of it which makes the resulting stack as simple as possible. The resulting
stack gives an alternative definition of reduced invariants, which is not
intrinsic; the relation between these two invariants is similar in spirit to a
Quantum Lefschetz theorem. The definition we give is more intrinsic, but
working with maps with fields instead of maps is more suited to approaching
1.0.2 and 1.0.3. See [CL15, LO22, LO21] for the proof of 1.0.3 in genus one
and two.

###  7.1. Moduli spaces of stable maps

Let $\mathfrak{M}_{g,n}$ denote the stack of genus $g$ pre-stable curves with
$n$ marked points and let $\underline{\mathfrak{C}}_{g,n}$ denote its
universal curve. Let $\mathfrak{Pic}_{g,n,d}$ denote the Artin stack which
parameterises genus $g$ pre-stable curves, with $n$ marked points, together
with a line bundle of degree $d$. Let $\mathfrak{Pic}_{g,n,d}^{\rm st}$ denote
the open subset of $\mathfrak{Pic}_{g,n,d}$ consisting of $(C,p_{1}\ldots
p_{n},L)$ which satisfy the stability condition

$L^{\otimes 3}\otimes\omega_{C}\left(\sum_{i=1}^{n}p_{i}\right)$

is ample. Notice that $\mathfrak{M}_{g,n}$ and $\mathfrak{Pic}_{g,n,d}$ are
not separated, but they are smooth and irreducible. The stack
$\mathfrak{Pic}_{g,n,d}$ is locally Noetherian and the stack
$\mathfrak{Pic}_{g,n,d}^{\rm st}$ is Noetherian. From now on we fix $g,n,d$
and the stability condition and we drop all the indices.

We define $\mathfrak{C}$ the universal curve over $\mathfrak{Pic}$ by the
Cartesian diagram. Notice that we also have a universal line bundle
$\mathfrak{L}$ over $\mathfrak{C}$.

${\mathfrak{L}}$${\mathfrak{C}}$${\underline{\mathfrak{C}}}$${\mathfrak{Pic}}$${\mathfrak{M}\text{.}}$$\scriptstyle{\pi}$${\ulcorner}$

We form the cone of sections of $\mathfrak{L}$ as in Chang-Li ([CL15], Section
2)

(19)
$S(\pi_{*}\mathfrak{L}):=\mathrm{Spec}\,\mathrm{Sym}\,(R^{1}\pi_{*}(\mathfrak{L}^{\vee}\otimes\omega_{\mathfrak{C}/\mathfrak{Pic}}))\to\mathfrak{Pic}.$

The definition requires further explanations as follows.

  1. (1)

We use the notation $S(\pi_{*}\mathfrak{L})$ for the cone of sections of
$\mathfrak{L}$, which in [CL15] is denoted by $C(\pi_{*}\mathfrak{L})$; for us
$C(\pi_{*}\mathfrak{L})$ denotes
$\mathrm{Spec}\,\mathrm{Sym}\,\pi_{*}\mathfrak{L}$. In [CL15] the authors
prove that $S(\pi_{*}\mathfrak{L})$ is a stack which parameterises $(C,L,s)$,
where $(C,L)\in\mathfrak{Pic}$ and ${s}\in H^{0}(C,L)$.

  2. (2)

Usually, the total space of a locally free sheaf $\mathfrak{E}$ is
$\mathrm{Spec}\,\mathrm{Sym}\,(\mathfrak{E}^{\vee})$. Here we would like to
take $\mathfrak{E}:=R^{0}\pi_{*}\mathfrak{L}$. By Serre duality, we have
$\mathfrak{E}^{\vee}=R^{1}\pi_{*}(\mathfrak{L}^{\vee}\otimes\omega_{\mathfrak{C}/\mathfrak{Pic}})$.

For the rest of the section, let

(20)
$\displaystyle\mathfrak{F}:=R^{1}\pi_{*}(\mathfrak{L}^{\vee}\otimes\omega_{\mathfrak{C}/\mathfrak{Pic}}).$

Note that since $\pi$ is proper, we have that $\mathfrak{F}$ is a coherent
sheaf on $\mathfrak{Pic}$. As defined in Section 2.3, we consider the stack
$\mathrm{Spec}\,\mathrm{Sym}\,\mathfrak{F}$, which is an abelian cone stack
over $\mathfrak{Pic}$.

Let $\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$ be the moduli space of
genus $g$, degree $d$ stable maps, with $n$ marked points. As in [CFK10], we
have that $\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$ is an open sub-
stack of the stack

(21) $\displaystyle S(\pi_{*}\mathfrak{L}^{\oplus
r+1})=\mathrm{Spec}\,\mathrm{Sym}\,(\oplus_{i=0}^{r}\mathfrak{F})\to\mathfrak{Pic}.$

As before, a point of this cone over $(C,L)\in\mathfrak{Pic}$ is
$(C,L,\underline{s})$ with $\underline{s}\in H^{0}(C,L)^{\oplus r+1}$. Note
that

$S(\pi_{*}\mathfrak{L}^{\oplus
r+1})=\overbrace{S(\pi_{*}\mathfrak{L})\times_{\mathfrak{Pic}}\cdots\times_{\mathfrak{Pic}}S(\pi_{*}\mathfrak{L})}^{r+1\rm\
times}\to\mathfrak{Pic}.$

We define $\mathcal{L},\mathcal{C}$ by the following Cartesian diagram

${\mathcal{L}}$${\mathfrak{L}}$${\mathcal{C}}$${\mathfrak{C}}$${S(\pi_{*}\mathfrak{L}^{\oplus
r+1})}$${\mathfrak{Pic}\text{.}}$${\ulcorner}$$\scriptstyle{\overline{\pi}}$${\ulcorner}$$\scriptstyle{\pi}$

The complex $\oplus_{i=0}^{r}R^{\bullet}\overline{\pi}_{*}\mathcal{L}$ is a
dual obstruction theory for the natural projection
$S(\pi_{*}\mathfrak{L}^{\oplus
r+1})=\mathrm{Spec}\,\mathrm{Sym}\,(\oplus_{i=0}^{r}\mathfrak{F})\to\mathfrak{Pic}$
(see [CL15]).

######  Lemma 7.1.1.

In notation as above we have an isomorphism of sheaves

$(\mathfrak{F})^{\vee}\simeq\pi_{*}\mathfrak{L}$

over $\mathfrak{Pic}$.

