2.2.1 The statement of the formula

Let ${\overline {\mathscr M}}_{0,n,A}(X)$ be the moduli space of genus zero, degree $A\in H_2(X;\mathbf {Z})$, $n$ marked, stable maps into $X$. Let $\alpha _j\in H^*(X)$ for $1\leqslant j\leqslant n$. A genus zero Gromov–Witten invariant of $X$ is

\[ \big\langle \alpha_1,\ldots,\alpha_n \big\rangle^X_{0,n,A}:=\int_{[{\overline{\mathscr M}}_{0,n,A}(X)]^{\text{vir}}}\prod_{j=1}^n {ev}_j^\ast(\alpha_j), \]

where ${ev}_j$ is the $j$th evaluation map associated to the $j$th marking.

Recall that $p\colon \thinspace {\widehat {X}}\rightarrow X$ is the blowup projection. Then we have the following Gromov–Witten invariant of ${\widehat {X}}$

\[ \big\langle p^\ast\alpha_1,\ldots,p^\ast\alpha_n \big\rangle^{\widehat{X}}_{0,n,p^!A}:=\int_{[{\overline{\mathscr M}}_{0,n,p^!A}({\widehat{X}})]^{\text{vir}}}\prod_{j=1}^n {ev}_j^\ast(p^\ast\alpha_j). \]

Our first main result on the blowup formula of Gromov–Witten invariants is stated as follows.

Theorem 2.3 Let $X$ be a smooth projective variety and $S\subseteq X$ be a smooth subvariety. Let ${\widehat {X}}$ be the blowup of $X$ along $S$. Suppose the normal bundle $N$ of $S$ in $X$ is positive (see Definition 1.1) and $S$ has codimension $\kappa \geqslant 2$. Then we have the following equality of genus zero Gromov–Witten invariants

(10)\begin{equation} \big\langle \alpha_1,\ldots,\alpha_n\big\rangle^X_{0,n,A}=\big\langle p^\ast\alpha_1,\ldots,p^\ast\alpha_n\big\rangle^{\widehat{X}}_{0,n,p^!A}. \end{equation}

Remark 2.4 The virtual dimension of ${\overline {\mathscr M}}_{0,n,A}(X)$ is the same as that of ${\overline {\mathscr M}}_{0,n,p^!A}({\widehat {X}})$. Therefore, a priori, we assume that the total degree of insertions in (10) matches the dimension constraint, i.e.

(11)\begin{equation} \frac{1}2\sum_{j=1}^n \deg \alpha_j=\mathrm{vdim}\, {\overline{\mathscr M}}_{0,n,A}(X). \end{equation}

In fact, when the dimension constraint is not satisfied, both Gromov–Witten invariants in (10) are zero and the blowup formula (10) holds trivially.

Remark 2.5 (Remark on psi-class)

Here the blowup formula (10) deals with primary Gromov–Witten invariants. The computation in § 2.4 shows that this formula also holds for certain descendent invariants, as we now explain. First, we have the natural projection

\[ \pi\colon\thinspace{\overline{\mathscr M}}_{0,n,p^!A}({\widehat{X}})\rightarrow{\overline{\mathscr M}}_{0,n,A}(X). \]

Let $\bar \psi _j$ be the psi-class of ${\overline {\mathscr M}}_{0,n,A}(X)$ associated to the $j$th marking, and $\pi ^\ast \bar \psi _j$ be the pullback class. Then under the assumption of Theorem 2.3, we have the equality

\[ \big\langle {\bar\tau}_{d_1}\alpha_1,\ldots,{\bar \tau}_{d_n}\alpha_n\big\rangle^X_{0,n,A}= \big\langle {\bar\tau}_{d_1}p^\ast\alpha_1,\ldots, {\bar\tau}_{d_n}p^\ast\alpha_n \big\rangle^{\widehat{X}}_{0,n,p^!A}, \]

where ${\bar \tau }_{d_j}$ means the additional insertion $\bar \psi ^{d_j}_j$ for $X$ and $\pi ^\ast \bar \psi ^{d_j}_j$ for ${\widehat {X}}$.

2.2.2 Proof of the blowup formula

We prove the blowup formula modulo Propositions 2.8, 2.9 and 2.10, which are proved in § 2.4. The main ingredient is the vanishing of a certain Gromov–Witten invariant of $W_r$ for all $r$.

Viewing $A$ as a homology class of $X\times \mathbf{P}^1$ via the inclusion $X\rightarrow X\times \{0\}\subseteq X\times \mathbf{P}^1$, we get a homology class $p^!_WA\in H_2(W;\mathbf {Z})$ with $p_W\colon \thinspace W\rightarrow X\times \mathbf{P}^1$ the blowup projection. Recall that $W_r$ is the $r$th root stack of $W$ along $F_\infty$. Its coarse space is $W$. Consider the moduli space ${\overline {\mathscr M}}_{0,n,p_W^!A}(W_r)$ of genus zero, degree $p_W^!A$, orbifold stable maps into $W_r$ with $n$ smooth markings (cf. [Reference Chen, Ruan, Adem, Morava and RuanCR02, Reference Abramovich, Graber and VistoliAGV08]), and the following Gromov–Witten invariant of $W_r$

(12)\begin{equation} \big\langle q^\ast\alpha_1,\ldots,q^\ast\alpha_n \big\rangle^{W_r}_{0,n,p_W^!A}:= \int_{[{\overline{\mathscr M}}_{0,n,p_W^!A}(W_r)]^{\text{vir}}} \prod_{j=1}^n {ev}_j^\ast(q^\ast\alpha_j), \end{equation}

where $q$ is the projection $q\colon \thinspace W_r\rightarrow W \rightarrow X\times \mathbf{P}^1\rightarrow X$.

Proof. Note that $c_1^{W_r}(p_W^! A)=c_1^X(A)$, $\text {vdim}\,{\overline {\mathscr M}}_{0,n,p_W^!A}(W_r)=\text {vdim}\,{\overline {\mathscr M}}_{0,n,A}(X)+1$. Therefore, by the dimension constraint assumption (11), we have

\[ \frac{1}{2}\sum_{j=1}^n \deg \alpha_j=\text{vdim}\,{\overline{\mathscr M}}_{0,n,A}(X)<\text{vdim}\,{\overline{\mathscr M}}_{0,n,p_W^!A}(W_r) \]

and, consequently, the Gromov–Witten invariant (12) vanishes.

This lemma says that the Gromov–Witten invariant (12) vanishes. By Remark 2.6, we will compute this vanishing invariant by the virtual localization technique via the $\mathbf {C}^\ast$-action on $W_r$. We denote by $t$ the equivariant parameter of the $\mathbf {C}^\ast$-action.

