves within E separating the nodes of $\Sigma $ from the core of E all have the same length (for a more general, not necessarily regular, one-parameter family, being balanced can be phrased in terms of the tropical distance from the core to the components adjacent E).

Proof. The first fact goes back to Knudsen’s thesis [Reference Knudsen21, Lemma 1.6]. The second one is basically the first claim of [Reference Smyth35, Prop. 3.6 and Cor. 3.7]: consider the short exact sequence

$$\begin{align*}0\to \omega_{1}\to\rho^*\omega_2\to\rho^*\omega_{2|\mathcal{E}}=\mathcal{O}_{\mathcal{E}}\to 0;\end{align*}$$

pushing it down along $\pi _1=\pi _2\circ \rho $ , and applying the projection formula (since $\rho $ is a birational contraction, we have $\rho _*\mathcal {O}_{C_1}=\mathcal {O}_{C_2}$ ), we obtain:

$$\begin{align*}0\to\mathcal{H}_1\to\mathcal{H}_2\to\mathcal{O}_S(\Delta)\to 0,\end{align*}$$

where right-exactness follows because the $\operatorname {R}^1\pi _{1,*}\omega _{C_i}$ , for $i=1,2$ , are isomorphic (by Serre duality, both of them can be identified with $\mathcal {O}_S$ ).

We will also use the following fact [Reference Smyth35, Proposition 3.4], comparing the $\lambda $ -class of a one-parameter family of $\ell $ -elliptic singularities with the $\psi $ -classes of its pointed normalisation:

2.7. Proof of Theorem 2.16

The locus of elliptic m-bridges $\mathbf {T}_m(m-1)$ decomposes into the disjoint union of $\mathbf {T}_S(m-1)$ , each of which is a closed regular substack of codimension $k=k(S)$ , with split normal bundle:

(2.4) $$ \begin{align} N_{\mathbf{T}_S/\overline{\mathcal{M}}_{1,n}(m-1)}=\bigoplus_{i=1}^k\mathbb L^\vee_{1,i}\boxtimes \mathbb L^\vee_{0,\star_i}. \end{align} $$

Now, recall from Proposition 2.20 that on the first factor, which is isomorphic to $\overline {\mathcal {M}}_{1,m}(m-1)$ , we may identify every cotangent line bundle $\mathbb L_{1,i}$ with the Hodge line bundle $\mathcal {H}(m-1)$ , so

(2.5) $$ \begin{align} N_{\mathbf{T}_S/\overline{\mathcal{M}}_{1,n}(m-1)}=\mathcal{H}(m-1)^\vee\boxtimes\left(\bigoplus_{i=1}^k \mathbb L^\vee_{0,\star_i}\right). \end{align} $$

Recall that $\mathbf {Ell}_{m}(m) \subset \overline {\mathcal {M}}_{1,n}(m)$ parametrises curves with an elliptic m-fold point: this locus decomposes into the disjoint union of the components $\mathbf {Ell}_{S}(m)$ with $\#S=m$ . We may similarly decompose the exceptional divisor of the blow-up $\rho _{m}\colon \overline {\mathcal {M}}_{1,n}\left (m-\frac {1}{2}\right )\to \overline {\mathcal {M}}_{1,n}(m)$ into a disjoint union indexed by partitions, and we obtain the following identification:

(2.6) $$ \begin{align} \mathbf{Exc}_S\left(m-\frac{1}{2}\right)\simeq \overline{\mathcal{M}}_{1,m}(m-1)\times\mathbb{P}_{\overline{\mathcal{M}}_{0,S}}\left(\bigoplus_{i=1}^k \mathbb L^\vee_{0,\star_i}\right), \end{align} $$

with normal bundle

$$\begin{align*}\mathcal N_{\mathbf{Exc}(m-\frac{1}{2})/\overline{\mathcal{M}}_{1,n}(m-\frac{1}{2})}= \mathcal{H}(m-1)^\vee \boxtimes \mathcal{O}(-1). \end{align*}$$

The morphism $\rho _{m}$ , restricted to $\mathbf {Exc}_S(m-\frac {1}{2})$ , coincides with the projection onto the second factor of (2.6). Thanks to Proposition 2.10, we can conclude by applying the criterion for smooth weighted blow-downs [Reference Arena, Di Lorenzo, Inchiostro, Mathur, Obinna and Pernice1, Theorem 1.1]. Finally, observe that given any weighted blow-up $Bl_Y X \to X$ , the exceptional divisor is (locally on the centre Y of the weighted blow-up) isomorphic to a product of the base with a weighted projective stack $\mathcal {P}(a_0,\ldots , a_m)$ . The weights of the blow-up are precisely the values $a_0,\ldots , a_m$ . They can be recovered from the orders of the automorphism groups of the torus-fixed points. Therefore, we can deduce the weights of $\rho _m$ by looking at the weights appearing in the description of $\overline {\mathcal {M}}_{1,m}(m-1)$ as a weighted projective stack. This concludes the proof of Theorem 2.16.

2.8. Fundamental classes and normal bundles

From Section 2.2 and Section 2.5 we know that we need to compute the fundamental class and the normal bundle of the elliptic m-fold loci $\mathbf {Ell}_{S}(m)$ in $\overline {\mathcal {M}}_{1,n}(m)$ . It is easy to perform such a calculation in $\widetilde {\mathcal {M}}_{1,m}$ . We would like to pull it back along the map forgetting $n-m$ markings. In general this is only a rational map, but for $n\leq 4$ it is actually regular. We defer a general discussion of forgetful maps to Appendix A.

We start from Lekili and Polishchuk’s description of the space of almost m-stable curves.

Proof. From the descriptions of Proposition 2.9, we deduce that $\widetilde {\mathcal {M}}_{1,m}$ is a vector bundle over $\mathcal {B}\mathbb {G}_{\mathrm {{m}}}$ . Therefore, by homotopy invariance of Chow groups we get

$$\begin{align*}A^*(\widetilde{\mathcal{M}}_{1,m+1}) \simeq A^*(\mathcal{B}\mathbb{G}_{\mathrm{{m}}}) \simeq \mathbb{Z}[t].\end{align*}$$

We also have $[\mathcal {Z}]=[\{0\}]_{\mathbb {G}_{\mathrm {{m}}}}$ , that is, the fundamental class of the zero section of a vector bundle over $\mathcal {B}\mathbb {G}_{\mathrm {{m}}}$ . The latter is equal to the top Chern class of the vector bundle itself. Using the fact that $c_i^{\mathbb {G}_{\mathrm {{m}}}}(V_d)=dt$ , we obtain the claimed results.

Finally, to prove that $c_1(\widetilde {\mathcal {H}}(m))=t$ , observe that a generator of the fibre of the Hodge line bundle over a point of $\widetilde {\mathcal {M}}_{1,m+1}$ is given by the global differential form $d(X/Y)$ , on which $\mathbb {G}_{\mathrm {{m}}}$ acts with weight $1$ . Here X and Y are inhomogeneous coordinates on the affine universal curve over $\widetilde {\mathcal {M}}_{1,m+1}$ , which is planar in the given range of m (cf. [Reference Lekili and Polishchuk24, Equations (1.1.3-4)]).

Consider the forgetful map $\overline {\mathcal {M}}_{1,n}(m)\dashrightarrow \overline {\mathcal {M}}_{1,n-1}(m)$ . Suppose that the n-th marking lies on the core, and that the latter has level $m+1$ . Then forgetting $p_n$ makes the curve m-unstable. The solution is to contract the core after sprouting the markings lying directly on it; this will increase the level, since there must be at least one rational tail containing at least two special points. The problem is that the contraction is not well-defined as soon as there are at least two rational tails, because the resulting singularity has a nontrivial crimping space. In order to resolve the indeterminacy, it is enough to blow up the loci $\mathbf {T}_S(m)$ , where S is a partition of $[n]$ into $m+1$ parts, of which at least two have $\lvert S_i\rvert \geq 2$ , and one is the singleton $\{n\}$ ; see [Reference Smyth36, §2.1]. Fortunately for us, in this paper we only need the cases $1\leq m<n\leq 4$ . Even for $n=4$ , there is not enough room for two rational tails and one marking on the core. So, the forgetful morphism $\overline {\mathcal {M}}_{1,n}(m)\dashrightarrow \overline {\mathcal {M}}_{1,n-1}(m)$ is well-defined in this range.