###### Proof.

The claim can be obtained by Verdier duality we have

$R\mathcal{H}om(R\pi_{*}\mathfrak{L}^{\vee}\otimes\omega_{\mathfrak{C}/\mathfrak{Pic}},\mathcal{O})\simeq
R\pi_{*}R\mathcal{H}om(\mathfrak{L}^{\vee}\otimes\omega_{\mathfrak{C}/\mathfrak{Pic}},\omega_{\mathfrak{C}/\mathfrak{Pic}}[1]).$

We can obtain an explicit resolution of $\mathfrak{F}$ as follows. Let $A$ be
a sufficiently high power of a very ample section of the morphism
$\mathfrak{C}\to\mathfrak{Pic}$ that is we have
$R^{1}\pi_{*}\mathfrak{L}(A)=0$. Then, we have a morphism

(22) $0\to\mathfrak{L}\to\mathfrak{L}(A)\to\mathfrak{L}(A)|_{A}\to 0.$

This induces a long exact sequence

(23) $0\to R^{0}\pi_{*}\mathfrak{L}\to R^{0}\pi_{*}\mathfrak{L}(A)\to
R^{0}\pi_{*}\mathfrak{L}(A)|_{A}\to R^{1}\pi_{*}\mathfrak{L}\to 0.$

Similarly, we can resolve
$\mathfrak{F}=R^{1}\pi_{*}\mathfrak{L}^{\vee}\otimes\omega_{\mathfrak{C}/\mathfrak{Pic}}$
considering the dual exact sequence

$0\to\mathfrak{L}^{\vee}(-A)\otimes\omega_{\mathfrak{C}/\mathfrak{Pic}}\to\mathfrak{L}^{\vee}\otimes\omega_{\mathfrak{C}/\mathfrak{Pic}}\to\mathfrak{L}^{\vee}\otimes\omega_{\mathfrak{C}/\mathfrak{Pic}}|_{A}\to
0.$

Pushing forward to $\mathfrak{Pic}$ we get

(24)
$R^{0}\pi_{*}\mathfrak{L}^{\vee}\otimes\omega_{\mathfrak{C}/\mathfrak{Pic}}|_{A}\to
R^{1}\pi_{*}\mathfrak{L}^{\vee}(-A)\otimes\omega_{\mathfrak{C}/\mathfrak{Pic}}\to
R^{1}\pi_{*}\mathfrak{L}^{\vee}\otimes\omega_{\mathfrak{C}/\mathfrak{Pic}}\to
0.$

By Serre duality we have isomorphisms
$R^{0}\pi_{*}\mathfrak{L}(A)\simeq(R^{1}\pi_{*}\mathfrak{L}^{\vee}(-A)\otimes\omega_{\mathfrak{C}/\mathfrak{Pic}})^{\vee}$
and
$R^{0}\pi_{*}\mathfrak{L}\simeq(R^{1}\pi_{*}\mathfrak{L}^{\vee}\otimes\omega_{\mathfrak{C}/\mathfrak{Pic}})^{\vee}$.
Applying the functor $\mathcal{H}om(-,\mathcal{O})$ to 24 and using that it is
left-exact, we get

(25) $0\to(\mathfrak{F})^{\vee}\to R^{0}\pi_{*}\mathfrak{L}(A)\to
R^{0}\pi_{*}\mathfrak{L}(A)|_{A}$

Comparing (23) with (25) we get that
$(\mathfrak{F})^{\vee}\simeq\pi_{*}\mathfrak{L}$.

The last claim follows from the fact that taking the dual of vector bundle
commutes with base change. ∎

######  Remark 7.1.2.

The proof of Lemma 7.1.1 shows that we have a resolution to the left given by
(24). The isomorphisms after Equation 23 show that the morphism

$R^{0}\pi_{*}\mathfrak{L}^{\vee}\otimes\omega_{\mathfrak{C}/\mathfrak{Pic}}|_{A}\to
R^{1}\pi_{*}\mathfrak{L}^{\vee}(-A)\otimes\omega_{\mathfrak{C}/\mathfrak{Pic}}$

is dual to

$R^{0}\pi_{*}\mathfrak{L}(A)\to R^{0}\pi_{*}\mathfrak{L}(A)|_{A}.$

We will use this duality in Section 8. Hu and Li work with the resolution to
the right we have in 23. In the previous sections we used the resolution to
the left Equation 23. Since dual morphisms have the same fitting ideals, both
morphisms give the same Hu–Li blow-up.

######  Definition 7.1.3.

Consider the Zariski closure in
$\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$ of the locus where the curve
is smooth of genus $g$. We call this component the main component and we
denote it by $\overline{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)$.

Note that on $\overline{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)$, the
universal curve is generically smooth and $\pi_{*}\mathcal{L}$ is generically
a vector bundle.

######  Proposition 7.1.4.

We have that $\overline{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)$ is an
open substack of

$\mathrm{Spec}\,(\mathrm{Sym}\,\oplus_{i=0}^{r}\mathfrak{F})^{\mathrm{tf}}=\prod_{\mathfrak{Pic}}^{r+1}\mathrm{Spec}\,(\mathrm{Sym}\,\mathfrak{F})^{\mathrm{tf}}.$

###### Proof.

We have a surjective morphism
$\mu:\mathrm{Sym}\,\mathfrak{F}\rightarrow(\mathrm{Sym}\,\mathfrak{F})^{\mathrm{tf}}$.
The torsion of $\mathrm{Sym}\,\mathfrak{F}$ is in the kernel of $\mu$.

This gives a closed embedding

$\mathrm{Spec}\,(\mathrm{Sym}\,\pi_{*}\mathfrak{F})^{\mathrm{tf}}\hookrightarrow\mathrm{Spec}\,\mathrm{Sym}\,\pi_{*}\mathfrak{F}.$

$\overline{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)$ is the open substack
of $\mathrm{Spec}\,(\mathrm{Sym}\,\pi_{*}\mathfrak{F}^{\oplus
r+1})^{\mathrm{tf}}$ given by imposing the stability condition for stable
maps. Moreover, for a module over an integral domain $R$, we have that taking
torsion commutes with direct sums. This gives the equality in the statement. ∎

######  Remark 7.1.5.

For $g\geq 1$ the moduli space
$\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$ has several irreducible
components (see for example [HL10]). While for $g=1$ these components are
understood, in general we do not have a complete picture. However, after
desingularizing $\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$, theorem
5.2.1 describes its components.

###  7.2. Maps with fields

We recall the construction of maps with fields in [CL12] and its properties.

In the following we fix $k\in\mathbb{Z}$, $k>1$ and we consider the sheaf

$\pi_{*}(\mathfrak{L}^{\oplus
r+1}\oplus(\mathfrak{L}^{\otimes-k}\otimes\omega_{\mathfrak{C}/\mathfrak{Pic}}))$

on $\mathfrak{Pic}$ with corresponding abelian cone

$S\left(\pi_{*}(\mathfrak{L}^{\oplus
r+1}\oplus(\mathfrak{L}^{{}^{\otimes-k}}\otimes\omega_{\mathfrak{C}/\mathfrak{Pic}})\right)=\mathrm{Spec}\,\mathrm{Sym}\,R^{1}\pi_{*}\left(\left(\mathfrak{L}^{\vee}\otimes\omega_{\mathfrak{C}/\mathfrak{Pic}}\right)^{\oplus
r+1}\oplus\mathfrak{L}^{\otimes
k}\right)\stackrel{{\scriptstyle\mu^{p}}}{{\rightarrow}}\mathfrak{Pic}.$

Imposing stable map stability we obtain the moduli space of maps with fields
$\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}$. So an element of
$\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}$ over
$(C,L,\underline{s})\in\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$ is
given by a choice of a section $p\in H^{0}(C,L^{\otimes-k}\otimes\omega_{C})$.
Consider the Cartesian diagram

(26) ${\mathcal{C}^{p}}$${\
\mathcal{C}}$${\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}}$${\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)\text{.}}$$\scriptstyle{\overline{\pi}^{p}}$$\scriptstyle{\nu^{p}}$${\ulcorner}$$\scriptstyle{\overline{\pi}}$

The complex

$\mathbb{E}^{\bullet}:=R^{\bullet}\overline{\pi}^{p}_{*}(\oplus_{i=0}^{r}\mathcal{L}\oplus\mathcal{L}^{\otimes-k}\otimes\omega_{\overline{\pi}^{p}})$

is a dual obstruction theory for the morphism $\mu^{p}$.