By the virtual localization technique, the invariant (12) is contributed from the fixed locus of the $\mathbf {C}^\ast$-action on ${\overline {\mathscr M}}_{0,n,p_W^!A}(W_r)$. As usual, each component of the fixed locus of ${\overline {\mathscr M}}_{0,n,p_W^!A}(W_r)$ is indexed by a decorated graph $\Phi$ and the component is denoted by ${\overline {\mathscr M}}_\Phi$. Then the total contributions to the invariant (12) of all components of the fixed locus of ${\overline {\mathscr M}}_{0,n,p_W^!A}(W_r)$ are

\[ 0= \int_{[{\overline{\mathscr M}}_{0,n,p_W^!A}(W_r)]^{\text{vir}}}\prod_{j=1}^n {ev}_j^\ast(q^\ast\alpha_j) =\sum_\Phi\int_{{\overline{\mathscr M}}_\Phi} \prod_{j=1}^n {ev}_j^\ast(q^\ast\alpha_j)\cdot \frac{1}{e_{\mathbf{C}^\ast} (\mathscr N_\Phi)}=:\sum_\Phi \mathrm{Cont}(\Phi), \]

where $\mathscr N_\Phi$ is the virtual normal bundle of ${\overline {\mathscr M}}_\Phi$ in ${\overline {\mathscr M}}_{0,n,p_W^!A}(W_r)$.

All components ${\overline {\mathscr M}}_\Phi$ are described explicitly in § 2.3. The contribution $\mathrm {Cont}(\Phi )$ of each component ${\overline {\mathscr M}}_\Phi$ to the invariant (12) is computed in § 2.4. In particular, there are two special graphs, $\Phi _0$ and $\Phi _\infty$, for which every stable map in ${\overline {\mathscr M}}_{\Phi _0}$ (and ${\overline {\mathscr M}}_{\Phi _\infty }$, respectively) has images in $F_0$ (and $\sqrt [r]{F_\infty }$ , respectively). We will prove the following propositions in §§ 2.4.1, 2.4.2 and 2.4.3, respectively.

Proposition 2.8 The contribution of ${\overline {\mathscr M}}_{\Phi _0}$ to the invariant (12) is

\[ \mathrm{Cont}(\Phi_0)=-\frac{1}t\cdot \big\langle \alpha_1,\ldots,\alpha_n\big\rangle^X_{0,n,A}. \]

Proposition 2.9 The contribution of ${\overline {\mathscr M}}_{\Phi _\infty }$ to the invariant (12) is

\[ \mathrm{Cont}(\Phi_\infty)=\frac{1}t\cdot\big\langle p^\ast\alpha_1,\ldots,p^\ast\alpha_n \big\rangle^{\widehat{X}}_{0,n,p^!A}. \]

With these three propositions, we finish the proof of Theorem 2.3.

Proof of Theorem 2.3 Under the assumption of the theorem, by Propositions 2.8, 2.9 and 2.10 we see that, after the transformation ${\mathfrak s}=tr$, when $r\gg 1$,

\begin{align*} 0 &= \mathrm{Cont}(\Phi_0)+\mathrm{Cont}(\Phi_\infty)+ \sum_{\Phi\neq\Phi_0,\Phi_\infty}\mathrm{Cont}(\Phi)\\ &= \frac{r}{{\mathfrak s}}\big(\big\langle p^\ast\alpha_1,\ldots,p^\ast\alpha_n \big\rangle^{\widehat{X}}_{0,n,p^!A}-\big\langle \alpha_1,\ldots,\alpha_n\big\rangle^X_{0,n,A}\big)+ \sum_{\Phi\neq\Phi_0,\Phi_\infty}\mathrm{Cont}(\Phi) \end{align*}

with the right-hand side being a polynomial in $r$. Thus, the vanishing of the coefficient of $r$ on the right-hand side implies the blowup formula.

2.3.1 Decorated graphs

It is standard to associate connected components of fixed locus of ${\overline {\mathscr M}}_{0,n,p_W^!A}(W_r)$ with decorated graphs. See, for example, [Reference Liu, Farkas and MorrisonLiu13]. We assume $r\gg 1$ in this subsection.

Recall that the fixed locus $W_r^{\mathbf {C}^\ast }$ consists of the disjoint union of $F_0\cong X$, $\sqrt [r]{F_\infty }\cong {\sqrt [r]{\widehat {X}}}$ and $F_\ast \cong S$, whose coarse spaces are $|F_0|=F_0$, $|\sqrt [r]{F_\infty }|=F_\infty$ and $|F_\ast |=F_\ast$. Now consider a $\mathbf {C}^\ast$-fixed stable map $f\colon \thinspace (C,x_1,\ldots,x_n)\rightarrow W_r$ in ${\overline {\mathscr M}}_{0,n,p_W^!A}(W_r)$. We associate it with a decorated graph $\Phi$ which consists of the set $V(\Phi )$ of vertices with labels, the set $E(\Phi )$ of edges with labels, and the set of $n$ tails $T=\{\iota _1,\ldots,\iota _n\}$. The decorated graph $\Phi$ is described as the following.

Vertices. Assign a vertex $v$ to a connected component $C_v$ in $f^{-1}(W_r^{\mathbf {C}^\ast })$. Moreover, for each $v$ we associate it with the following labels:

• – label $c_v=0$, $\infty$ or $\ast$ depending on whether $|f(C_v)|\subseteq F_0,F_\infty$ or $F_\ast$, respectively;

• – label $A_v=f_\ast [|C_v|]\in H_2(F_{{c}_v};\mathbf {Z})$, the homology class of $f_\ast ([|C_v|])$;

• – label $T_v\subseteq T$, the set of markings on $C_v$.

We define three subsets $V_0$, $V_\infty$ and $V_\ast$ of $V(\Phi )$ by $V_{\bullet }:=\{v\mid c_v=\bullet \}$.

Edges. Assign an edge $e$ to each connected component of $C\setminus \bigcup _{v\in V(\Gamma )}C_v$. Let $C_e$ be the closure of the corresponding component. Then $C_e$ is an orbifold $\mathbf{P}^1$ and the map $f\colon \thinspace C_e\rightarrow D_r$ is a branched covering of a fiber of $D_r\rightarrow S_\ast$ ramified over $S_\ast$ and ${\sqrt [r]Z}$. We assume $0=f^{-1}(S_\ast )$, $\infty =f^{-1}(I\sqrt [r]Z)$. We associate $e$ the following labels:

• – label $d_e$ to be the degree of $f$, i.e. $f_\ast [|C_e|]=d_e[F]\in H_2(D;\mathbf {Z})$, where $[F]$ is the fiber class of $D\rightarrow S_\ast$;

• – label twisted sectors ${\sqrt [r]Z}[d_e]$ of ${\sqrt [r]Z}$, which is the image of $\infty$; here $d_e< r$ since $r\gg 1$.