Finally, there always is a morphism $\overline {\mathcal {M}}_{1,m+1}(m)\to \widetilde {\mathcal {M}}_{1,m}$ . Indeed, $\overline {\mathcal {M}}_{1,m+1}(m)$ sits as an open inside $\widetilde {\mathcal {M}}_{1,m+1}$ , and the latter can be identified with the quotient of $\widetilde {\mathcal {U}}^{sns}_{1,m+1}$ by $\mathbb {G}_{\mathrm {{m}}}$ , see Section 2.4. Now, [Reference Lekili and Polishchuk24, Proposition 1.1.5] ensures that there always is a $\mathbb {G}_{\mathrm {{m}}}$ -equivariant morphism $\mathcal {U}^{sns}_{1,m+1}\to \mathcal {U}^{sns}_{1,m}$ , identifying the former with the affine universal curve over the latter.

Concretely, we may think of the morphism $\overline {\mathcal {M}}_{1,n}(m)\to \widetilde {\mathcal {M}}_{1,m}$ as follows. Consider the universal curve $C'$ over $\overline {\mathcal {M}}_{1,n}(m)$ , marked with the first m markings only. Rational tails and bridges made unstable by forgetting the last markings may be harmlessly contracted. Now, the core of $C'$ is m-unstable if and only if at least two of the first m markings belong to a rational tail. If this is the case, we blow up the core at its markings, and then contract it (by Smyth’s contraction lemma [Reference Smyth34, Lemma 2.13] or [Reference Ranganathan, Santos-Parker and Wise32, Proposition 3.7.6.1]). By stability, this will increase the core level, so we can reiterate until it becomes $\geq m$ . For $n\geq 4$ , we need to perform this operation only if $n=4$ and $m=2$ . Indeed, the following two conditions must be met:

1. 1. there must be a partition S having at least $m+1$ parts, and

2. 2. by restricting S to the subset $[m]=\{1,\ldots ,m\}$ we do not get the discrete partition.

For $n\leq 4$ , this happens if and only if $n=4$ and $m=2$ .

This is important, because contracting a genus one subcurve alters the Hodge bundle according to Proposition 2.21.

Moreover observe that, whenever a core contraction takes place, the resulting singularity is an elliptic $\ell $ -fold point with $\ell <m$ . So, the preimage of the m-fold point in $\widetilde {\mathcal {M}}_{1,m}$ remains away from the correction loci. We conclude that the normal bundle to the m-fold loci in $\overline {\mathcal {M}}_{1,n}(m)$ has the same expression as the normal bundle to the m-fold point in $\widetilde {\mathcal {M}}_{1,m}$ .

We summarise the previous discussion in the following:

Corollary 2.26. With the notations established in the corollaries of Theorem 2.16, we have:

where $\tau _{12}$ is the fundamental class of the locus of curves having a rational tail marked with $p_1$ and $p_2$ .

The class $\tau _{12}$ in the corollary above appears precisely because for $n=4$ and $m=2$ , in order to define the morphism to $\widetilde {\mathcal {M}}_{1,2}$ , we need to contract the elliptic core of the universal curve over $\mathbf {T}_{\{1,2\}}(2)$ . This affects the pullback of the Hodge bundle, as explained in Proposition 2.21.

3.1. The Chow ring of $\overline {\mathcal {M}}_{1,3}(2)$ and $\overline {\mathcal {M}}_{1,3}(1)$

We start from the following:

Lemma 3.1. Let $\lambda _{(2)}$ be the first Chern class of the Hodge line bundle $\mathcal {H}(2)$ on $\overline {\mathcal {M}}_{1,3}(2)$ . Then we have:

$$\begin{align*}A^*(\overline{\mathcal{M}}_{1,3}(2))=\mathbb{Z}[\lambda_{(2)}]/(12\lambda_{(2)}^4).\end{align*}$$

We now want to apply the weighted blow-up formula (2.1), leveraging the fact that $\rho _1\colon \overline {\mathcal {M}}_{1,3}(1)\to \overline {\mathcal {M}}_{1,3}(2)$ is a weighted blow-up centred at the three points $\mathbf {Tac}_i$ for $i=1,2,3$ (Corollary 2.18). As explained in §2.2, for this we need to compute the following:

1. 1. the fundamental class of the centre, that is, the three points $\mathbf {Tac}_{i}$ ;

2. 2. the $\mathbb {G}_{\mathrm {{m}}}$ -equivariant top Chern class of the normal bundle of $\mathbf {Tac}_{i}$ ;

3. 3. surjectivity and the kernel of the pullback homomorphism $A^*(\overline {\mathcal {M}}_{1,3}(2))\to A^*(\mathbf {Tac}_{i})$ .

By Corollary 2.26 we know that $[\mathbf {Tac}_1]+[\mathbf {Tac}_2]=24\lambda _{(2)}^3$ , hence by symmetry each $\mathbf {Tac}_i$ has class $12\lambda _{(2)}^3$ .

Point (3) is straightforward since $\mathbf {Tac}_i$ is a point for every i (Corollary 2.18). By the same token, the normal bundle is trivial with $\mathbb {G}_{\mathrm {{m}}}$ -action of weights $(2,3,4)$ :

(3.1) $$ \begin{align} c_3^{\mathbb{G}_{\mathrm{{m}}}}(\mathcal{N}_{\mathbf{Tac}_i}) = 24t^3. \end{align} $$

Let us proceed by blowing up one tacnodal point at a time: We start by computing the Chow ring of the weighted blow-up $\overline {\mathcal {M}}_{1,3}(2)'$ of $\overline {\mathcal {M}}_{1,3}(2)$ at the point $\mathbf {Tac}_1$ . The pullback homomorphism $A^*(\overline {\mathcal {M}}_{1,3}(2))\to A^*(\mathbf {Tac}_1)$ is surjective, and its kernel is generated by $\lambda _{(2)}$ . We may hence apply the simplified weighted blow-up formula (2.1):

$$ \begin{align*} A^*(\overline{\mathcal{M}}_{1,3}(2)') &= A^*(\overline{\mathcal{M}}_{1,3}(2))[\delta_1]/(\lambda_{(2)}\delta_1,12\lambda_{(2)}^3-24\delta_1^3). \end{align*} $$

where we denoted by $\delta _1$ the fundamental class of the exceptional divisor, representing curves with a node separating $p_1$ on the core from $p_2$ and $p_3$ on a rational tail.

When we blow up $\mathbf {Tac}_2$ (respectively $\mathbf {Tac}_3$ ) we get a similar result, with the only difference that now the kernel of the pullback homomorphism is generated by $\lambda _{(2)}$ and $\delta _1$ (respectively $\lambda _{(2)}$ , $\delta _1$ and $\delta _2$ ). In the end we obtain the following:

Lemma 3.3. The Chow ring of $\overline {\mathcal {M}}_{1,3}(1)$ can be written as follows:

$$\begin{align*}A^*(\overline{\mathcal{M}}_{1,3}(1))=\mathbb{Z}[\lambda_{(2)},\delta_1,\delta_2,\delta_3]/(12\lambda_{(2)}^4,\{\delta_i\delta_j\}_{i\neq j},\{\lambda_{(2)}\delta_i,-24\delta_i^3+12\lambda_{(2)}^3\}_{i=1,2,3}). \end{align*}$$

Remark 3.4. The first relation is now redundant, as for any i we can write

$$\begin{align*}12\lambda_{(2)}^4 = \lambda_{(2)} (-24\delta_i^3+12\lambda_{(2)}^3) + 24\delta_i^2 (\delta_i\lambda_{(2)}).\end{align*}$$

On $\overline {\mathcal {M}}_{1,3}(1)$ there are two families of curves, the universal $1$ -stable curve, and the $2$ -stable curve pulled back from $\overline {\mathcal {M}}_{1,3}(2)$ . Their Hodge line bundles are related by Proposition 2.21. Accordingly, we make a change of variable:

$$ \begin{align*}\lambda_{(1)}=\lambda_{(2)}-\delta_1-\delta_2-\delta_3,\end{align*} $$

where $\lambda _{(1)}$ is the first Chern class of the Hodge line bundle $\mathcal {H}(1)$ of the $1$ -stable curve. Moreover, recall that $\overline {\mathcal {M}}_{1,3}(1)$ is the same as Schubert’s stack of pseudo-stable curves.