The stack $\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}$ is not proper,
but the perfect obstruction theory admits a cosection $\sigma$. This data
gives a cosection localised virtual class
$[\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}]^{\mathrm{vir}}_{\sigma}$.

For $X$ a smooth subvariety cut out by any regular section of
$\mathcal{O}_{\mathbb{P}^{r}}(k)$ the authors of [CL12] prove that

(27)
$[\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}]^{\mathrm{vir}}_{\sigma}=(-1)^{(r+1)d+1-g}[\overline{\mathcal{M}}_{g,n}(X,d)]^{\mathrm{vir}}\in
A_{d^{\mathrm{vir}}(\overline{\mathcal{M}}_{g,n}(X,d))}(\overline{\mathcal{M}}_{g,n}(X,d))$

###  7.3. Reduced Gromov–Witten invariants

We start with a technical lemma below, which allows us to apply the results in
Section 4.

######  Lemma 7.3.1 ([Sta22, Tag 0ESR]).

Let $\mathfrak{E}$ be any vector bundle on $\mathfrak{C}_{\mathfrak{P}}$, the
universal curve on some locally Noetherian integral stack $\mathfrak{P}$ over
$\mathfrak{M}$. Let $\mathfrak{F}=R^{1}{\pi_{\mathfrak{P}}}_{*}\mathfrak{E}$.
There exists a blow-up $b:\mathfrak{P}^{\prime}\to\mathfrak{P}$ such that
$b^{*}\mathfrak{F}$ is a perfect $\mathcal{O}_{\mathfrak{P}^{\prime}}$ module
of tor dimension $\leq 1$.

###### Proof.

Consider
$b:\mathfrak{P}^{\prime}=\mathrm{Bl}^{HL}_{\mathfrak{F}}(\mathfrak{P})\to\mathfrak{P}$.
Locally on $\mathfrak{P}^{\prime}$, $F_{r}(b^{*}\mathfrak{F})$, is a principal
ideal defined by a non-zero divisor and $F_{r-1}(b^{*}\mathfrak{F})=0$. Then
$b^{*}\mathfrak{F}$ has tor dimension $\leq 1$ by [Sta22, Tag 0F7M]. Note that
$b^{*}\mathfrak{F}=R^{1}\pi_{\mathfrak{P}^{\prime}*}(\overline{b}^{*}\mathfrak{E})$,
for
$\overline{b}=\mathfrak{C}_{\mathfrak{P}^{\prime}}\to\mathfrak{C}_{\mathfrak{P}}$.
∎

######  Remark 7.3.2.

This allows us to apply theorem 4.1.14 to
$\mathrm{Bl}^{HL}_{R^{1}\pi_{*}\mathfrak{E}}\mathfrak{Pic}$, since now
$R^{1}\pi_{*}\mathfrak{E}$ is locally diagonalizable of tor dimension $\leq
1$.

In the following we fix $k\geq 1$. Let

$\mathfrak{E}_{k}:=R^{1}\pi_{*}\mathfrak{L}^{\otimes k}$

denote the sheaf on $\mathfrak{Pic}$ whose restriction to a point
$(C,L)\in\mathfrak{Pic}$ is

$H^{1}(C,L^{\otimes
k})\simeq(H^{0}(C,L^{\otimes-k}\otimes\omega_{C}))^{\vee}.$

We define

$\widetilde{\mathfrak{Pic}}_{k}:=\mathrm{Bl}^{HL}_{\mathfrak{E}_{k}}\mathrm{Bl}^{HL}_{\mathfrak{F}}\mathfrak{Pic}.$

Let $p_{k}:\widetilde{\mathfrak{Pic}}_{k}\to\mathfrak{Pic}$ be the natural
projection. Consider the Cartesian diagram

$\textstyle{\tilde{}\mathfrak{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q_{k}}$$\scriptstyle{\tilde{\pi}}$$\textstyle{\mathfrak{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{\widetilde{\mathfrak{Pic}}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p_{k}}$$\textstyle{\mathfrak{Pic}}$

where $\mathfrak{C}$ is the universal curve over $\mathfrak{Pic}$.

We define $\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$ as the following
Cartesian diagram:

(28) ${\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)}$${\
\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)}$${\widetilde{\mathfrak{Pic}}_{k}}$${\mathfrak{Pic}\text{.}}$$\scriptstyle{\widetilde{\mu}}$$\scriptstyle{\overline{p}_{k}}$${\ulcorner}$$\scriptstyle{\mu}$$\scriptstyle{p_{k}}$

One can see that we have an open embedding

$\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)\hookrightarrow\mathrm{Spec}\,\mathrm{Sym}\,\
\oplus_{i=0}^{r}p_{k}^{*}\mathfrak{F}.$

We define the main component of
$\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$ as the fiber product

(29) ${\widetilde{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)}$${\
\overline{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)}$${\widetilde{\mathfrak{Pic}}_{k}}$${\mathfrak{Pic}\text{.}}$$\scriptstyle{\widetilde{\mu}^{\circ}}$$\scriptstyle{\overline{p}_{k}}$${\ulcorner}$$\scriptstyle{\mu^{\circ}}$$\scriptstyle{p_{k}}$

Again, we have an open embedding

$\widetilde{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)\subset\mathrm{Spec}\,\mathrm{Sym}\,\
\oplus_{i=0}^{r}(p_{k}^{*}\mathcal{F})^{\mathrm{tf}}.$

By [CL12] the morphism $\mu$ has a perfect obstruction theory equal to
$\oplus_{i=0}^{r}R^{\bullet}\overline{\pi}_{*}\mathcal{L}$. Let
$\widehat{\pi}:\tilde{\mathcal{C}}\to\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$
be the universal curve and let
$\widehat{q_{k}}:\tilde{\mathcal{C}}\to\mathcal{C}$ the morphism induced by
$q_{k}$. The morphism $\tilde{\mu}$ has a dual perfect obstruction theory
equal to

$\oplus_{i=0}^{r}R^{\bullet}\widehat{\pi}_{*}\widehat{q_{k}}^{*}\mathcal{L}.$

These perfect obstruction theories induce virtual classes

$[\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)]^{\mathrm{vir}}:=\mu^{!}[\mathfrak{Pic}]\in
A_{*}(\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d))$

and

$[\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)]^{\mathrm{vir}}:=\widetilde{\mu}^{!}[\widetilde{\mathfrak{Pic}}]\in
A_{*}(\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)),$

where $\mu^{!}$ and $\widetilde{\mu}^{!}$ are defined as in [Man11].
Similarly, we define $\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}$ as
the following Cartesian diagram:

(30) ${\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}}$${\
\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}}$${\widetilde{\mathfrak{Pic}}_{k}}$${\mathfrak{Pic}\text{.}}$$\scriptstyle{\widetilde{\mu}^{p}}$${\ulcorner}$$\scriptstyle{\mu^{p}}$$\scriptstyle{p_{k}}$

This gives a localised virtual class
$[\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}]^{\mathrm{vir}}$. See
[CL12, Section 3] for details.