As previous subsection, for each decorated graph $\Phi$, we denote by ${\overline {\mathscr M}}_\Phi$ the corresponding component of the fixed locus of ${\overline {\mathscr M}}_{0,n,p_W^!A}(W_r)$ and the virtual normal bundle of ${\overline {\mathscr M}}_\Phi$ by $\mathscr N_\Phi$. Denote by ${\mathrm {Aut}}(\Phi )$ the group of automorphism of $\Phi$ that fixes all labels.

Definition 2.12 For a decorated graph $\Phi$, a vertex $v\in V(\Phi )$ is called stable if $val(v)+n_v\geqslant 3$ or $A_v\neq 0$, where $val(v)$ is the number of edges adjacent to $v$ and $n_v=|T_v|$ is the number of markings on $C_v$. Let $V^S(\Phi )$ be the set of stable vertices in $V(\Phi )$, and $V^{U}(\Phi )$ be the set of unstable vertices in $V(\Phi )$.

The set of unstable vertices $V^{U}(\Phi )$ is the union of the following three types of vertices:

\begin{align*} V^1(\Phi) &= \{v\in V(\Phi)\mid val(v)=1,\ n_v=0,\ A_v=0\},\\ V^{1,1}(\Phi) &= \{v\in V(\Phi)\mid val(v)=n_v=1,\ A_v=0\},\\ V^2(\Phi) &= \{v\in V(\Phi)\mid val(v)=2,\ n_v=0,\ A_v=0\}. \end{align*}

Proof. We show that any $v$ with $c_v=\infty$ must be stable. Let $v$ be a vertex with $c_v=\infty$. If it is not stable, then $v$ belongs to $V^1(\Phi ),V^{1,1}(\Phi )$ or $V^2(\Phi )$.

If $v\in V^1(\Phi )\sqcup V^{1,1}(\Phi )$, let $e$ be the unique edge adjacent to $v$. This $e$ then corresponds to an orbifold map $f: C_e\to D_r$ with $C_v$ being the point $\infty =f^{-1}(I\sqrt [r]Z)$ in $C_e$. Then we see that $C_v=\infty$ is mapped into $\sqrt [r]Z[d_e]$ since $r\gg 1$ and, hence, is an orbifold marking over the domain curve $C$. This contradicts to the assumption that stable maps in ${\overline {\mathscr M}}_{0,n,p_W^!A}(W_r)$ contain only smooth markings.

Similarly, an unstable vertex $v\in V^2(\Phi )$ with $c_v=\infty$ must correspond to a balanced orbifold nodal point whose two branches are mapped into ${\sqrt [r]Z}[d_{e_1}]$ and ${\sqrt [r]Z}[d_{e_2}]$, respectively, where $e_1$ and $e_2$ are the only two edges adjacent to $v$. Since for a balanced orbifold nodal point, its two branches must be mapped to two twisted sectors of $\sqrt [r]Z$ decorated by two elements in $\mu _r$ that are inverses of each other, therefore we must have $e^{2\pi i ({d_{e_1}}/r)}\cdot e^{2\pi i ({d_{e_2}}/r)}=1$, i.e.

\[ d_{e_1}+d_{e_2}\equiv 0\pmod r, \]

which is impossible when $r\gg 1$. This finishes the proof.

Definition 2.14 Let $\Phi _0$ (and $\Phi _\infty$, respectively) denote the graph consisting of one single vertex $v$ with $c_v=0$ (and $c_v=\infty$, respectively). Then

\[ {\overline{\mathscr M}}_{\Phi_0}={\overline{\mathscr M}}_{0,n,A}(F_0),\quad\text{and}\quad {\overline{\mathscr M}}_{\Phi_\infty}={\overline{\mathscr M}}_{0,n,p^!A}(\sqrt[r]{F_\infty}). \]

Now consider a graph $\Phi$ other than $\Phi _0,\Phi _\infty$. Since the homology class is $p_W^!A$, there is no vertex $v\in V(\Phi )$ with $c_v=0$. For this graph $\Phi$, suppose

\[ V_\infty=\{u_1,\ldots,u_{\mathfrak p}\},\quad V_\ast=\{v_1,\ldots,v_{\mathfrak q}\}, \]

and the set of edges is

\[ E=\{e_{jk} \mid 1\leqslant j\leqslant {\mathfrak p},\ 1 \leqslant k \leqslant {\mathfrak q}\}. \]

Set the degree of $e_{jk}$ to be $d_{jk}$. We remark that $V_\infty$ may be empty.

2.3.2 Explicit expression of ${\overline {\mathscr M}}_\Phi$

Roughly speaking, ${\overline {\mathscr M}}_\Phi$ is certain (fiber) products of moduli spaces associated to vertices and edges, which we denote by ${\overline {\mathscr M}}_v$ and ${\mathscr F}_e$, respectively.

The moduli space ${\overline {\mathscr M}}_u$ for each vertex $u\in V_\infty$. The labels on $u$ are $c_u=\infty$, $A_u$ and $T_u=\{\iota _{u1},\ldots,\iota _{ua}\}$. Suppose the set of edges adjacent to $u$ is $E_u=\{e_{ul_1},\ldots, e_{ul_b}\}$. Then $E_u$ provides $b$ nodal points on the domain curve $C$ whose branches on $C_u$ are viewed as (unordered) orbifold markings. These orbifold markings on $C_u$ are mapped into the twisted sectors

\[ {\sqrt[r]Z}[r-d_{ul_k}]\subseteq \sqrt[r]{F_\infty}[r-d_{ul_k}] \]

for $1\leqslant k\leqslant b$. Thus, we get

\[ {\overline{\mathscr M}}_u:={\overline{\mathscr M}}_{0, T_u, E_u, A_u}(\sqrt[r]{F_\infty})\cap \bigwedge_{k=1}^b [{\sqrt[r]Z}[r-d_{ul_k}]], \]

where ${\overline {\mathscr M}}_{0,T_u,E_u,A_u}(\sqrt [r]{F_\infty })$ is the moduli space of genus zero stable maps in $\sqrt [r]{F_\infty }$ with original $T_u$-markings and $E_u$ (unordered) orbifold markings, $[{\sqrt [r]Z}[r-d_{ul_k}]]$ is the class of ${\sqrt [r]Z}[r-d_{ul_k}]$ in $\sqrt [r]{F_\infty }[r-d_{ul_k}]$ which equals to the pullback of the class of $Z$ in $F_\infty$ via the natural projection $\sqrt [r]{F_\infty }[r-m_{ul_k}]\rightarrow F_\infty$.

Lemma 2.15 For each edge $e_{ul_k}\in E_u$, there is a natural map

\[ \widetilde{ev}_{{ul_k}}: {\overline{\mathscr M}}_u\to F_\ast=S_\ast \]

induced from the evaluation map $ev_{{ul_k}}\colon \thinspace {\overline {\mathscr M}}_{0, T_u, E_u, A_u}(\sqrt [r]{F_\infty })\rightarrow I\sqrt [r]{F_\infty }$.