Proposition 3.5. The Chow ring of the moduli stack of pseudo-stable $3$ -pointed curves of genus one is:

$$ \begin{align*} A^*(\overline{\mathcal{M}}_{1,3}^{ps})\simeq\mathbb{Z}[\lambda_{(1)},\delta_1,\delta_2,\delta_3]/I^{ps}_{1,3} \end{align*} $$

where the ideal of relations $I^{ps}_{1,3}$ is generated by:

$$ \begin{align*} &\delta_i^2+\lambda_{(1)}\delta_i=0 &\text{for }i=1,2,3,\\ &\delta_i\delta_j=0 &\text{for }i\neq j,\\ &24(\delta_i^3-\delta_j^3)=0 &\text{for }i\neq j,\\ &12(\lambda_{(1)}^3+\delta_1^3+\delta_2^3-\delta_3^3)=0. \end{align*} $$

3.2. The Chow ring of $\overline {\mathcal {M}}_{1,3}$

Recall from Corollary 2.18 that $\rho \colon \overline {\mathcal {M}}_{1,3}\to \overline {\mathcal {M}}_{1,3}(1)$ is a weighted blow-up of weights $(4,6)$ centred in the locus of cuspidal curves $\mathbf {Cu}$ . The exceptional divisor can be identified with the locus of unmarked elliptic tails $\Delta _{\emptyset }$ .

In order to apply the simplified weighted blow-up formula, we need the following ingredients:

1. 1. the surjectivity and kernel of the pullback homomorphism $A^*(\overline {\mathcal {M}}_{1,3}(1)) \to A^*(\mathbf {Cu})$ ;

2. 2. the fundamental class of $\mathbf {Cu}$ ;

3. 3. the $\mathbb {G}_{\mathrm {{m}}}$ -equivariant top Chern class of the normal bundle of $\mathbf {Cu}$ .

3.2.1. Pinching and pulling

We aim to study the Gysin pullback:

$$\begin{align*}\iota^*\colon A^*(\overline{\mathcal{M}}_{1,3}(1))\to A^*(\mathbf{Cu}). \end{align*}$$

Pinching (collapsing the tangent line) at the marking $\star $ identifies $\mathbf {Cu}$ with $\overline {\mathcal {M}}_{0,3\sqcup \star }$ , which in turn is isomorphic to $\mathbb {P}^1$ . Hence,

$$\begin{align*}A^*(\mathbf{Cu})=A^*(\overline{\mathcal{M}}_{0,3\sqcup\star})=A^*(\mathbb{P}^1)=\mathbb{Z}[h]/(h^2),\end{align*}$$

where any boundary point and any $\psi $ -class on $\overline {\mathcal {M}}_{0,3\sqcup \star }$ is identified with the generator h (see [Reference Kock22, §1.5]).

The boundary divisor $\delta _i$ intersects $\mathbf {Cu}$ transversely in one point, representing a cusp marked by $p_i$ with a rational tail marked by $p_j$ and $p_k$ : its class can therefore be identified with the generator of the Chow ring h; in particular, the pullback $\iota ^*$ is surjective. On the other hand, Proposition 2.22 gives us $\iota ^*\lambda _{(1)}= -\psi _{\star } = -h$ . Putting everything together, we obtain the following, which is ingredient (1) for applying the weighted blow-up formula.

Lemma 3.6. The pullback homomorphism $\iota ^*:A^*(\overline {\mathcal {M}}_{1,3}(1))\to A^*(\mathbf {Cu})$ is surjective and

$$\begin{align*}\ker(\iota^*) = (\lambda_{(1)}+\delta_1,\lambda_{(1)}+\delta_2,\lambda_{(1)}+\delta_3). \end{align*}$$

3.2.2. End of the computation

The normal bundle of $\mathbf {Cu}$ can be pulled back along $f\colon \overline {\mathcal {M}}_{1,3}(1) \to \widetilde {\mathcal {M}}_{1,1}$ , as explained in Proposition 2.25, and by Lemma 2.23 it is equal to

$$ \begin{align*}f^*( \widetilde{\mathcal{H}}(1)^{\otimes 4}\oplus \widetilde{\mathcal{H}}(1)^{\otimes 6})|_{\mathbf{Cu}}=\mathcal{H}(1)^{\otimes 4} \oplus \mathcal{H}(1)^{\otimes 6}|_{\mathbf{Cu}}.\end{align*} $$

This provides ingredient (3). Finally, ingredient (2) is provided to us by Corollary 2.26. After a straightforward computation, we obtain

$$ \begin{align*} A^*(\overline{\mathcal{M}}_{1,3})=\mathbb{Z}[\lambda_1,\delta_1,\delta_2,\delta_3,\delta_{\emptyset}]/ I^{\prime}_{1,3} \end{align*} $$

where $I^{\prime }_{1,3}$ is generated by the relations

$$ \begin{align*} &\delta_i\delta_j&\text{ for }i,j\in \{1,2,3\},\\ &\delta_i^2+\lambda_{(1)}\delta_i&\text{ for }i\in\{1,2,3\},\\ &12\lambda_{(1)}^2(\lambda_{(1)}+\delta_i+\delta_j-\delta_k)&\text{ for }i,j,k\in\{1,2,3\},\\ &\delta_{\emptyset}(\lambda_{(1)}+\delta_i)&\text{ for }i\in\{1,2,3\},\\ &24(\lambda_{(1)}-\delta_{\emptyset})^2.& \end{align*} $$

Proposition 2.21 gives us the change of variables $\lambda =\lambda _{(1)}-\delta _{\emptyset }$ , under which we get:

Theorem 3.7. The integral Chow ring of the moduli stack of $3$ -marked stable curves of genus $1$ is

$$ \begin{align*} A^*(\overline{\mathcal{M}}_{1,3})=\mathbb{Z}[\lambda,\delta_1,\delta_2,\delta_3,\delta_{\emptyset}]/ I_{1,3} \end{align*} $$

where $I_{1,3}$ is generated by ten relations in degree $2$ :

(1) $$ \begin{align} &24\lambda^2=0 & \end{align} $$

(2) $$ \begin{align}&\delta_{\emptyset}^2=-\lambda\delta_{\emptyset}-\delta_3\delta_{\emptyset} & \end{align} $$

(2+i) $$ \begin{align} &\delta_i^2=-\lambda\delta_i-\delta_3\delta_{\emptyset}, & i=1,2,3 \end{align} $$

(5+k) $$ \begin{align} &\delta_i\delta_j=0 , & \{i,j,k\}=\{1,2,3\} \end{align} $$

(7+j) $$ \begin{align} &\delta_{\emptyset}\delta_1=\delta_{\emptyset}\delta_j, & j=2,3 \end{align} $$

and by one extra (torsion) relation in degree $3$ , namely

(11) $$ \begin{align} &12(\lambda+\delta_{\emptyset})^2(\lambda+\delta_{\emptyset}+\delta_i+\delta_j-\delta_k)=0 & \end{align} $$

Remark 3.8. Tensoring with $\mathbb {Q}$ , we recover [Reference Belorousski5, Theorem 3.3.2].

4.1. The Chow ring of $\overline {\mathcal {M}}_{1,4}(3)$ and $\overline {\mathcal {M}}_{1,4}(2)$

Our starting point is the Chow ring of $\overline {\mathcal {M}}_{1,4}(3)$ , which can be easily computed using the fact that $\overline {\mathcal {M}}_{1,4}(3)\simeq \mathcal {P}(1,1,1,2,2)$ (Proposition 2.10 and the formula for the Chow ring of a weighted projective stack [Reference Arena, Obinna and Abramovich2, Theorem 3.12]).

Lemma 4.1. We have

$$\begin{align*}A^*(\overline{\mathcal{M}}_{1,4}(3))=\mathbb{Z}[\lambda_{(3)}]/(4\lambda_{(3)}^5)\end{align*}$$

where $\lambda _{(3)}$ is the first Chern class of the Hodge line bundle $\mathcal {H}(3)$ .

To compute the Chow ring of $\overline {\mathcal {M}}_{1.4}(2)$ , we leverage Corollary 2.19: there is a morphism

$$\begin{align*}\rho_2\colon \overline{\mathcal{M}}_{1,4}(2) \longrightarrow \overline{\mathcal{M}}_{1,4}(3) \end{align*}$$

which is a weighted blow-up of weights $2$ , $3$ and $4$ centred at the locus of elliptic curves with a triple point. To apply the weighted blow-up formula we need, as usual, the following ingredients:

1. 1. the surjectivity and kernel of the pullback homomorphism $A^*(\overline {\mathcal {M}}_{1,4}(3)) \to A^*(\mathbf {Tri}_{ij})$ ;

2. 2. the fundamental class of $\mathbf {Tri}_{ij}$ ;

3. 3. the $\mathbb {G}_{\mathrm {{m}}}$ -equivariant top Chern class of the normal bundle of $\mathbf {Tri}_{ij}$ .