######  Remark 7.3.3.

While $\mathfrak{Pic}$ is smooth, $\widetilde{\mathfrak{Pic}}_{k}$ does not
need to be smooth. This is not a problem, all we need for a well-defined
virtual class is that $\widetilde{\mathfrak{Pic}}$ has pure dimension. This is
true, since $\mathfrak{Pic}$ has pure dimension and $p_{k}$ is birational.

######  Theorem 7.3.4.

In notation as above, let

$\displaystyle\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}=$
$\displaystyle\cup_{\lambda\in\Lambda}\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p,\lambda}$
$\displaystyle\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)=$
$\displaystyle\cup_{\theta\in\Theta}\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{\theta},$

with $\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p,\lambda}$ irreducible
components of $\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}$ and
$\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{\theta}$ irreducible
components of $\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$. Let

$\displaystyle\widehat{\pi}^{p,\lambda}:\tilde{\mathcal{C}}^{\lambda}$
$\displaystyle\to\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p,\lambda}$
$\displaystyle\tilde{\pi}^{\theta}:\tilde{\mathcal{C}}^{\theta}$
$\displaystyle\to\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{\theta}.$

The following statements hold.

  1. (1)

The morphism $\bar{p}_{k}$ is birational and proper.

  2. (2)

The irreducible components
$\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p,\lambda}$ and
$\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{\theta}$ are smooth over
their image in $\widetilde{\mathfrak{Pic}}_{k}$. In particular,
$\widetilde{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)$ is smooth over
$\widetilde{\mathfrak{Pic}}$.

  3. (3)

The sheaf $\widehat{\pi}^{p,\lambda}_{*}ev^{*}\mathcal{O}(k)$ is a locally
free sheaf on $\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p,\lambda}$,
the sheaf $\tilde{\pi}^{\theta}_{*}ev^{*}\mathcal{O}(k)$ is a locally free
sheaf on $\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{\theta}$. In
particular $\tilde{\pi}^{\circ}_{*}ev^{*}\mathcal{O}(k)$ is a locally free
sheaf on $\widetilde{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)$.

###### Proof.

1\. We have that $\overline{p}_{k}$ is proper, as $p_{k}$ is proper.

2\. Consider the following diagram

(31)
${\mathrm{Spec}\,\mathrm{Sym}\,p_{k}^{*}(\mathfrak{F}\oplus\mathfrak{E}_{k})}$${\
\mathrm{Spec}\,\mathrm{Sym}\,\mathfrak{F}\oplus\mathfrak{E}_{k}}$${\widetilde{\mathfrak{Pic}}_{k}}$${\mathfrak{Pic}\text{.}}$${\ulcorner}$$\scriptstyle{p_{k}}$

We have that $\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}$ is an open
sub-stack of
$\mathrm{Spec}\,\mathrm{Sym}\,p_{k}^{*}(\mathfrak{E}_{k}\oplus\oplus_{i=0}^{r}\mathfrak{F})$
and by Theorem 5.2.1 we have that the irreducible components of the stacks

$\mathrm{Spec}\,\mathrm{Sym}\,\
\oplus_{i=0}^{r}\mathcal{F}\oplus\mathfrak{E}_{k}\text{\ \ \ and \ \ \
}\mathrm{Spec}\,\mathrm{Sym}\,\ \oplus_{i=0}^{r}\mathfrak{F}$

are smooth over their image in $\widetilde{\mathfrak{Pic}}_{k}$.

This shows that $\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p,\lambda}$
and $\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{\theta}$ are smooth over
their image in $\widetilde{\mathfrak{Pic}}_{k}$. In particular

$\widetilde{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)=\mathrm{Spec}\,\mathrm{Sym}\,(\oplus_{i=0}^{r}p_{k}^{*}\mathfrak{F})^{\mathrm{tf}}\mbox{
is smooth.}$

3\. Let
$\mu^{\lambda}:\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p,\lambda}\to\mathfrak{Pic}$
be the restriction of $\mu^{p}$. Let $Z^{\lambda}$ be the image of
$\mu^{\lambda}$. Let $\pi^{\lambda}:\mathfrak{C}^{\lambda}\to Z^{\lambda}$ be
the restriction of $\pi$. Let $\tilde{Z}^{\lambda}$ be the fibre product

(32) ${\tilde{Z}^{\lambda}}$${\
Z^{\lambda}}$${\widetilde{\mathfrak{Pic}}_{k}}$${\mathfrak{Pic}\text{.}}$$\scriptstyle{p_{k}^{\lambda}}$${\ulcorner}$$\scriptstyle{p_{k}}$

Let $\tilde{\pi}^{\lambda}:\tilde{\mathfrak{C}}^{\lambda}\to Z^{\lambda}$ be
the restriction of $\tilde{\pi}$ and let
$q_{k}^{\lambda}:\tilde{\mathfrak{C}}^{\lambda}\to\mathfrak{C}^{\lambda}$ be
the restriction of $q_{k}$. By commutativity of proper push-forwards with
base-change we have that

$(p_{k}^{\lambda})^{*}R^{\bullet}\pi^{\lambda}_{*}\mathfrak{L}\simeq
R^{\bullet}\tilde{\pi}^{\lambda}_{*}(q_{k}^{\lambda})^{*}\mathfrak{L}$

Again, cohomology and base change in the Cartesian diagram

(33) ${\tilde{\mathcal{C}}^{\lambda}}$${\
\tilde{\mathfrak{C}}^{\lambda}}$${\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{\lambda}}$${\tilde{Z}^{\lambda}\text{.}}$$\scriptstyle{\widehat{\pi}^{\lambda}}$$\scriptstyle{\nu^{\lambda}}$${\ulcorner}$$\scriptstyle{\tilde{\pi}^{\lambda}}$$\scriptstyle{\mu^{\lambda}}$

gives

(34)
$R^{\bullet}\widehat{\pi}^{\lambda}_{*}ev^{*}\mathcal{O}(k)=(\mu^{\lambda})^{*}R^{\bullet}\tilde{\pi}^{\lambda}_{*}\mathfrak{L}^{k}.$

We have a short exact sequence

$0\to R^{0}\tilde{\pi}^{\lambda}_{*}\mathfrak{L}^{k}\to
E^{0}\stackrel{{\scriptstyle\phi}}{{\rightarrow}}E^{1}\to
R^{1}\tilde{\pi}^{\lambda}_{*}\mathfrak{L}^{k}\to 0.$

By construction we have that $\phi$ is locally diagonal. By eq. 34 we have
that

$R^{\bullet}\widehat{\pi}^{\lambda}_{*}ev^{*}\mathcal{O}(k)\simeq[(\mu^{\lambda})^{*}E^{0}\stackrel{{\scriptstyle(\mu^{\lambda})^{*}\phi}}{{\longrightarrow}}(\mu^{\lambda})^{*}E^{1}].$

Since $(\mu^{\lambda})^{*}\phi$ is locally diagonal, proposition 4.1.2 implies
that $R^{0}\widehat{\pi}^{\lambda}_{*}ev^{*}\mathcal{O}(k)$ is locally free.