The moduli space ${\overline {\mathscr M}}_v$ for each vertex $v\in V_\ast$. If $v$ is unstable, then ${\overline {\mathscr M}}_v=F_\ast =S_\ast$. Now suppose $v$ is stable, and is labelled by $c_v=\ast$, $A_v$ and $T_v=\{\iota _{v1},\ldots,\iota _{va}\}$. Similarly, suppose the set of edges adjacent to $v$ is $E_v=\{e_{l_1v},\ldots, e_{l_bv}\}$. Then we get a moduli space

\[ {\overline{\mathscr M}}_v:={\overline{\mathscr M}}_{0,T_v, E_v,A_v}(F_\ast)={\overline{\mathscr M}}_{0,T_v, E_v,A_v}(S_\ast). \]

Here $T_v$ stands for the original marked points on $C_v$ and $E_v$ stands for the unordered markings on $C_v$ from the nodal points determined by edges in $E_v$ as above.

The moduli space ${\mathscr F}_{e}$ for each edge $e\in E$. Let $e=e_{jk}$ be the edge connecting $u_j\in V_\infty$ and $v_k\in V_\ast$ with degree $d_{jk}$. If $v_k\notin V^1(\Phi )$, ${\mathscr F}_{e_{jk}}$ is the fixed locus of

\[ {\overline{\mathscr M}}_{0,1,[d_{jk}], d_{jk}[F]}(D_r)\subseteq {\overline{\mathscr M}}_{0,1,[d_{jk}], d_{jk}[F]} (W_r), \]

where $0$ is the genus, $1$ is the number of smooth markings, $[d_{jk}]$ indicates the twisted sector $\sqrt [r]Z[d_{jk}]$ of the unique orbifold marking, $d_{jk}[F]$ is the degree of maps. If $v_k\in V^1(\Phi )$, ${\mathscr F}_{e_{jk}}$ is the fix locus of

\[ {\overline{\mathscr M}}_{0,[d_{jk}], d_{jk}[F]}(D_r)\subseteq {\overline{\mathscr M}}_{0,[d_{jk}], d_{jk}[F]} (W_r). \]

Furthermore, ${\mathscr F}_{e_{jk}}$ is a fibration over $S_\ast$ with fiber being the $\mathbf {C}^\ast$-fixed locus of the moduli space stable maps into $\mathbf{P}_{(r,\ldots,r,1)}(\mathbf {C}^{\kappa +1})$ with topological data $(0,1, [d_{jk}], d_{jk}[F])$ or $(0,[d_{jk}], d_{jk}[F])$. This fixed locus ${\mathscr F}_{e_{jk}}$ is determined by the marking mapped into $\sqrt [r]Z[d_{jk}]$. It is straightforward to see that ${\mathscr F}_{e_{jk}}$ is a $\mathbf {Z}_{d_{jk}}$-gerbe over $Z$.

The moduli space ${\overline {\mathscr M}}_\Phi$

Proposition 2.16 For a graph $\Phi$ other than $\Phi _0$ and $\Phi _\infty$, set $d_\Phi :=\prod _{e\in E} d_e$. Then

(13)\begin{equation} {\overline{\mathscr M}}_\Phi=\frac{r^{|E|}}{d_\Phi\cdot |{\mathrm{Aut}}(\Phi)|}\cdot \prod_{u_j\in V_\infty} {\overline{\mathscr M}}_{u_j} \times_{S_\ast^{|E|}} \prod_{v_k\in V_\ast} {\overline{\mathscr M}}_{v_k}. \end{equation}

Here we use the map defined in Lemma 2.15 for the fiber product.

Proof. First, we have

(14)\begin{equation} {\overline{\mathscr M}}_\Phi=\frac{1}{|{\mathrm{Aut}}(\Phi)|}\cdot \prod_{u_j\in V_\infty} {\overline{\mathscr M}}_{u_j} \times_{(I{\sqrt[r]Z})^{|E|}} \prod_{e_{jk}\in E} {\mathscr F}_{e_{jk}}\times_{S_\ast^{|E|}} \prod_{v_k\in V_\ast} {\overline{\mathscr M}}_{v_k} \end{equation}

where $I{\sqrt [r]Z}$ is the inertia space of ${\sqrt [r]Z}$.

Let $\{\beta _l\}$ be a basis of $H^\ast (F_\ast )=H^\ast (S_\ast )$ and $\{\check \beta _l\}$ be its dual basis. Let $H$ be the hyperplane class of $Z\rightarrow S_\ast$, then

\[ \{\gamma_{lk}=\beta_l H^k\mid 1\leqslant k\leqslant \kappa-1\} \]

is a basis of $H^\ast (Z)$ with $\{\check \gamma _{lk}\}$ being its dual basis. We now simplify (14). For simplicity, we only consider the case $V_\infty =\{u\}$, $V_\ast =\{v\}$, $E=\{e_{uv}\}$, and omit the factor $ {1}/{|{\mathrm {Aut}}(\Phi )|}$. For the general cases, the computation is identical. From (14), we have

\[ {\overline{\mathscr M}}_\Phi= r\cdot \sum_{\gamma_{lk},\beta_{l'}}([{\overline{\mathscr M}}_{u}\cap ev^\ast_{uv}\check \gamma_{lk}]\cdot[{\mathscr F}_{e_{uv}}\cap(\gamma_{lk}\cup\check \beta_{l'})] \cdot [{\overline{\mathscr M}}_{v}\cap ev^\ast_{uv}\beta_{l'}]), \]

where $r$ comes from the orbifold Poincaré dual of ${\sqrt [r]Z}$. Since ${\mathscr F}_{e_{uv}}$ is a $\mathbf {Z}_{d_{e_{uv}}}$-gerbe over $Z$, only when $\gamma _{lk}=\gamma _{l'(\kappa -1)}=\beta _{l'} H^{\kappa -1}$ the term $[{\mathscr F}_{e_{uv}}\cap (\gamma _{lk}\cup \check \beta _{l'})]$ is nonzero, and contributes $ {1}/{d_{e_{uv}}}$. Therefore,

\[ {\overline{\mathscr M}}_\Phi=\frac{r}{d_{e_{uv}}}\sum_{\beta_{l'}}[{\overline{\mathscr M}}_u\cap \widetilde{ev}^\ast_{uv} \check\beta_{l'}]\cdot[{\overline{\mathscr M}}_v\cap ev^\ast_{uv}\beta_{l'}]=\frac{r}{d_{e_{uv}}} \cdot({\overline{\mathscr M}}_u\times_{S_\ast} {\overline{\mathscr M}}_v). \]

Here, we use the facts that $\check \gamma _{lk}=\check \beta _{l'}$ when $\gamma _{lk}=\beta _{l'}H^{\kappa -1}$, and $ev^\ast _{uv}\check \gamma _{lk}= ev^\ast _{uv}\check \beta _{l'}=\widetilde {ev}^\ast _{uv} \check \beta _{l'}$.