The second ingredient is given to us by Corollary 2.26: the flat forgetful morphism $\overline {\mathcal {M}}_{1,4}(3)\to \widetilde {\mathcal {M}}_{1,3}$ pulls $[\mathcal {Z}]=12t^4$ back to $[\mathbf {Tri}_{14}]+[\mathbf {Tri}_{24}]+[\mathbf {Tri}_{34}]=12\lambda _{(3)}^4$ , so by the action of the symmetric group

$$\begin{align*}[\mathbf{Tri}_{ij}] = 4\lambda_{(3)}^4 \text{ for all }\{i,j\}\subseteq[4]. \end{align*}$$

Being points, the normal bundle is trivial (with $\mathbb {G}_{\mathrm {{m}}}$ -action of weights $(2,3,4)$ ), and restriction is surjective on Chow groups. From the blow-up formula (2.1) we get:

(4.1) $$ \begin{align} A^*(\overline{\mathcal{M}}_{1,4}(2))=\mathbb{Z}[\lambda_{(3)},\{\tau_{ij}\}_{\{i,j\}\subset [4]}]/I_{1,4}(2)', \end{align} $$

where $\tau _{ij}$ is the class of the exceptional divisor $\mathbf {T}_{ij}$ , which can be identified with the locus of curves with a node separating the markings $h,k$ on the core from the markings $i,j$ on a rational tailFootnote ³, and $I_{1,4}(2)'$ is generated by

(1–6) $$ \begin{align} &\lambda_{(3)}\tau_{ij}=0, & \{i,j\}\subset [4], \end{align} $$

(7–21) $$ \begin{align} &\tau_{ij}\tau_{hk}=0, & \{i,j \}\neq \{h,k\}, \end{align} $$

(22–27) $$ \begin{align}&12\tau_{ij}^4+4\lambda_{(3)}^4=0. & \end{align} $$

The relation $4\lambda _{(3)}^5$ is redundant and we thus omit it.

Proposition 2.21 yields the change of variables $\lambda _{(3)}=\lambda _{(2)}+\sum _{1\leq i<j\leq 4}\tau _{ij}$ . After performing this substitution, we obtain the following.

Proposition 4.2. The Chow ring of the moduli stack of $4$ -marked $2$ -stable curves of genus one is:

$$ \begin{align*} A^*(\overline{\mathcal{M}}_{1,4}(2))=\mathbb{Z}\left[\lambda_{(2)},\{\tau_{ij}\}_{\{i,j\}\subset [4]}\right]/I_{1,4}(2) \end{align*} $$

where the ideal of relations $I_{1,4}(2)$ is generated by:

(1–6) $$ \begin{align} \lambda_{(2)}\tau_{ij}+\tau_{ij}^2, &&\{i,j\}\subset [4], \end{align} $$

(7–21) $$ \begin{align} \tau_{ij}\tau_{hk}, &&\{ i,j \}\neq \{h,k\}, \end{align} $$

(22–27) $$ \begin{align} 12\tau_{ij}^4 + 4(\lambda_{(2)}^4-\sum_{r,s}\tau_{rs}^4) && \end{align} $$

The rational Chow ring of the moduli space $\overline {M}_{1,4}(2)$ has Hilbert series $1+7t+7t^2+7t^3+t^4$ .

4.2. The Chow ring of $\overline {\mathcal {M}}_{1,4}(1)$

We distinguish the tacnodal loci in $\overline {\mathcal {M}}_{1,4}(2)$ into two types: those separating one marking i from the other three (there are four of them, call them $\mathbf {Tac}_i$ ), and those separating two markings $i,j$ from the other two (there are three, call them $\mathbf {Tac}_{ij}$ ). The birational map $\overline {\mathcal {M}}_{1,4}(1)\dashrightarrow \overline {\mathcal {M}}_{1,4}(2)$ can be resolved by a suitable blow-up (Corollary 2.19):

Here $\rho _2$ is a blow-up of weights $(2,3,4)$ in the tacnodal locus. We denote the classes of the exceptional divisors by $\beta $ in Proposition 4.7 below. The morphism $\rho _{\frac {3}{2}}$ is an ordinary blow-up of the elliptic bridges $\mathbf {T}_{\{ij\},\{hk\}}$ , whose exceptional divisors can be identified with those over $\mathbf {Tac}_{ij}$ ; the loci $\mathbf {T}_{\{ij\},\{hk\}}$ are thus flipped. In particular, $\beta _{ij}$ pushes forward to $0$ along $\rho _{\frac {3}{2}}$ , while $\beta _i$ does not.

We will first compute the Chow ring of $\overline {\mathcal {M}}_{1,4}(\frac {3}{2})$ via the weighted blow-up formula, and then identify the Chow ring of $\overline {\mathcal {M}}_{1,4}(1)$ as a subring of the former.

4.2.1. The Chow ring of $\overline {\mathcal {M}}_{1,4}(\frac {3}{2})$

We need the following ingredients:

1. 1. the surjectivity and kernel of the pullback homomorphism to $A^*(\mathbf {Tac}_{ij})$ and $A^*(\mathbf {Tac}_{i})$ ;

2. 2. the fundamental classes of $\mathbf {Tac}_{ij}$ and $\mathbf {Tac}_i$ ;

3. 3. the $\mathbb {G}_{\mathrm {{m}}}$ -equivariant top Chern class of the normal bundle of $\mathbf {Tac}_{ij}$ and $\mathbf {Tac}_i$ .

For the first point, we need to compute the pullback of the generators of $A^*(\overline {\mathcal {M}}_{1,4}(2))$ in $A^*(\mathbf {Tac}_i)$ (respectively $A^*(\mathbf {Tac}_{ij})$ ). Let $\varphi _i\colon \mathbb {P}^1\simeq \overline {\mathcal {M}}_{0,\{\star ,h,j,k\}}\to \overline {\mathcal {M}}_{1,4}(2)$ be the pinching morphism, defined as follows: first, consider the auxiliary stack $\mathcal {X}$ whose objects are triples $(C\to S,D\to S,\alpha )$ , where

1. (i) $C\to S$ is a stable curve of genus zero with markings indexed by $\star $ , h, j and k,

2. (ii) $D\to S$ is a smooth genus zero curve with two markings indexed by $\star '$ and i, and

3. (iii) $\alpha \colon (\mathfrak {m}_{\star }/\mathfrak {m}^2_{\star })^\vee \simeq (\mathfrak {m}_{\star '}/\mathfrak {m}^2_{\star '})^\vee $ is an isomorphism.

There is a natural morphism $\mathcal {X} \to \overline {\mathcal {M}}_{1,4}(2)$ whose image is $\mathbf {Tac}_i$ : this morphism is given by attaching $C\to S$ and $D\to S$ at the markings indexed by $\star $ and $\star '$ by gluing the two tangent spaces via $\alpha $ . On the other hand, the projection $\mathcal {X} \to \overline {\mathcal {M}}_{0,\{\star , h,j,k\}}$ is an isomorphism: indeed, the fibres consist of the choice of a smooth $2$ -marked genus zero curve $D\to S$ and an isomorphism $\alpha $ . There is a unique isomorphism class of such Ds, with automorphism group isomorphic to $\mathbb {G}_{\mathrm {{m}}}$ . The latter acts with weight one on the tangent space at $\star '$ , and hence it acts freely and transitively on the set of isomorphisms $\alpha $ . By composing the inverse of the projection with the gluing map, we obtain the morphism $\varphi _i\colon \overline {\mathcal {M}}_{0,\{\star ,h,j,k\}}\to \overline {\mathcal {M}}_{1,4}(2)$ with image $\mathbf {Tac}_i$ .

Similarly, let $\varphi _{ij}\colon \mathbb {P}^1\to \overline {\mathcal {M}}_{1,4}(2)$ with image $\mathbf {Tac}_{ij}$ be the crimping morphism: the pointed normalisations of these tacnodal curves have no moduli, since they consist of two $3$ -pointed rational curves $C\to S$ and $D\to S$ , marked with $\{i,j,\star \}$ and $\{h,k,\star '\}$ respectively. Let $\mathbb {P}^1=\mathbb {P}(T_{\star }C \oplus T_{\star '}D)$ be the compactified moduli space of attaching data: the points away from $0$ and $\infty $ correspond to the choice of an isomorphism $(\mathfrak {m}_{{\star }}/\mathfrak {m}_{{\star }}^2)^\vee \simeq (\mathfrak {m}_{{\star '}}/\mathfrak {m}_{{\star '}}^2)^\vee $ ; over $0$ and $\infty $ , a rational tail is sprouted, supporting the markings $\{i,j\}$ and $\{h,k\}$ respectively, so the automorphism group of the curve acts transitively on the attaching data, as in the previous paragraph. More precisely, we blow up $C\times \mathbb {P}^1$ at $(\star ,0)$ and $D\times \mathbb {P}^1$ at $(\star ',\infty )$ , and then glue the proper transforms of $\star $ and $\star '$ according to their common projection in $\mathbb {P}^1$ .