A similar argument shows that $\tilde{\pi}^{\theta}_{*}ev^{*}\mathcal{O}(k)$
is a locally free sheaf on
$\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{\theta}$ and in particular
$\tilde{\pi}^{\circ}_{*}ev^{*}\mathcal{O}(k)$ is a locally free sheaf on
$\widetilde{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)$. ∎

######  Remark 7.3.5.

In [VZ08, HL10] the authors desingularize $\pi_{*}\mathfrak{L}$ for genus one
and, as a consequence, the sheaf $p_{k}^{*}(\pi_{*}\mathfrak{L}^{\otimes
k})^{\mathrm{tf}}$ becomes locally free on $\widetilde{\mathfrak{Pic}}_{1}$
for any $k\geq 1$. In the following we give a proof of this fact.

In a neighbourhood of
$(\tilde{C},\tilde{L})\in\mathrm{Bl}_{\pi_{*}\mathfrak{L}}\mathfrak{Pic}$ we
can choose $Z$ a section of $\mathfrak{L}^{\otimes k-1}$. This gives an exact
sequence

(35) $0\to R^{0}\tilde{\pi}_{*}\mathfrak{L}\to
R^{0}\tilde{\pi}_{*}(\mathfrak{L}^{\otimes k})\to
R^{0}\tilde{\pi}_{*}(\mathfrak{L}^{\otimes k}|_{Z})\to
R^{1}\tilde{\pi}_{*}\mathfrak{L}\to R^{1}\tilde{\pi}_{*}(\mathfrak{L}^{\otimes
k})\to 0.$

Notice that for $\tilde{C}$ a curve of genus one, and for any $k\geq 1$, we
have an isomorphism

(36) $H^{1}(\tilde{C},\tilde{L})\simeq H^{1}(\tilde{C},\tilde{L}^{\otimes
k}).$

This implies that the last arrow in sequence (35) is an isomorphism and thus
we get a short exact sequence

$0\to R^{0}\tilde{\pi}_{*}\mathfrak{L}\to
R^{0}\tilde{\pi}_{*}(\mathfrak{L}^{\otimes k})\to
R^{0}\tilde{\pi}_{*}(\mathfrak{L}^{\otimes k}|_{Z})\to 0.$

Since $R^{0}\tilde{\pi}_{*}(\mathfrak{L}^{\otimes k}|_{Z})$ and
$R^{0}\tilde{\pi}_{*}\mathfrak{L}$ are locally free sheaves on
$\mathrm{Bl}_{\pi_{*}\mathfrak{L}}\mathfrak{Pic}$, we get that
$R^{0}\tilde{\pi}_{*}(\mathfrak{L}^{\otimes k})$ is locally free.

######  Proposition 7.3.6.

We have the following equality

$(\overline{p}_{k})_{*}[\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)]^{\mathrm{vir}}=[\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)]^{\mathrm{vir}}$

###### Proof.

Note that by cohomology and base change we have that
$R^{\bullet}\tilde{\pi}_{*}\overline{p}_{k}^{*}\mathcal{L}=\overline{p}_{k}^{*}R^{\bullet}\overline{\pi}_{*}\mathcal{L}$.
As $p_{k}$ is birational and projective we have
$(p_{k})_{*}[\widetilde{\mathfrak{Pic}}_{k}]=[\mathfrak{Pic}]$. We now apply
Costello’s Pushforward theorem ([HW22]) to (28) and we get

$(\overline{p}_{k})_{*}[\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)]^{\mathrm{vir}}=[\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)]^{\mathrm{vir}}.\qed$

In the following we define reduced Gromov–Witten invariants.

######  Assumption 7.3.7.

In the following we fix $d>2g-2$. For $C$ a smooth genus $g$ curve, $L$ a line
bundle of degree $d$, and $d>2g-2$, we have that $H^{1}(C,L)=0$. This shows
that for $d>2g-2$, the locus of stable maps with smooth domain is smooth and
irreducible, so its closure is an irreducible component of
$\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$.

######  Construction 7.3.8.

Let $X\subset\mathbb{P}^{r}$ be a smooth complete intersection, given by a
section

$s:\mathcal{O}_{\mathbb{P}}(1)^{\oplus r+1}\to\mathcal{O}_{\mathbb{P}}(k).$

We define

(37) ${\overline{\mathcal{M}}^{\circ}_{g,n}(X,d)}$${\
\overline{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)}$${\overline{\mathcal{M}}_{g,n}(X,d)}$${\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)\text{.}}$${\ulcorner}$

The main component $\overline{\mathcal{M}}^{\circ}_{g,n}(X,d)$ does not have a
perfect obstruction theory. In order to define a perfect obstruction theory on
it, we define

(38) ${\widetilde{\mathcal{M}}^{\circ}_{g,n}(X,d)}$${\
\widetilde{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)}$${\widetilde{\mathcal{M}}_{g,n}(X,d)}$${\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)\text{.}}$${\ulcorner}$$\scriptstyle{i}$

By theorem 7.3.4 we get that $(\pi_{*}\mathcal{L}^{\otimes k})^{\mathrm{tf}}$
is locally free on $\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$. We
define

(39)
$[\widetilde{\mathcal{M}}^{\circ}_{g,n}(X,d)]^{\mathrm{vir}}=i^{!}[\widetilde{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)].$

Note that for any $1\leq i\leq n$ we have morphisms

$\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)\to\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)\stackrel{{\scriptstyle
ev_{i}}}{{\longrightarrow}}\mathbb{P}^{r}$

and

$\widetilde{\mathcal{M}}_{g,n}(X,d)\to\overline{\mathcal{M}}_{g,n}(X,d)\stackrel{{\scriptstyle
ev_{i}}}{{\longrightarrow}}X.$

By abuse of notation we denote these compositions by $ev_{i}$.

######  Proposition 7.3.9.

Let assume $d>2g-2$. Let
$p^{\prime}:\widetilde{\mathfrak{Pic}}^{\prime}\to\mathfrak{Pic}$ and
$p^{\prime\prime}:\widetilde{\mathfrak{Pic}}^{\prime\prime}\to\mathfrak{Pic}$
be birational projective maps such that
$(p_{i}^{*}\pi_{*}\mathcal{F})^{\mathrm{tf}}$ and
$p_{i}^{*}(\pi_{*}\mathfrak{L}^{\otimes k})^{\mathrm{tf}}$ are locally free.
Consider $\widetilde{\mathcal{M}}^{\circ}_{g,n}(X,d)^{\prime}$ and
$\widetilde{\mathcal{M}}^{\circ}_{g,n}(X,d)^{\prime\prime}$ defined
analogously to $\widetilde{\mathcal{M}}^{\circ}_{g,n}(X,d)$ above. Then we
have

$\int_{[\widetilde{\mathcal{M}}^{\circ}_{g,n}(X,d)^{\prime}]^{\mathrm{vir}}}\prod
ev^{*}\gamma_{i}=\int_{[\widetilde{\mathcal{M}}^{\circ}_{g,n}(X,d)^{\prime\prime}]^{\mathrm{vir}}}\prod
ev^{*}\gamma_{i}$

This proposition permits us to define the reduced Gromov–Witten invariants as
they are independent of the blowing-up of $\mathfrak{Pic}$.