2.4.1 Proof of Proposition 2.8

For the graph $\Phi _0$ we have ${\overline {\mathscr M}}_{\Phi _0}={\overline {\mathscr M}}_{0,n,A}(F_0)$. Let $\mathcal {C}\rightarrow {\overline {\mathscr M}}_{\Phi _0}$ be the universal curve and $f\colon \thinspace \mathcal {C}\rightarrow F_0$ be the universal map. The virtual normal bundle of ${\overline {\mathscr M}}_{\Phi _0}$ in ${\overline {\mathscr M}}_{0,n,p_W^!A}(W_r)$ is the induced index bundle $\mathcal {R}\pi _\ast f^\ast L_0$ by the normal line bundle $L_0$ of $F_0\cong X$ in $W_r$. We denote this index bundle simply by $(L_0)_{\Phi _0}$. It is a line bundle with action weight $-1$. Thus, since $(q^*\alpha _j)|_{F_0}=\alpha _j$, by the dimension constraint (11), the contribution of $\Phi _0$ is

\begin{align*} \mathrm{Cont}(\Phi_0) &= \int_{[{\overline{\mathscr M}}_{0,n,A}(F_0)]^{\text{vir}}} \frac{\prod_{j=1}^n {ev}_j^*((q^*\alpha_j)|_{F_0})} {c_1((L_0)_{\Phi_0}\otimes \mathcal{O}(-1))}=\int_{[{\overline{\mathscr M}}_{0,n,A}(F_0)]^{\text{vir}}} \frac{\prod_{j=1}^n {ev}_j^*((q^*\alpha_j)|_{F_0})}{-t+c_1((L_0)_{\Phi_0})}\\ &=\frac{1}{-t}\cdot \int_{[{\overline{\mathscr M}}_{0,n,A}(F_0)]^{\text{vir}}} \prod_{j=1}^n {ev}_j^*(\alpha_j) =-\frac{1}{t}\cdot\big\langle \alpha_1,\ldots,\alpha_n \big\rangle^X_{0,n,A}. \end{align*}

This finishes the proof of Proposition 2.8.

2.4.2 Proof of Proposition 2.9

For the graph $\Phi _\infty$, we have ${\overline {\mathscr M}}_{\Phi _\infty }={\overline {\mathscr M}}_{0,n,p^!A}(\sqrt [r]{F_\infty })$. The virtual normal bundle of ${\overline {\mathscr M}}_{\Phi _\infty }$ in ${\overline {\mathscr M}}_{0,n,p_W^!A}(W_r)$ is the induced index bundle $\mathcal {R}\pi _\ast f^\ast \sqrt [r]{L_\infty }$ by the normal bundle of $\sqrt [r]{F_\infty }$ in $W_r$, i.e. the $r$th root $\sqrt [r]{L_\infty }$ of the line bundle $L_\infty$. We denote this index bundle simply by $(\!\sqrt[r]{L_\infty })_{\Phi _\infty }$. It is a line bundle with action weight $ {1}/r$ since $r\gg 1$. Moreover, we have the following commutative diagram

and $q^*\alpha _j|_{\sqrt [r]{F_\infty }}=(q|_{\sqrt [r]{F_\infty }})^*\alpha _j =\pi ^*p^*(\alpha _j)$. Thus, the contribution of $\Phi _\infty$ is

\begin{align*} \mathrm{Cont}(\Phi_\infty)&= \int_{[{\overline{\mathscr M}}_{0,n,p^!A}(\!\sqrt[r]{F_\infty})]^{\text{vir}}} \frac{\prod_{j=1}^n {ev}_j^*((q^*\alpha_j)|_{\sqrt[r]{F_\infty}})}{ c_1\big((\!\sqrt[r]{L_\infty})_{\Phi_\infty} \otimes\mathcal{O}({1}/r)\big)}\\ &= \int_{[{\overline{\mathscr M}}_{0,n,p^!A}(\!\sqrt[r]{F_\infty})]^{\text{vir}}} \frac{\prod_{j=1}^n{ev}_j^*(\pi^*p^*(\alpha_j))}{{t}/{r}+c_1((\!\sqrt[s]{L_\infty})_{\Phi_\infty})}\\ &= \frac{r}{t}\cdot \int_{[{\overline{\mathscr M}}_{0,n,p^!A}(\!\sqrt[r]{F_\infty})]^{\text{vir}}} \prod_{j=1}^n{ev}_j^*(\pi^*p^*(\alpha_j))\\ &= \frac{r}{t}\cdot\frac{1}{r}\cdot \int_{[{\overline{\mathscr M}}_{0,n,p^!A}({F_\infty})]^{\text{vir}}} \prod_{j=1}^n {ev}_j^*(p^*\alpha_j)\\ &= \frac{1}{t}\cdot \big\langle p^*\alpha_1,\ldots,p^*\alpha_n\big\rangle^{\widehat{X}}_{0,n,p^!A}, \end{align*}

where the third equality follows from the dimension constraint assumption (11) and

\[ \text{vdim}\,{\overline{\mathscr M}}_{0,n,p^!A}(\!\sqrt[r]{F_\infty})= \text{vdim}\,{\overline{\mathscr M}}_{0,n,p^!A}(F_\infty)=\text{vdim}\,{\overline{\mathscr M}}_{0,n,A}(X), \]

and the fourth equality follows from the computation of Gromov–Witten invariants of root gerbes in [Reference Andreini, Jiang and TsengAJT15, Theorem 4.3], i.e. $\epsilon _*([{\overline {\mathscr M}}_{0,n,p^!A}(\!\sqrt [r]{F_\infty })]^{\text {vir}})= ({1}/r) [{\overline {\mathscr M}}_{0,n,p^!A}(F_\infty )]^{\text {vir}}$. This finishes the proof of Proposition 2.9.

2.4.3 Proof of Proposition 2.10

We now consider a general graph $\Phi$ other than $\Phi _0$ and $\Phi _\infty$. For such a graph, the contribution is

\[ \mathrm{Cont}(\Phi)=\frac{1}{|{\mathrm{Aut}}(\Phi)|}\cdot \prod_{j=1}{ev}_j^\ast(q^\ast\alpha_j)\cap \frac{[{\overline{\mathscr M}}_\Phi]^{\text{vir}} } {e_{\mathbf{C}^\ast}(\mathscr N_\Phi)} =\frac{1}{|{\mathrm{Aut}}(\Phi)|}\cdot \prod_{j=1}{ev}_j^\ast(q^\ast\alpha_j)\cap \epsilon_\ast \bigg(\frac{[{\overline{\mathscr M}}_\Phi]^{\text{vir}} } {e_{\mathbf{C}^\ast}(\mathscr N_\Phi)}\bigg), \]

with $\epsilon \colon \thinspace {\overline {\mathscr M}}_{0,n,p_W^!A}(W_r)\rightarrow {\overline {\mathscr M}}_{0,n,p_W^!A}(W)$ the natural projection.