Therefore, since both $\mathbf {Tac}_i$ and $\mathbf {Tac}_{ij}$ are isomorphic to $\mathbb {P}^1$ , their Chow ring can be identified with $\mathbb {Z}[h]/(h^2)$ .

Lemma 4.3. Fix $1\leq i<j\leq 4$ , and let $\mathbf {T}_{ij}$ denote the locus of curves with a rational tail marked by $i,j$ inside $\overline {\mathcal {M}}_{1,4}(2)$ . Then:

1. 1. $\int _{\mathbf {Tac}_\bullet }\varphi _\bullet ^*\lambda _{(2)}=-1$ for both $\bullet =i$ and $\bullet =ij$ ;

2. 2. for $1\leq h<k\leq 4$ , $\mathbf {Tac}_{ij}$ intersects the divisor $\mathbf {T}_{hk}$ if and only if $\{h,k\}=\{i,j\}$ or $[4]\setminus \{i,j\}$ , in which cases they intersect transversely at one point;

3. 3. $\mathbf {Tac}_i$ intersects the divisor $\mathbf {T}_{hk}$ if and only if $\{h,k\}\subset [4]\setminus \{i\}$ , in which cases they intersect transversely at one point.

In particular, the kernel of the Gysin pullback to $\mathbf {Tac}_i$ and $\mathbf {Tac}_{ij}$ is determined as follows:

$$ \begin{align*} \ker(A^*(\overline{\mathcal{M}}_{1,4}(2)) \longrightarrow A^*(\mathbf{Tac}_{i})) &= (\{\tau_{ij}\}_{j\neq i}, \{\lambda_{(2)} + \tau_{hk}\}_{h,k\neq i},\lambda_{(2)}^2 ). \\ \ker(A^*(\overline{\mathcal{M}}_{1,4}(2)) \longrightarrow A^*(\mathbf{Tac}_{ij})) &= (\{\tau_{ih}\}_{h\neq j}, \{\tau_{jk}\}_{k\neq i},\lambda_{(2)} + \tau_{hk}, \lambda_{(2)} + \tau_{ij},\lambda_{(2)}^2 ). \end{align*} $$

Proof. Point (1) is Proposition 2.22, the others follow from the deformation theory of nodes.

Now, we want to determine the classes of $\mathbf {Tac}_i$ and $\mathbf {Tac}_{ij}$ in $A^3(\overline {\mathcal {M}}_{1,4}(2))$ in terms of $\lambda _{(3)}$ and $\tau $ ’s. From Corollary 2.26 we know that:

(4.4) $$ \begin{align} [\mathbf{Tac}_1] + [\mathbf{Tac}_2] + [\mathbf{Tac}_{13}] + [\mathbf{Tac}_{14}] = f_{4,2}^*[\mathcal{Z}] = 24(\lambda_{(2)}+\tau_{12})^3=24\lambda_{(2)}^3+24\tau_{12}^3. \end{align} $$

We exploit the $S_4$ -action (by permuting the markings) to simplify the intersection-theoretic calculations. As an $S_4=S(1,2,3,4)$ -module, $A^3(\overline {\mathcal {M}}_{1,4}(2))$ can be identified with $\operatorname {triv}(\ell )\oplus \Lambda ^2\mathbb C^4_{\text {std}}$ , the former with basis $\lambda _{(2)}^3=:\ell $ , and the latter with basis $\tau _{ij}^3=:e_{ij}$ . Note that $[\mathbf {Tac}_1]$ is invariant under $S_3=S(2,3,4)$ , hence its class can be written as:

$$\begin{align*}[\mathbf{Tac}_1]=u_1\ell+a_1(e_{12}+e_{13}+e_{14})+b_1(e_{23}+e_{24}+e_{34}).\end{align*}$$

Similarly

$$\begin{align*}[\mathbf{Tac}_2]=u_2\ell+a_2(e_{12}+e_{23}+e_{24})+b_2(e_{13}+e_{14}+e_{14}).\end{align*}$$

Note that $[\mathbf {Tac}_{13}]$ is invariant under $S_2\times S_2=S(1,3)\times S(2,4)$ , hence its class can be written as:

$$\begin{align*}[\mathbf{Tac}_{13}]=v_1\ell+c_1e_{13}+d_1e_{24}+f_1(e_{12}+e_{14}+e_{23}+e_{34}).\end{align*}$$

Similarly

$$\begin{align*}[\mathbf{Tac}_{14}]=v_2\ell+c_2e_{14}+d_2e_{23}+f_2(e_{12}+e_{13}+e_{24}+e_{34}).\end{align*}$$

By the $S_4$ -action we see that

$$ \begin{align*}u_1=u_2=:u,v_1=v_2=:v,a_1=a_2=:a, b_1=b_2=:b,c_1=d_1=c_2=d_2=:c,f_1=f_2=:f.\end{align*} $$

Grouping the coefficients of $e_{ij}$ , Equation (4.4) reduces to the following linear system:

$$\begin{align*}\begin{pmatrix} 2 & 0 & 0 & 2 \\ 1 & 1 & 1 & 1 \\ 0 & 2 & 0 & 2 \end{pmatrix} \begin{pmatrix} a\\b\\c\\f\end{pmatrix}= \begin{pmatrix} 24\\0\\0\end{pmatrix}\end{align*}$$

Furthermore, by intersecting $\mathbf {Tac}_i$ with $\tau _{ij}$ we obtain $u=a$ : indeed, we know from Lemma 4.3 that $\tau _{ij}$ is in the kernel of the pullback to $\mathbf {Tac}_i$ , hence $[\mathbf {Tac}_i]\cdot \tau _{ij}=0$ ; on the other hand, by Proposition 4.2 we have $\tau _{ij}\tau _{hk}=0$ unless $\{i,j\}=\{h,k\}$ , and $\tau _{ij}\lambda _{(2)}=-\tau _{ij}^2$ . Putting everything together, we deduce that $[\mathbf {Tac}_i]\cdot \tau _{ij}=(a-u)\tau _{ij}^4$ , hence $a-u=0$ because $A^4(\overline {\mathcal {M}}_{1,4}(2))\otimes \mathbb {Q}\simeq \mathbb {Q}\cdot \tau _{ij}^4$ .

With a similar argument, by intersecting $\mathbf {Tac}_{ij}$ with $\tau _{ik}$ we obtain $v=f$ . We need one more equation, which we are going to obtain by computing the pullback of $\lambda _{(3)}$ from $\overline {\mathcal {M}}_{1,4}(3)$ to $\mathbf {Tac}_i$ : this will determine u quite directly. Recall the morphism $\rho _3\colon \overline {\mathcal {M}}_{1,4}(2)\to \overline {\mathcal {M}}_{1,4}(3)$ .

Lemma 4.4. The composition $\rho _3\circ \varphi _i:\mathbf {Tac}_i\simeq \overline {\mathcal {M}}_{0,4}\to \overline {\mathcal {M}}_{1,4}(3)$ maps the three special points to the $3$ -elliptic loci $\mathbf {Tri}_{jk}$ , $i\notin \{j,k\}\subset [4]$ . The following hold:

(4.5) $$ \begin{align} \int_{\mathbf{Tac}_i}\varphi_i^*\psi_j&=1, \end{align} $$

(4.6) $$ \begin{align} \int_{\mathbf{Tac}_i}\varphi_i^*\lambda_{(2)}&=-1, \end{align} $$

(4.7) $$ \begin{align} \int_{\mathbf{Tac}_i}\varphi_i^*\rho_3^*\lambda_{(3)}&=2.\end{align} $$

Therefore, $u=8$ and $v=4$ .

Proof. Equation (4.5) is a genus zero calculation, since crimping happens away from the marking.

Equation (4.6) follows from Proposition 2.22 and a genus zero computation.Footnote ⁴

From Proposition 2.21 it follows that $\int _{\mathbf {Tac}_i}\varphi _i^*\rho _3^*\lambda _{(3)}=\int _{\mathbf {Tac}_i}\varphi _i^*\lambda _{(2)}+3$ , since the universal $3$ -stable family is obtained by sprouting the lonely markings and contracting the tacnode over the three special points. In view of this, Equation (4.7) follows from Equation (4.6).