######  Definition 7.3.10.

For $d>2g-2$, we call _reduced_ Gromov–Witten invariants of $X$, the following
numbers

$\int_{[\widetilde{\mathcal{M}}^{\circ}_{g,n}(X,d)]^{\mathrm{vir}}}\prod
ev^{*}\gamma_{i}.$

###### Proof of Proposition 7.3.9.

Consider the fiber product

$\textstyle{\widehat{\mathfrak{Pic}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\mathfrak{Pic}}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime}}$$\textstyle{\widetilde{\mathfrak{Pic}}^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{p^{\prime\prime}}$$\textstyle{\mathfrak{Pic}}$

This gives

${\widehat{\mathcal{M}}(X)^{\circ}}$${\widetilde{\mathcal{M}}^{\circ}_{g,n}(X,d)^{\prime}}$${\widetilde{\mathfrak{Pic}}^{\prime}}$${\overline{\mathcal{M}}^{\circ}_{g,n}(X,d)^{\prime\prime}}$${\overline{\mathcal{M}}^{\circ}_{g,n}(X,d)}$${\mathfrak{Pic}}$${\widetilde{\mathfrak{Pic}}^{\prime\prime}}$${\mathfrak{Pic}}$$\scriptstyle{q^{\prime\prime}}$$\scriptstyle{q^{\prime}}$$\scriptstyle{p^{\prime}}$$\scriptstyle{p^{\prime\prime}}$

and thus $q^{\prime}$ and $q^{\prime\prime}$ are birational and projective.

Let
$\mu^{\prime}:\widetilde{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)^{\prime}\to\widetilde{\mathfrak{Pic}}^{\prime}$,
$\mu^{\prime\prime}:\widetilde{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)^{\prime\prime}\to\widetilde{\mathfrak{Pic}}^{\prime\prime}$
and
$\hat{\mu}:\widehat{\mathcal{M}}(\mathbb{P}^{r})\to\widehat{\mathfrak{Pic}}$
denote the maps which forget sections. Then we have Cartesian diagrams

$\textstyle{\widehat{\mathcal{M}}(X)^{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q^{\prime\prime}}$$\textstyle{\widehat{\mathcal{M}}(\mathbb{P}^{r})^{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\mu}}$$\scriptstyle{r^{\prime}}$$\textstyle{\widehat{\mathfrak{Pic}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\mathcal{M}}^{\circ}_{g,n}(X,d)^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime}}$$\textstyle{\widetilde{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{\prime}}$$\textstyle{\widetilde{\mathfrak{Pic}}^{\prime}}$

and

$\textstyle{\widehat{\mathcal{M}}(X)^{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{q^{\prime}}$$\textstyle{\widehat{\mathcal{M}}(\mathbb{P}^{r})^{\circ}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\widehat{\mu}}$$\scriptstyle{r^{\prime\prime}}$$\textstyle{\widehat{\mathfrak{Pic}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{\mathcal{M}}^{\circ}_{g,n}(X,d)^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{i^{\prime\prime}}$$\textstyle{\widetilde{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)^{\prime\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mu^{\prime\prime}}$$\textstyle{\widetilde{\mathfrak{Pic}}^{\prime\prime}}$

This implies that we have a virtual class

(40)
$[\widehat{\mathcal{M}}(X)^{\circ}]^{\mathrm{vir}}:=(i^{\prime})^{!}[\widehat{\mathcal{M}}(\mathbb{P}^{r})^{\circ}].$

Diagram 38 implies that
$[\widehat{\mathcal{M}}(X)^{\circ}]^{\mathrm{vir}}=i^{!}[\widehat{\mathcal{M}}(\mathbb{P}^{r})^{\circ}]$.
Similarly we also get that
$(i^{\prime\prime})^{!}[\widehat{\mathcal{M}}(\mathbb{P}^{r})^{\circ}]=i^{!}[\widehat{\mathcal{M}}(\mathbb{P}^{r})^{\circ}]$
and so

(41)
$[\widehat{\mathcal{M}}(X)^{\circ}]^{\mathrm{vir}}=(i^{\prime\prime})^{!}[\widehat{\mathcal{M}}(\mathbb{P}^{r})^{\circ}].$

Since $r^{\prime}$ and $r^{\prime\prime}$ are birational and proper we have

(42) $\displaystyle
r^{\prime}_{*}[\widehat{\mathcal{M}}(\mathbb{P}^{r})^{\circ}]$
$\displaystyle=[\widetilde{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)^{\prime}]\text{
and}$ (43) $\displaystyle
r^{\prime\prime}_{*}[\widehat{\mathcal{M}}(\mathbb{P}^{r})^{\circ}]$
$\displaystyle=[\widetilde{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)^{\prime\prime}].$

Using 40,41, 42, 43 and commutativity of pull-backs with push-forwards, we get

(44)
$\displaystyle\begin{split}q^{\prime\prime}_{*}[\widehat{\mathcal{M}}(X)^{\circ}]^{\mathrm{vir}}&=[\widetilde{\mathcal{M}}^{\circ}_{g,n}(X,d)^{\prime}]^{\mathrm{vir}}\text{\
and}\\\
q^{\prime}_{*}[\widehat{\mathcal{M}}(X)^{\circ}]^{\mathrm{vir}}&=[\widetilde{\mathcal{M}}^{\circ}_{g,n}(X,d)^{\prime\prime}]^{\mathrm{vir}}.\end{split}$

Intersection both equations above with $\prod ev^{*}\gamma_{i}$ we get the
conclusion. ∎

We also have the following.

######  Proposition 7.3.11.

The localised invariants do not depend on the blow-up of $\mathfrak{Pic}$,
more precisely,

$\deg[\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}]^{\mathrm{vir}}=\deg[\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}]^{\mathrm{vir}}.$

###### Proof.

It follows from eq. 27 and proposition 7.3.9. ∎

######  Remark 7.3.12.

The results of proposition 7.3.9 and proposition 7.3.11 can be stated at level
of virtual classes. The statement of proposition 7.3.9 with virtual classes is
given by eq. 44.

Let
$\mathfrak{C}_{\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}/\widetilde{\mathfrak{Pic}}}=\cup_{i}\mathfrak{C}_{i}$
where $\mathfrak{C}_{i}$ denotes an irreducible component and let
$\mathfrak{C}_{0}$ denote the component supported on the main component of
$\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}$. Let
$[\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}]_{i}^{\mathrm{vir}}$
denote the class corresponding to the component $\mathfrak{C}_{i}$.