As usual, the contribution from the virtual normal bundle can be decomposed into a combination of contributions of vertices, edges and nodal points. Before we give explicit computations, we introduce some notation:

• – the set $V_\ast$ is decomposed into

  \[ V_\ast=V_\ast^S\sqcup V_\ast^1\sqcup V_\ast^{1,1}\sqcup V_\ast^2, \]
  the disjoint union of stable vertices and three types of unstable vertices (see Definition 2.12);

• – the set $E$ of edges is decomposed into

  \[ E= E^S \sqcup E^1\sqcup E^{1,1}\sqcup E^2 \]
  according to the vertices in $V_\ast$ that the edges are adjacent to;

• – nodal points attached to $C_u, u\in V_\infty$ are denoted by ${\mathsf n}$ with lower indices; nodal points attached to $C_v, v\in V_\ast$ are denoted by $\bar {\mathsf n}$ with lower indices.

(1) Contributions from stable vertices. For a stable vertex $v\in V^S(\Phi )$, ${\overline {\mathscr M}}_v$ is a moduli space of stable maps into the fixed component $F_0,\sqrt [r]{F_\infty }$ or $F_\ast \subseteq W_r$ according to $c_v=0$, $\infty$ or $\ast$. Let $\mathscr N_v$ be the virtual normal bundle of ${\overline {\mathscr M}}_v$ in the moduli space of stable maps into $W_r$ with the same topological data. By the contribution of a stable vertex we mean the equivariant Euler class of $\mathscr N_v$. The explicit formula for the contribution of each stable vertex is given as follows.

1. (1a) If $u\in V_\infty$,

   \[ \mathrm{cont}_u:= e_{\mathbf{C}^*}((\mathcal{R}\pi_\ast f^\ast\sqrt[r]{L_\infty})_u) ^{-1}= c_{\text{rk}(u)}\bigg(\!-(\mathcal{R}\pi_\ast f^\ast\sqrt[r]{L_\infty})_u\otimes \mathcal{O}\bigg(\frac{1}{r}\bigg)\bigg), \]
   where $(\mathcal {R}\pi _\ast f^\ast \sqrt [r]{L_\infty })_u$ means the induced index bundle over ${\overline {\mathscr M}}_u$, and $\text {rk}(u)=|E_u|-1$ since $r\gg 1$.

2. (1b) If $v\in V_\ast ^S$,

   \[ \mathrm{cont}_v:=e_{\mathbf{C}^\ast}((\mathcal{R}\pi_\ast f^\ast (N\oplus \mathcal{O}_S))_v)^{-1}, \]
   where $(\mathcal {R}\pi _\ast f^\ast (N\oplus \mathcal {O}_S))_v$ means the induced index bundle over ${\overline {\mathscr M}}_v$ whose rank is $\text {rk}(v)=\int _{\mathbf{P}^1}c_1(f^*_vN)+\kappa +1$ for a stable map $f_v\colon \thinspace \mathbf{P}^1\rightarrow S_*$ in ${\overline {\mathscr M}}_v$, hence $\text {rk}(v)=c_1(N)(A_v)+ \kappa +1$.

(2) Contributions from edges. Let $e\in E$ be an edge. By the contribution of the edge $e$, we mean the equivariant Euler class of the virtual normal bundle of ${\mathscr F}_{e}$ in the corresponding moduli space of $W_r$. Suppose the edge $e$ connects the vertex $u\in V_\infty$ and the vertex $v\in V_\ast$ and has degree $d_e$. By the standard computations, when $v\not \in V^1_\ast$, the contribution of $e$ is

(15)\begin{equation} \mathrm{cont}_{e}:=\bigg(\bigg(\prod_{0\leqslant j\leqslant d_{e}-1} \frac{(d_e-j)t}{d_e}\bigg)^\kappa+\cdots\bigg)^{-1}\cdot \bigg(\prod_{1\leqslant l\leqslant d_e-1}\frac{-lt}{d_e}+\cdots\bigg), \end{equation}

where those ‘${\cdots }$’ represent lower degree (but nonnegative) terms in $t$. When $v\in V^1_\ast$, in addition to $\mathrm {cont}_{e}$ there is an extra contribution $ {t}/{d_e}$ coming from the automorphisms of the domain curve.

(3) Contributions from nodal points. Nodal points correspond to some half edges or vertices with exactly two adjacent edges. Be precise, there are three cases.

1. (3a) Let $u\in V_\infty$ and $e=e_{uv}$ be an edge adjacent to $u$ with degree $d_e$. Then we have a half edge $(u,e)$ corresponding to a nodal point ${\mathsf n}_e$ attached to $C_u$. The contribution is

   (16)\begin{equation} \mathrm{cont}_{{\mathsf n}_e}:=\frac{1}{({t+{ev}_\infty^\ast (c_1(L_\infty))})/ {d_e}-\bar\psi_{u}} = \frac{d_e} {t+{ev}_\infty^\ast (c_1(L_\infty))-d_e\bar\psi_{u}}, \end{equation}
   where $\infty$ in $ev_\infty$ is the $\infty$ marking on $C_e$, $\bar \psi _{u}$ is the first Chern class of the cotangent line bundle over ${\overline {\mathscr M}}_u$ associated to the unordered orbifold marking corresponding to the nodal point ${\mathsf n}_e$, whose fiber is the cotangent space of the coarse space of $C_u$ at ${\mathsf n}_e$.

2. (3b) Let $v\in V^S_\ast$ and $e=e_{uv}$ be an edge adjacent to $v$ with degree $d_e$. Then we have a half edge $(v,e)$ corresponding to a nodal point $\bar {\mathsf n}_e$ attached to $C_v$. The contribution is

   \[ \mathrm{cont}_{\bar{\mathsf n}_e}:=\frac{(-1)^\kappa t^{\kappa+1}}{({-t-{ev}_\infty^\ast (c_1(L_\infty))})/{d_e} -\bar\psi_{v}}=\frac{d_e\cdot (-1)^\kappa t^{\kappa+1}}{-t-{ev}_\infty^\ast (c_1(L_\infty)) -d_e\bar\psi_{v}}, \]
   where $\infty$ in $ev_\infty$ is the $\infty$ marking on $C_e$, $\bar \psi _{v}$ is the first Chern class of the cotangent line bundle over ${\overline {\mathscr M}}_v$ associated to the unordered marking corresponding to the nodal point $\bar {\mathsf n}_e$, whose fiber is the cotangent space of $C_v$ at $\bar {\mathsf n}_e$ and the $t^{\kappa +1}$ corresponds to the deformation of stable maps at the nodal point ${\mathsf n}_e$ along the normal direction of $S_\ast$ in $W_r$.