We obtain $2u+2v=24$ by looking at the coefficients of $\lambda _{(2)}^3$ in Equation (4.4), hence v is determined by u. Now observe that, because of the relations $\lambda _{(3)}\tau _{ij}=0$ (see (4.1)) and $\lambda _{(3)}=\lambda _{(2)}+\sum \tau _{ij}$ (see Proposition 2.21), the projection formula gives

$$\begin{align*}\rho_{3,*}(\rho_3^*\lambda_{(3)} \cdot [\mathbf{Tac}_{i}]) = u\lambda_{(3)}^4.\end{align*}$$

On the other hand, we deduce from Equation (4.7) that

$$\begin{align*}\rho_{3,*}(\rho_3^*\lambda_{(3)} \cdot [\mathbf{Tac}_{i}]) = \rho_{3,*}\rho_3^*\varphi_{i,*}\varphi_i^*\lambda_{(3)} = \rho_{3,*}\varphi_{i,*}(2[\text{pt}]) = 2[\mathbf{Tri}_{jk}] = 8\lambda_{(3)}^4.\\[-37pt] \end{align*}$$

We may now solve the linear system above and find $u=a=-c=8,\ v=-b=f=4$ .

Proposition 4.5. The classes of the tacnodal loci are

$$ \begin{align*} [\mathbf{Tac}_i]&=8\lambda_{(2)}^3-4(\tau_{hj}^3+\tau_{hk}^3+\tau_{jk}^3) +8(\tau_{ih}^3+\tau_{ij}^3+\tau_{ik}^3),\\ [\mathbf{Tac}_{ij}]&=4\lambda_{(2)}^3-8(\tau_{ij}^3+\tau_{hk}^3)+4(\tau_{ih}^3+\tau_{ik}^3+\tau_{jh}^3+\tau_{jk}^3). \end{align*} $$

The equivariant normal bundle is given by Lemma 2.23. We can now compute the Chow ring of $\overline {\mathcal {M}}_{1,4}(\frac {3}{2})$ by applying the weighted blow-up formula (2.1).

Proposition 4.6. The Chow ring of the moduli stack of aligned $4$ -marked $1$ -stable curves of genus $1$ is generated by the symbols $\lambda _{(2)},\tau _{ij},\beta _i,\beta _{ij}=\beta _{hk}$ (where $[4]=\{i,j,h,k\}$ ), subject to the following relations:

$$ \begin{align*} &(1-6) &&\lambda_{(2)}\tau_{ij}+\tau_{ij}^2, &&\{i,j\}\subset [4],\\&(7-21) &&\tau_{ij}\tau_{hk}, &&\{h,k\}\neq \{i,j\},\\&(22-27) &&12\tau_{ij}^4 + 4(\lambda_{(2)}^4-\sum_{r,l}\tau_{rl}^4),&& \\&(28-39) &&\beta_i\tau_{ij} &&j\in[4]\setminus\{i\},\\&(40-51) &&\beta_i(\lambda_{(2)}+\tau_{jk}), &&j,k\in[4]\setminus\{i\}, \\&(52-57) &&\beta_i\beta_j, &&j\in[4]\setminus\{i\}, \\&(58-61) &&24\beta_i(3\lambda_{(2)}\beta_i-\beta_i^2)+[\mathbf{Tac}_i],&& \\& (62-67) &&\beta_{ij}\tau_{ih}, &&h\in[4]\setminus\{i,j\}, \\& (68-73) &&\beta_{ij}(\lambda_{(2)}+\tau_{ij}), && \\& (74-85) &&\beta_{ij}\beta_h, &&h\in[4], \\& (86-91) &&\beta_{ij}\beta_{hk},&&\{h,k\}\neq\{i,j\},\\& (92-94) &&24\beta_{ij}(3\lambda_{(2)}\beta_{ij}-\beta_{ij}^2)+[\mathbf{Tac}_{ij}]. && \end{align*} $$

The Hilbert series of the rational Chow ring of the moduli space $\overline {M}_{1,4}(\frac {3}{2})$ is equal to $1+14t+21t^2+14t^3+t^4$ .

According to Proposition 2.21, we make the substitution $\lambda _{(1)}=\lambda _{(2)}-\sum _i\beta _i-\sum _{i,j}\beta _{ij}$ .

Proposition 4.7. The Chow ring of $\overline {\mathcal {M}}_{1,4}(\frac {3}{2})$ is generated by $\lambda _{(1)},\tau _{ij},\beta _i,\beta _{ij}=\beta _{hk}$ with:

$$ \begin{align*} &(1-6) &&\tau_{ij}(\lambda_{(1)}+\beta_h+\beta_k+\beta_{ij}+\tau_{ij}), &&\{i,j\}\subset [4],\\ &(7-21) &&\tau_{ij}\tau_{hk}, &&\{h,k\}\neq \{i,j\},\\ &(22-27) &&12\tau_{ij}^4 + 4((\lambda_{(1)}+\sum_i\beta_i+\sum_{i,j}\beta_{ij})^4-\sum_{r,l}\tau_{rl}^4),&& \\ &(28-39) &&\beta_i\tau_{ij} &&j\in[4]\setminus\{i\},\\ &(40-51) &&\beta_i(\lambda_{(1)}+\beta_i+\tau_{jk}), &&j,k\in[4]\setminus\{i\}, \\ &(52-57) &&\beta_i\beta_j, &&j\in[4]\setminus\{i\}, \\ &(58-61) &&24\beta_i(3\lambda_{(1)}\beta_i+2\beta_i^2)+[\mathbf{Tac}_i],&& \\ & (62-67) &&\beta_{ij}\tau_{ih}, &&h\in[4]\setminus\{i,j\}, \\ & (68-73) &&\beta_{ij}(\lambda_{(1)}+\beta_{ij}+\tau_{ij}), && \\ & (74-85) &&\beta_{ij}\beta_h, &&h\in[4], \\ & (86-91) &&\beta_{ij}\beta_{hk},&&\{h,k\}\neq\{i,j\},\\ & (92-94) &&24\beta_{ij}(3\lambda_{(1)}\beta_{ij}+2\beta_{ij}^2)+[\mathbf{Tac}_{ij}]. && \end{align*} $$

4.2.2. The Chow ring of $\overline {\mathcal {M}}_{1,4}(1)$

The pullback homomorphism

$$\begin{align*}\rho_{\frac{3}{2}}^*:A^*\left(\overline{\mathcal{M}}_{1,4}(1)\right) \longrightarrow A^*\left(\overline{\mathcal{M}}_{1,4}\left(\frac{3}{2}\right)\right) \end{align*}$$

is injective [Reference Fulton16, Proposition 6.7(b)], hence if we know a set of generators of $A^*(\overline {\mathcal {M}}_{1,4}(1))$ , we can deduce a full presentation by looking at the subring of $A^*(\overline {\mathcal {M}}_{1,4}(\frac {3}{2}))$ generated by their pullback. We recall the following useful lemma.

Lemma 4.8. Let $\iota \colon Z\hookrightarrow X$ be a regularly embedded subvariety of a smooth ambient space. If Gysin pullback along $\iota $ is surjective, then $\operatorname {im}(\iota _*)$ is generated as an ideal by the fundamental class $[Z]$ .

Proof. If $\alpha =\iota ^*\widetilde \alpha $ , the projection formula implies that $\iota _*\alpha =\widetilde \alpha \cdot [Z]$ .

We recall our running convention: for $S\subseteq [4]$ , the boundary divisor $\Delta (1)_S\subset \overline {\mathcal {M}}_{1,4}(1)$ is the substack of curves with a separating node such that the markings $p_i$ , $i\in S$ , lie on the genus one side. Its fundamental class is denoted by $\delta _{S}$ . On the other hand, we have denoted by $\mathbf {T}_{ij}$ (and $\tau _{ij}$ ) the locus (and its class) of rational tails marked by i and j in $\overline {\mathcal {M}}_{1,4}(2)$ .

Proposition 4.9. The Chow ring of $\overline {\mathcal {M}}_{1,4}(1)$ is generated by $\lambda _{(1)}$ , $\{\delta _{i}\}_{i\in [4]}$ and $\{\delta _{ij}\}_{\{i,j\}\subset [4]}$ . We have:

$$ \begin{align*}\rho_{\frac{3}{2}}^*(\lambda_{(1)}) = \lambda_{(1)},\quad \rho_{\frac{3}{2}}^*(\delta_i)=\beta_i, \quad \rho_{\frac{3}{2}}^*(\delta_{ij})=\tau_{hk}+\beta_{hk}.\end{align*} $$

Proof. The second statement is a simple computation of the total transform of $\Delta (1)_i$ and $\Delta (1)_{ij}$ .