######  Proposition 7.3.13.

We have

$\deg[\widetilde{\mathcal{M}}^{\circ}_{g,n}(X,d)]^{\mathrm{vir}}=(-1)^{kd-g+1}\deg[\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}]_{0}^{\mathrm{vir}}.$

###### Proof.

This follows the lines of proof of Corollary 4.4 in [CL15]. Let

$\mathbb{E}_{1}^{\bullet}:=R^{\bullet}\widehat{\pi}^{p}_{*}(\oplus_{i=0}^{r}\mathcal{L}),\hskip
14.22636pt\mathbb{E}_{2}^{\bullet}:=R^{\bullet}\widehat{\pi}^{p}_{*}(\mathcal{L}^{\otimes-k}\otimes\omega_{\widehat{\pi}^{p}})$

and let $\mathcal{E}_{i}=h^{1}/h^{0}(\mathbb{E}_{i}^{\bullet})$ and
$\mathcal{E}=h^{1}/h^{0}(\mathbb{E}^{\bullet})$. We have that
$\mathcal{E}_{i}$ is a vector bundle stack on
$\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}$ and
$\mathcal{E}\simeq\mathcal{E}_{1}\oplus\mathcal{E}_{2}$. Let $\mathcal{U}$ be
the open subset of the main component of
$\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}$, with consists of maps
with fields with irreducible source. On $\mathcal{U}$ we have
$R^{1}\widehat{\pi}_{*}f^{*}\mathcal{O}(k)=0$, and thus $\mathcal{U}$ is also
an open subset of $\widetilde{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)$.
Using that $\mathcal{U}$ is smooth and unobstructed, we see that
$\mathfrak{C}_{\mathcal{U}/\widetilde{\mathfrak{Pic}}}$ is isomorphic to the
vector bundle stack $\mathcal{E}_{1}|_{\mathcal{U}}$. Since the embedding
$\mathfrak{C}_{\mathcal{U}/\widetilde{\mathfrak{Pic}}}\hookrightarrow
h^{1}/h^{0}(\mathcal{E}|_{\mathcal{U}})$ is

$(\mathcal{E}_{1}\oplus
0)|_{\mathcal{U}}\hookrightarrow(\mathcal{E}_{1}\oplus\mathcal{E}_{2})|_{\mathcal{U}}$

and
$\mathfrak{C}_{\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}/\widetilde{\mathfrak{Pic}}}\hookrightarrow\mathcal{E}$
is a closed embedding, we get that $\mathfrak{C}_{0}\simeq\mathcal{E}_{1}$. By
the definition of the localised cosection virtual class, we get

$[\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}]^{\mathrm{vir}}_{0}=0^{!}_{\sigma,loc}[\mathfrak{C}_{0}]=0^{!}[0_{\mathcal{E}_{2}}],$

where $0_{\mathcal{E}_{2}}$ is the zero section of
$\mathcal{E}_{2}|_{\widetilde{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)}$.
By Lemma 4.3 in [CL15] with the complex
$R:=R^{\bullet}\widehat{\pi}^{p,\circ}_{*}\mathcal{L}^{\otimes k}$ and Theorem
7.3.4, part 3 we get

$[\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}]^{\mathrm{vir}}_{0}=c_{\rm
top}(R^{1}\widehat{\pi}^{p,\circ}_{*}(\mathcal{L}^{\otimes-k}\otimes\omega_{\widehat{\pi}^{p,\circ}}))\cdot[\widetilde{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)].$

By Serre duality we have that

$c_{1-g+kd}(R^{1}\widehat{\pi}^{p,\circ}_{*}(\mathcal{L}^{\otimes-k}\otimes\omega_{\widehat{\pi}^{p}}))=(-1)^{1-g+kd}c_{1-g+kd}(R^{0}\widehat{\pi}^{p}_{*}\mathcal{L}^{\otimes
k})$

and thus

$[\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)^{p}]^{\mathrm{vir}}_{0}=(-1)^{1-g+kd}c_{1-g+kd}(R^{0}\widehat{\pi}^{p}_{*}\mathcal{L}^{\otimes
k})\cdot[\widetilde{\mathcal{M}}^{\circ}_{g,n}(\mathbb{P}^{r},d)].$

This proves the claim. ∎

######  Conjecture 7.3.14.

Let $X$ be a threefold. Then

$\deg[\widetilde{\mathcal{M}}_{g,n}(X,d)^{p}]_{i}^{\mathrm{vir}}=c_{i}\deg[\widetilde{\mathcal{M}}^{\circ}_{g_{i},n}(X,d)]^{\mathrm{vir}},$

for some $c_{i}\in\mathbb{Q}$ and $g_{i}<g$.

######  Remark 7.3.15.

The conjecture has been proved for genus one [Zin09a, Zin09b, Zin08], [CL15],
[LO21, LO22] and genus two [LLO22].

In genus $g=1$ and $X$ a Calabi–Yau three-fold, the conjecture translates into

$\deg[\overline{\mathcal{M}}_{g,n}(X,d)]=\deg[\widetilde{\mathcal{M}}^{\circ}_{g,n}(X,d)]^{\mathrm{vir}}+\frac{1}{12}\deg[\overline{\mathcal{M}}_{0,n}(X,d)]^{\mathrm{vir}}.$

##  8\. Desingularizations in genus one

In genus one, reduced Gromov–Witten invariants were originally defined using
the desingularization constructed in [VZ08]. It consists of a sequence of
blow-ups determined by the geometry of the moduli space
$\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)$. In [HL10], local equations
for the blowup are determined. We aim to compare this desingularization with
the one obtained using the Rossi–Villamayor blow-up
$\mathrm{Bl}_{\mathfrak{F}}\mathfrak{Pic}$, with $\mathfrak{F}$ as in eq. 20.
In particular, we describe $\mathrm{Bl}_{\mathfrak{F}}\mathfrak{Pic}$ in
locally in the spirit of [HL10].

####  8.0.1. Charts

In genus one, the original definition of refined Gromov–Witten invariants
comes from [VZ08]. The main is idea is to apply a sequence of blow ups to
$\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$ in order to desingularize the
main component. Strictly speaking, the sequence of blow ups takes place in the
stack $\mathfrak{M}_{1}^{wt}$ of genus-1 prestable curves endowed with a
weight. Let $\Theta_{k}$ denote the closure of the loci in
$\mathfrak{M}_{1}^{wt}$ of curves with $k$ trees of rational curves attached
to the core. Then one should blow up $\mathfrak{M}_{1}^{wt}$ along the loci
$\Theta_{1}$, $\Theta_{2}$, $\Theta_{3}$ and so on in order to produce a stack
$\widetilde{\mathfrak{M}}_{1}^{wt}$. This process induces a blowup
$\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$ of
$\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$ via fibre product.

Given a stratum $\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)_{\gamma}$
corresponding to a weighted graph $\gamma$, local equations of
$\widetilde{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$ and the local description of
$\Theta_{k}$ in that stratum are described explicitly in [HL10]. The purpose
of this section is to summarize such local description, give coordinates for
the new approach locally and compare both.

It may be helpful to keep in mind the following diagram, described below.