3. (3c) Let $v\in V_\ast ^2$ and $e_1=e_{u_1v}$, $e_2=e_{u_2v}$ be the only two adjacent edges with degrees $d_{e_1}$ and $d_{e_2}$, respectively. The nodal point $\bar {\mathsf n}_v$ is the intersection of the two components $C_{e_1}$ and $C_{e_2}$ corresponding to the two edges. The contribution is

   \[ \mathrm{cont}_{\bar{\mathsf n}_v} :=\frac{ (-1)^\kappa t^{\kappa+1}}{({-t-{ev}_\infty^\ast c_1(L_\infty)})/{d_{e_1}}+ ({-t-{ev}_\infty^\ast c_1(L_\infty)})/{d_{e_2}}}, \]
   where $\infty$ in $ev_\infty$ is the $\infty$ marking on $C_{e_1}$ and $C_{e_2}$, respectively, and the $t^{\kappa +1}$ corresponds to the deformation of stable maps at $\bar {\mathsf n}_v$ along the normal direction of $S_\ast$ in $W_r$.

We conclude that

\[ \frac{1}{e_{\mathbf{C}^\ast}(\mathscr N_\Phi)}= \prod_{u\in V_\infty}\bigg(\mathrm{cont}_u\prod_{e\in E_u}\mathrm{cont}_{{\mathsf n}_e}\bigg) \prod_{v\in V^S_\ast}\bigg(\mathrm{cont}_v\prod_{e\in E_v}\mathrm{cont}_{\bar{\mathsf n}_e}\bigg) \prod_{v\in V^2_\ast} \mathrm{cont}_{\bar{\mathsf n}_v}\prod_{e\in E}\mathrm{cont}_e \prod_{\substack{e\in E_v\\ v\in V_\ast^1}}\frac{t}{d_e}. \]

For a stable vertex $u\in V_\infty$, we set

(17)\begin{equation} \mathrm{Cont}_u:=\frac{r^{|E_u|}}{d_u}\bigg(\mathrm{cont}_u\cdot\prod_{e\in E_u}\mathrm{cont}_{{\mathsf n}_e}\bigg)\cap [{\overline{\mathscr M}}_u]^{\text{vir}}, \end{equation}

with $d_u=\prod _{e\in E_u}d_e$. For a vertex $v\in V_\ast$, we set

(18) \begin{equation} \mathrm{Cont}_v:= [{\overline{\mathscr M}}_v]^{\text{vir}}\cap\left\{\begin{array}{@{}lll} \displaystyle\mathrm{cont}_v\cdot\prod_{e\in E_v}\mathrm{cont}_{\bar{\mathsf n}_e} \cdot\prod_{e\in E_v}\mathrm{cont}_e & \mbox{if}\!\!\!\!\! & v\in V^S_\ast,\\ \displaystyle\mathrm{cont}_{\bar{\mathsf n}_v}\cdot\prod_{e\in E_v}\mathrm{cont}_e & \mbox{if}\!\!\!\!\! & v\in V^2_\ast,\\ \mathrm{cont}_e & \mbox{if}\!\!\!\!\! &v\in V^{1,1}_\ast,\\ \displaystyle\dfrac{t}{d_e}\cdot \mathrm{cont}_e & \mbox{if}\!\!\!\!\! & v\in V^{1}_\ast, \end{array}\right. \end{equation}

where when $v\in V^{1,1}_\ast \sqcup V^{1}_\ast$, $E_v=\{e\}$. Then we have the following.

Proposition 2.18 When $r\gg 1$, the contribution $\mathrm {Cont}(\Phi )$ is

\[ \frac{1}{|{\mathrm{Aut}}(\Phi)|}\cdot \sum_{l_e,e\in E} \prod_{j=1}^n ev_j^\ast(q^\ast\alpha_j) \prod_{e\in E}\widetilde{ev}_{e,+}^\ast\beta_{l_e}{ev}_{e,-}^\ast \check\beta_{l_e} \prod_{u\in V_\infty}\epsilon_\ast \mathrm{Cont}_u \prod_{v\in V_\ast} \mathrm{Cont}_v, \]

where $\{\beta _l\}$ is a basis of $H^\ast (F_\ast )=H^\ast (S_\ast )$ with $\{\check \beta _l\}$ the dual basis as in the proof of Proposition 2.16, ${ev}_{e,+}$ and $ev_{e,-}$ are the evaluation maps corresponding respectively to the two special points $\infty$ and $0$ on $C_e$ for an edge $e$, and $\widetilde {ev}_{e,+}$ is obtained from ${ev}_{e,+}$ via Lemma 2.15.

To prove Proposition 2.10, we need the following two technical lemmas. The first was proved in [Reference Janda, Pandharipande, Pixton and ZvonkineJPPZ20, § 3].

Proof. For a polynomial in $t^{-1}$, denote its lowest degree in $t^{-1}$ by $\deg _{t^{-1}}^{\text {low}}(\cdot )$. We next compute $\deg _{t^{-1}}^{\text {low}}(\mathrm {Cont}_v)$ for all $v\in V_\ast$.

For a vertex $v\in V_\ast ^S$, we have

\begin{align*} \deg_{t^{-1}}^{\text{low}}(\mathrm{Cont}_v) &= \deg_{t^{-1}}^{\text{low}}(\mathrm{cont}_v)+ \sum_{e\in E_v}(\deg_{t^{-1}}^{\text{low}}(\mathrm{cont}_{\bar{\mathsf n}_e}) +\deg_{t^{-1}}^{\text{low}}(\mathrm{cont}_e) )\\ &= (c_1(N)(A_v)+\kappa+1)+ \sum_{e\in E_v} (1-(\kappa+1)+\mathrm{rk}_{e})\\ &= (c_1(N)(A_v)+\kappa+1)+ \sum_{e\in E_v} (1-(\kappa+1)+d_{e}(\kappa-1)+1)\\ &\geqslant (1+1)+ \sum_{e\in E_v}(d_e-1)(\kappa-1) \geqslant 2. \end{align*}

Here we use the key assumption $c_1(N)(A_v)+\kappa =\int _{\mathbf{P}^1}c_1(f_v^*N)+\kappa >0$.