For proving the first statement, let $\mathbf {B}(\frac {3}{2})=\cup \mathbf {B}_{ij}$ be the exceptional divisor of $\rho _{\frac {3}{2}}$ , where $\mathbf {B}_{ij}$ is the exceptional divisor of $\rho _{2}$ that lies over $\mathbf {Tac}_{ij}$ . We claim that $A^*(\overline {\mathcal {M}}_{1,4}(\frac {3}{2})\smallsetminus \mathbf {B}(\frac {3}{2}))$ is generated by $\lambda _{(1)}$ , $\{\beta _i\}_{i\in [4]}$ and $\{\tau _{ij}\}_{\{i,j\}\subset [4]}$ . This claim follows from the previous lemma and the surjectivity of the pullback homomorphism for every $i,j$ :

$$ \begin{align*}\iota^*_{ij}:A^*\left(\overline{\mathcal{M}}_{1,4}\left(\frac{3}{2}\right)\right)\longrightarrow A^*(\mathbf{B}_{ij}).\end{align*} $$

Recall the isomorphism:

$$ \begin{align*}\mathbf{B}_{ij}\simeq \overline{\mathcal{M}}_{1,2}(1)\times\mathbb{P}^1\simeq \mathcal{P}(2,3,4)\times\mathbb{P}^1,\end{align*} $$

So, the Chow ring of $\mathbf {B}_{ij}$ is generated (as a ring) by the two hyperplane classes $h_{\mathcal {P}}$ and $h_{\mathbb {P}^1}$ . We have $\iota _{ij}^*\beta _{ij}=-h_{\mathbb {P}^1}$ and $\iota _{ij}^*(\rho _{\frac {3}{2}}^*\lambda _{(1)}) = h_{\mathcal {P}}$ , so $\iota _{ij}^*$ is surjective.

Observe that $\rho _{\frac {3}{2}}$ induces an isomorphism of $\overline {\mathcal {M}}_{1,4}(\frac {3}{2})\smallsetminus \mathbf {B}(\frac {3}{2})$ with $\overline {\mathcal {M}}_{1,4}(1)\smallsetminus \mathbf {EB}(1)$ , where $\mathbf {EB}(1)=\cup \mathbf {EB}_{ij}$ is the codimension two substack of elliptic bridges, and $\mathbf {EB}_{ij}\simeq \mathcal {P}(2,3,4)$ , whose Chow ring is generated by the restriction of $\lambda _{(1)}$ . Moreover, the locus $\mathbf {EB}_{ij}$ is the transversal intersection of the divisors $\Delta (1)_{ij}$ and $\Delta (1)_{hk}$ , thus $[\mathbf {EB}_{ij}]=\delta _{ij}\delta _{hk}$ . We conclude that $A^*(\overline {\mathcal {M}}_{1,4}(1))$ is generated by $\lambda _{(1)}$ , $\{\delta _{i}\}_{i\in [4]}$ , $\{\delta _{ij}\}_{\{i,j\}\subset [4]}$ as a ring.

Corollary 4.10. $A^*(\overline {\mathcal {M}}_{1,4}(1))$ is generated by $\lambda _{(1)}$ , $\{\beta _{i}\}_{i\in [4]}$ , $\{\beta _{ij}+\tau _{ij}\}_{\{i,j\}\subset [4]}$ as a subring of $A^*(\overline {\mathcal {M}}_{1,4}(\frac {3}{2}))$ .

We find a presentation of $A^*(\overline {\mathcal {M}}_{1,4}(1))$ by eliminating the remaining variables from $A^*(\overline {\mathcal {M}}_{1,4}(\frac {3}{2}))$ . The following has been produced by Sage [38]; we do not know if it is a minimal presentation.

Proposition 4.11. The Chow ring of the moduli stack of $4$ -marked pseudo-stable curves of genus one is:

$$\begin{align*}A^*(\overline{\mathcal{M}}_{1,4}^{ps})=\mathbb{Z}[\lambda_{(1)},\{\delta_{ij}\}_{\{i,j\}\subset [4]},\{\delta_i\}_{i\in [4]}]/I_{1,4}^{ps}\end{align*}$$

where $I_{1,4}^{ps}$ is generated by the following relations:

$$ \begin{align*} &(1-6) &&\delta_i\delta_j, &&i\neq j,\\&(7-18) &&\delta_{ij}\delta_{ih}, &&i\neq j\neq h, \\&(19-30) &&\delta_{i}\delta_{hj}, && i\neq j\neq h, \\&(31-42) &&\delta_{i}(\delta_{ij}-\delta_{ih}), &&i\neq j \neq h, \\&(43-48) &&\delta_{ij}(\lambda_{(1)}+\delta_{ij}+\delta_{i}+\delta_{j}), &&i\neq j, \\&(49-60) &&\delta_i(\lambda_{(1)}+\delta_{ij}+\delta_i), &&i \neq j, \\&(61-72) &&\delta_i\delta_{ij}^2, &&i\neq j \\&(73-75) &&12(\delta_{ij}^3-\delta_{ik}^3-\delta_{jh}^3+\delta_{hk}^3), &&\{i,j,h,k\}=[4],\\&(76-87) &&12(2\delta_{i}^3+6\delta_{i}^2\delta_{ik}+\delta_{ik}^3-\delta_{ik}\delta_{jh}^2-\delta_{ih}\delta_{jk}^2-\delta_{jk}^3-\delta_{ij}\delta_{hk}^2-\delta_{hk}^3), &&\{i,j,h,k\}=[4],\\&(88-93) && f_{ij} \text{ (see below)}, &&\{i,j,h,k\}=[4], \end{align*} $$

$$ \begin{align*}f_{ij}&=4(\lambda_{(1)}^3 + \delta_{i}^3 + \delta_{j}^3 + \delta_{h}^3 + \delta_{k}^3 + \delta_{ij}^3 + \delta_{ih}^3 \\& \quad + 3\delta_{i}^2\delta_{ik} - 2\delta_{ik}^3 - \delta_{ik}\delta_{jh}^2 - 2\delta_{jh}^3 + 3\delta_{j}^2\delta_{jk} - \delta_{ih}\delta_{jk}^2 + \delta_{jk}^3 + 3\delta_{h}^2\delta_{hk} + 3\delta_{k}^2\delta_{hk} - \delta_{ij}\delta_{hk}^2 + \delta_{hk}^3). \end{align*} $$

The Hilbert series of the rational Chow ring of $\overline {M}_{1,4}^{ps}$ is equal to $1+11t+18t^2+11t^3+t^4$ .

4.3. The Chow ring of $\overline {\mathcal {M}}_{1,4}$

We know from Corollary 2.19 that $\overline {\mathcal {M}}_{1,4}$ is a weighted blow-up of $\overline {\mathcal {M}}_{1,4}(1)$ centred at $\mathbf {Cu}$ , which is the closed substack of cuspidal curves. As usual, we need the following three ingredients in order to apply the weighted blow-up formula for Chow rings:

1. 1. surjectivity and the kernel of the pullback homomorphism $\iota ^*\colon A^*(\overline {\mathcal {M}}_{1,4}(1))\to A^*(\overline {\mathcal {M}}_{0,5})$ .

2. 2. the fundamental class of $\mathbf {Cu}\subset \overline {\mathcal {M}}_{1,4}(1)$ ,

3. 3. the normal bundle of $\mathbf {Cu}$ , and

Recall that the locus of cuspidal curves $\mathbf {Cu}_{1,4}$ is isomorphic to $\overline {\mathcal {M}}_{0,5}$ by pinching the fifth section. In turn, $\overline {\mathcal {M}}_{0,5}$ is isomorphic to the blow-up of $\mathbb {P}^1\times \mathbb {P}^1$ in three points $(0,0)$ , $(1,1)$ and $(\infty ,\infty )$ . The calculation of its Chow rings follows from this explicit description. We take it directly from [Reference Keel20, Theorem 1]: let S be a subset of the markings set $[5]$ of cardinality between $2$ and $3$ , and let $D^S$ be (the class of) the divisor in $\overline {\mathcal {M}}_{0,5}$ of curves with a node separating the markings indexed by S from those indexed by $S^c$ . We say that two sets of indices S and T are incomparable if $S\nsubseteq T, S^c\nsubseteq T,T\nsubseteq S, T^c\nsubseteq S$ . Keel’s result is the following:

Theorem 4.12 (Keel).