For a vertex $v\in V_\ast ^2$, let $e_1$ and $e_2$ be the only two adjacent edges. Then the degrees of these two edges satisfy $d_{e_1}, d_{e_2}\geqslant 1$. Consequently, we have

\begin{align*} \deg_{t^{-1}}^{\text{low}}(\mathrm{Cont}_v)&= \mathrm{rk}_{{e_1}} +\mathrm{rk}_{{e_2}}+1-(\kappa+1)\\ &=(d_{e_1}(\kappa-1)+1+d_{e_2}(\kappa-1)+1)-\kappa \\ &=(d_{e_1}+d_{e_2}-1)(\kappa-1)+1\geqslant \kappa\geqslant 2. \end{align*}

For a vertex $v\in V_\ast ^{1,1}$ with the unique adjacent edge $e$ of degree $d_e$, we have $d_e\geqslant 1$ and

\[ \deg_{t^{-1}}^{\text{low}}(\mathrm{Cont}_v) =\mathrm{rk}_{e}=d_e(\kappa-1)+1 \geqslant \kappa\geqslant 2. \]

For a vertex $v\in V_\ast ^1$ with the unique adjacent edge $e$ of degree $d_e$, we have $d_e\geqslant 1$ and

\[ \deg_{t^{-1}}^{\text{low}}(\mathrm{Cont}_v)=\mathrm{rk}_{e}-1 =d_e(\kappa-1)+1-1\geqslant \kappa-1\geqslant 1. \]

The lemma follows.

Now we prove Proposition 2.10.

Proof of Proposition 2.10 By Proposition 2.18, when $r\gg 1$, $\mathrm {Cont}(\Phi )$ is

\[ \mathrm{Cont}(\Phi)=\frac{1}{|{\mathrm{Aut}}(\Phi)|}\cdot\sum_{l_e,e\in E} \prod_{j=1}^n {ev}_j^\ast q^\ast\alpha_j \prod_{e\in E} \widetilde{ev}_{e,+}^\ast\beta_{l_e} {ev}_{e,-}^\ast\check\beta_{l_e} \prod_{u\in V_\infty}\epsilon_\ast\mathrm{Cont}_u \prod_{v\in V_\ast}\mathrm{Cont}_v. \]

Under the transformation ${\mathfrak s}=tr$, i.e. $t^{-1}=r{\mathfrak s}^{-1}$, for $V_\ast$ we have

\[ \prod_{v\in V_\ast}\mathrm{Cont}_v=\bigg(\frac{1}{t}\bigg) ^{|V_\ast|}G(t^{-1})= \bigg(\frac{r}{{\mathfrak s}}\bigg)^{|V_\ast|}G(r,{\mathfrak s}), \]

where $G(r,{\mathfrak s})$ is a polynomial in $r$ by Lemma 2.20. Similarly, when $V_\infty$ is nonempty, by Lemma 2.19, for each vertex $u\in V_\infty$,

\[ \epsilon_\ast\mathrm{Cont}_u=\frac{1}{t} F_u(r,{\mathfrak s})=\frac{r}{{\mathfrak s}}F_u(r,{\mathfrak s}), \]

where $F_u(r,{\mathfrak s})$ is also a polynomial in $r$ when $r\gg 1$, which implies that

\[ \prod_{u\in V_\infty}\epsilon_\ast\mathrm{Cont}_u=\bigg(\frac{r}{{\mathfrak s}}\bigg)^{|V_\infty|} F(r,{\mathfrak s}), \]

with $F(r,{\mathfrak s})$ being a polynomial in $r$ when $r\gg 1$. Put them together, we conclude that when $V_\infty \neq \varnothing$ and $r\gg 1$, $\mathrm {Cont}(\Phi )$ is a polynomial in $r$ with lowest degree at least $|V_\ast |+|V_\infty |\geqslant 2$. If $V_\infty$ is empty, $V_\ast$ contains at least one stable vertex, i.e. $V_\ast ^S\neq \varnothing$. By Lemma 2.20, we have the fact that $\mathrm {Cont}(\Phi )$ is a polynomial in $r$ with lowest degree at least $|V_*|+1\geqslant 2$. This finishes the proof of Proposition 2.10.

3.3.1 Decorated graphs

As in § 2.3.1 we use decorated graphs to describe connected components of the fixed locus of ${\overline {\mathscr M}}_{0,{\mathfrak g},p_W^!A}({\mathsf W}_r)$. The principle of the assignment of decorated graphs to connected components of fixed locus is the same as that in § 2.3.1. We omit it here.

However, we remark that the case is slightly complicated due to the labels of twisted sectors on half edges as we now explain. As in § 2.3.1, each edge $e$ corresponds to a fixed curve represented by an orbifold ${\mathsf C}_e$ mapped onto a fiber of $\mathsf D_r\to {\mathsf S}_\ast$:

\[ \mathsf u\colon\thinspace{\mathsf C}_e\to \mathsf D_r. \]

Here ${\mathsf C}_e$ is an orbifold $\mathbf {P}^1$. Then $\mathsf u(0)\in \sf IS_\ast$ and $\mathsf u(\infty )\in \mathsf I\sqrt [r]{{\mathsf Z}}$. The labels associated to $e$ then are:

• – label $d_e$ to be the degree of $\mathsf u$, i.e. $|\mathsf u({\mathsf C}_e)|=d_e[F]$, where $[F]$ is the fiber class of ${\mathsf D}\rightarrow {\mathsf S}$;

• – label twisted sectors $[g_e]$ and $[h_e]$ associated to $\mathsf u(0)$ and $\mathsf u(\infty )$, respectively.

On the other hand, by projecting to ${\mathsf S}_*$ we get a stable map ${\mathsf C}_e\rightarrow {\mathsf S}_*$ with two orbifold markings which is constant on coarse space, therefore $I({\mathsf S}_*[g_e])={\mathsf S}_*[\Lambda ([h_e])]$ where $I\colon \thinspace \sf IS_*\rightarrow IS_*$ is the canonical involution of inertia orbifolds. Consequently, these labels are related by

\[ [h_e]=[g_e^{-1}, e^{2\pi i{ d_e}/{r}}]. \]

This suggests that the label $[h_e]$ encodes all information of $d_e, g_e$ and $h_e$. It also implies that

\[ d_e\in \mathcal{R}_{[g_e]}. \]

We follow the lines in § 2.3.1 and use same notation. As Lemma 2.13 we have following result.

Let $\Phi _0$ be the graph consisting of only one single vertex $v$ with $c_v=0$. Then

\[ {\overline{\mathscr M}}_{\Phi_0}={\overline{\mathscr M}}_{0, {\mathfrak g},A}({\mathsf F}_0)\cong{\overline{\mathscr M}}_{0, {\mathfrak g},A}({\mathsf X}). \]

It is the $\Phi _0$ in Proposition 3.12.

Now let $\Phi$ be a graph other than $\Phi _0$. It then has no vertex with $c_v=0$. Suppose

\[ V_\infty=\{u_1,\ldots,u_{\mathfrak p}\},\quad V_*=\{v_1,\ldots,v_{\mathfrak q}\} \]

and the set of edges is

\[ E=\{e_{jk}\mid 1\leqslant j\leqslant {\mathfrak p},1 \leqslant k \leqslant {\mathfrak q}\}. \]