The integral Chow ring of $\overline {\mathcal {M}}_{0,5}$ is:

$$\begin{align*}A^*(\overline{\mathcal{M}}_{0,5})=\mathbb{Z}[D^S]/\left(D^S-D^{S^c},\sum_{i,j\in S,h,k\in S^c}D^S-\sum_{i,h\in S,j,k\in S^c}D^S; D^SD^T\right)_{\substack{h,i,j,k\in[5]\\S,T\text{ incomparable}}}\end{align*}$$

We observe that the following relations hold under pullback:

(4.8) $$ \begin{align} \iota^*\delta_{hk}=D^{\{i,j\}},&\qquad \{i,j\}\subseteq [4],\end{align} $$

(4.9) $$ \begin{align} \iota^*\delta_i=D^{\{i,5\}},&\qquad i\in [4]. \end{align} $$

This already shows the surjectivity of $\iota ^*$ .

To compute the restriction of $\lambda _{(1)}$ , we first relate it to a $\psi $ -class on $\overline {\mathcal {M}}_{1,4}(1)$ , and then express every term in terms of Keel’s basis after pulling them back to $\overline {\mathcal {M}}_{0,5}$ . First, Proposition 2.20 identifies the difference between $\lambda _{(1)}$ and $\psi _1$ with the class of the divisor where the first marking lies on a rational tail:

(4.10) $$ \begin{align} \lambda_{(1)}=\psi_1-\sum_{\{i,j,k\}=\{2,3,4\}}(\delta_{ij}+\delta_k). \end{align} $$

Observe now that $\iota ^*\psi _1=\psi _1$ because the pinching occurs away from the first marking. Finally, we write $\psi _1\in A^1(\overline {\mathcal {M}}_{0,5})$ in terms of boundary classes. The comparison of $\psi $ classes under forgetting the last marking gives us:

(4.11) $$ \begin{align} \psi_1^{\{0,5\}} = f_5^*\psi_1^{\{0,4\}} + D^{\{1,5\}}. \end{align} $$

For the same reason, $\psi _1^{\{0,4\}}=D^{\{1,4\}}$ , so we conclude:

$$ \begin{align*} \iota^*\lambda_{(1)} &=D^{\{1,4\}}+D^{\{2,3\}}+D^{\{1,5\}}-(D^{\{1,2\}}+D^{\{1,3\}}+D^{\{1,4\}}+D^{\{2,5\}}+D^{\{3,5\}}+D^{\{4,5\}})\\ &= -(D^{\{1,3\}}+D^{\{2,5\}}+D^{\{4,5\}})=-\iota^*(\delta_{24}+\delta_2+\delta_4). \end{align*} $$

Putting this together with Equation (4.8), we deduce the following.

Finally, from Corollary 2.26 we know that:

$$ \begin{align*} [\mathbf{Cu}]&=24\lambda_{(1)}^2\in A^2(\overline{\mathcal{M}}_{1,4}(1)),\\ \mathcal{N}_{\mathbf{Cu}/\overline{\mathcal{M}}_{1,4}(1)}&=(\mathcal{H}(1)^{\otimes 4}\oplus \mathcal{H}(1)^{\otimes 6})_{|\mathbf{Cu}}. \end{align*} $$

Denoting by $\delta _{\emptyset }$ the fundamental class of the divisor in $\overline {\mathcal {M}}_{1,4}$ of curves with an unmarked elliptic tail, applying (2.1) gives us the following.

Proposition 4.14. We have

$$\begin{align*}A^*(\overline{\mathcal{M}}_{1,4})=\mathbb{Z}[\lambda_{(1)},\{\delta_{ij}\}_{\{i,j\}\subset [4]},\{\delta_i\}_{i\in [4]},\delta_{\emptyset}]/I_{1,4}'\end{align*}$$

where $I_{1,4}'$ is generated by the relations

$$ \begin{align*} & (1-93) && \text{as in Proposition}\ {4.11}, && \\ &(94) &&\delta_{\emptyset}((\delta_{12}+\delta_{34}) - (\delta_{13}+\delta_{24})), && \\ &(95) &&\delta_{\emptyset}((\delta_{12}+\delta_{34}) - (\delta_{14}+\delta_{23})), && \\ &(96-103) &&\delta_{\emptyset}((\delta_{hk}+\delta_h) - (\delta_{jk}+\delta_j)), && j,h,k\in[4],\\ &(104) &&\delta_{\emptyset}(\lambda_{(1)} + \delta_{24} + \delta_2 + \delta_4), && \\ &(105) &&24(\lambda_{(1)}-\delta_{\emptyset})^2. && \end{align*} $$

To conclude, because of Proposition 2.21 we have $\lambda =\lambda _{(1)}-\delta _{\emptyset }$ , where $\lambda $ is the first Chern class of the Hodge line bundle on $\overline {\mathcal {M}}_{1,4}$ . Applying this change of variable, we finally get to the following presentation for $A^*(\overline {\mathcal {M}}_{1,4})$ . As above, we do not know whether it is minimal.

Theorem 4.15. We have

$$\begin{align*}A^*(\overline{\mathcal{M}}_{1,4})=\mathbb{Z}[\lambda,\{\delta_{ij}\}_{\{i,j\}\subset [4]},\{\delta_i\}_{i\in [4]},\delta_{\emptyset}]/I_{1,4}\end{align*}$$

where $I_{1,4}$ is generated by the following relations:

$$ \begin{align*} &(1-6) &&\delta_i\delta_j, &&i\neq j,\\&(7-18) &&\delta_{ij}\delta_{ih}, &&j\neq h, \\&(19-30) &&\delta_{i}\delta_{hj}, && h,j \neq i \\&(31-42) &&\delta_{i}(\delta_{ij}-\delta_{ih}), &&i\neq j \neq h, \\&(43-48) &&\delta_{ij}(\lambda + \delta_{\emptyset} +\delta_{ij}+\delta_{i}+\delta_{j}), &&i\neq j, \\&(49-60) &&\delta_i(\lambda + \delta_{\emptyset}+\delta_{ij}+\delta_i), &&j,h \neq i, \\&(61-72) &&\delta_i\delta_{ij}^2, &&i\neq j \\&(73-75) &&12(\delta_{ij}^3-\delta_{ik}^3-\delta_{jh}^3+\delta_{hk}^3), &&\{i,j,h,k\}=[4],\\&(76-87) &&12(2\delta_{i}^3+6\delta_{i}^2\delta_{ik}+\delta_{ik}^3-\delta_{ik}\delta_{jh}^2-\delta_{ih}\delta_{jk}^2-\delta_{jk}^3-\delta_{ij}\delta_{hk}^2-\delta_{hk}^3), &&\{i,j,h,k\}=[4],\\&(88-93) && f_{ij}\text{ (see below)}, &&\{i,j,h,k\}=[4]. \\&(94) &&\delta_{\emptyset}((\delta_{12}+\delta_{34}) - (\delta_{13}+\delta_{24})), && \\&(95) &&\delta_{\emptyset}((\delta_{12}+\delta_{34}) - (\delta_{14}+\delta_{23})), && \\&(96-103) &&\delta_{\emptyset}((\delta_{hk}+\delta_h) - (\delta_{jk}+\delta_j)), && j,h,k\in[4],\\&(104) &&\delta_{\emptyset}(\lambda + \delta_{\emptyset} + \delta_{24} + \delta_2 + \delta_4), && \\&(105) &&24\lambda^2, && \end{align*} $$

$$ \begin{align*}f_{ij}&=4((\lambda+\delta_{\emptyset})^3 + \delta_{i}^3 + \delta_{j}^3 + \delta_{h}^3 + \delta_{k}^3 + \delta_{ij}^3 + \delta_{ih}^3 \\& \quad + 3\delta_{i}^2\delta_{ik} - 2\delta_{ik}^3 - \delta_{ik}\delta_{jh}^2 - 2\delta_{jh}^3 + 3\delta_{j}^2\delta_{jk} - \delta_{ih}\delta_{jk}^2 + \delta_{jk}^3 + 3\delta_{h}^2\delta_{hk} + 3\delta_{k}^2\delta_{hk} - \delta_{ij}\delta_{hk}^2 + \delta_{hk}^3). \end{align*} $$

The Hilbert series of the rational Chow ring of $\overline {M}_{1,4}$ is equal to $1+12t+23t^2+12t^3+t^4$ .