${\mathcal{E}_{\mathcal{V}}}$${E_{\gamma}}$${(F=0)}$${(\Phi_{\gamma}=0)}$${V_{\gamma}}$${\mathcal{U}}$${\mathcal{V}}$${\widetilde{\mathcal{V}}}$${U}$${\mathfrak{D}_{1}}$${\widetilde{\mathfrak{D}}_{1}}$${\mathfrak{M}_{1}^{wt}}$${\widetilde{\mathfrak{M}}_{1}^{wt}}$$\scriptstyle{\widetilde{\phi}}$$\scriptstyle{\phi}$${\square}$${\square}$

Fix a weighted graph $\gamma$ with root $o$. Let $\mathrm{Ver}(\gamma)$,
$\mathrm{Ver}(\gamma)^{t}$ and $\mathrm{Ver}(\gamma)^{*}$ denote the vertices,
the terminal vertices (or leaves) and the non-rooted vertices of $\gamma$,
respectively. We take the natural ordering in $\mathrm{Ver}(\gamma)$ making
the root $o$ the minimal element. We assume that the weight in $\gamma$ is
non-negative on every vertex and that $\gamma$ is terminally weighted, meaning
that the vertices with non-zero weights are exactly those in
$\mathrm{Ver}(\gamma)^{t}$. Let $\mathfrak{M}_{1}^{wt}$ is the stack of
genus-1 prestable curves endowed with a weight. Remember that every element
$C$ parametrized by $\mathfrak{M}_{1}^{wt}$ has a dual (weighted) graph
$\gamma$, which can be made terminally weighted and rooted by first declaring
the root to be the (contraction of) the core of $C$ and then pruning along all
non-terminal positively weighted nodes. We will denote by $o$ the root of any
terminally weighted rooted tree, and by $a,b,\ldots$ the remaining vertices.

In the diagram, $\widetilde{\mathfrak{M}}_{1}^{wt}$ denotes the blow up of
$\mathfrak{M}_{1}^{wt}$ described above and $\mathfrak{D}_{1}$ is the stack of
stable pairs $(C,D)$ with $D$ an effective Cartier divisors supported in the
smooth locus of $C$. Fix a point $(C,D)$ in $\mathfrak{D}_{1}$ and a map in
$\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$ with underlying curve $C$.
Then $U$ is a small open around the fixed map in
$\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$, $\mathcal{V}$ is a smooth
chart around the point $(C,D)$ in $\mathfrak{D}_{1}$ containing the image of
$U$ and $\mathcal{E}_{\mathcal{V}}$ is the total space of the sheaf
$\rho_{\ast}\mathcal{L}(\mathcal{A})^{\oplus n}$ on $\mathcal{V}$.

Let $V_{\gamma}=\prod_{v\in\mathrm{Ver}(\gamma)^{*}}\mathbb{A}^{1}$ be an
affine space that serves as model for local equations. We denote by
$z_{a},z_{b},\ldots$ the natural coordinates in $V_{\gamma}$. Similarly,
$E_{\gamma}=V_{\gamma}\times\prod_{v\in\mathrm{Ver}(\gamma)^{t}}(\mathbb{A}^{1})^{r}$
and the coordinates on the affine space $(\mathbb{A}^{1})^{r}$ corresponding
to $a\in\mathrm{Ver}(\gamma)^{t}$ will be denoted by $w_{a,1},\ldots,w_{a,r}$.
The ideal $\Phi_{\gamma}=(\Phi_{\gamma,1},\ldots,\Phi_{\gamma,r})$ will be
described explicitly in eq. 45. The smooth morphism $\phi$ comes from the
natural coordinates on $\mathcal{V}$, associated to the smoothing of each of
the disconnecting nodes in $C$ (which are in natural bijection with
$\mathrm{Ver}(\gamma)^{*}$). The map $\mathcal{U}\to(F=0)$ is an open
embedding. Finally, $\widetilde{\phi}$ is induced by $\phi$ and
$F=\widetilde{\phi}^{*}\Phi_{\gamma}$. It is in this sense that we can think
of $\Phi_{\gamma}$ as the equations of
$\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)_{\gamma}$.

Following [HL10], given a terminally weighted rooted graph $\gamma$, the ideal
$\Phi_{\gamma}=(\Phi_{\gamma,1},\ldots,\Phi_{\gamma,r})$ inside $V_{\gamma}$
can be described as

(45)
$\Phi_{\gamma,i}=\sum_{v\in\mathrm{Ver}(\gamma)^{t}}z_{\left[v,o\right]}w_{v,i}\quad
1\leq i\leq r,$

where

$z_{\left[v,o\right]}=\prod_{o\prec a\preceq v}z_{a}.$

Note that, for fixed $i$, the variables $w_{a,i}$ only appear in the $i$-th
equation $\Phi_{\gamma,i}$. Due to the symmetry of the equations and the fact
that all blow ups take place in $V_{\gamma}$, which has coordinates
$\\{z_{a}\\}_{a\in\mathrm{Ver}(\gamma)^{*}}$ (but not the $w_{a,i}$), in the
examples below we will not write down the index $i$ in the equations
$\Phi_{\gamma,i}$ nor in the variables $w_{a,i}$. For example, in the study of
the equations $\Phi_{\gamma,i}$ after blowing up, it will be clear that the
index $i$ is irrelevant, in the sense that the way that $\Phi_{\gamma,i}$
changes is independent of $i$.

###  8.1. Local equations of desingularizations

The local equations of the loci that must be blown up are described, following
[HL10].

Firstly, we describe how to assign an ideal $I_{\gamma}$ to any semistable
terminally weighted rooted tree $\gamma$. Here, semistability of $\gamma$
means that every non-root vertex with weight zero has at least two edges.

The trunk of $\gamma$ is the maximal chain $o=v_{0}\prec v_{1}\prec\ldots\prec
v_{r}$ of vertices in $\gamma$ such that each vertex $v_{i}$ with $1\leq i<r$
has exactly one immediate descendant and $v_{r}$ is called a branch vertex if
is it not terminal. Note that $\gamma$ is a path tree if and only if it has no
branch vertex.

######  Definition 8.1.1.

Let $\gamma$ be a semistable terminally weighted rooted tree with branch
vertex $v$ and let $a_{1},\ldots,a_{k}$ be the immediate descendants of $v$.
To $\gamma$ we associate the ideal

$I_{\gamma}=(z_{a_{1}},\ldots,z_{a_{k}})$

in $V_{\gamma}$.

First, we must blow up $V_{\gamma}$ along the ideal $I_{\gamma}$. To describe
the remaining steps we need to introduce the following operations.

######  Definition 8.1.2.

Let $\gamma$ be a terminally weighted semistable rooted tree.

  * •

The pruning of $\gamma$ along a vertex $v$ is the new tree obtained by
removing all the descendants of $v$ (and the corresponding edges) and
declaring the weight of $v$ to be the sum of the original weight of $v$ plus
the weights of all removed vertices.

  * •

The advancing of a vertex $v$ with immediate ascendant $\overline{v}$ in
$\gamma$ is a new tree obtained by replacing every edge
$(\overline{v},v^{\prime})$ with $v^{\prime}\neq v$ by an edge
$(\overline{v},v)$ and pruning along all positively weighted non-terminal
vertices as long as possible. In section 8.1.4 we will denote by $\gamma_{v}$
the advancing of $v$ in $\gamma$ and by $\gamma_{v}^{\prime}$ the same tree
before pruning.

  * •

Suppose $\gamma$ has a branch vertex $v$. A monoidal transform of $\gamma$ is
a tree obtained by advancing one of the immediate descendants of $v$. The set
of monoidal transforms of $\gamma$ is $\mathrm{Mon}(\gamma)$.

It turns out that the ideal $\Phi_{\gamma}$ behaves nicely under monoidal
transforms. Indeed, let $\gamma$ be a semistable terminally weighted rooted
tree with branch vertex $v$ and let $a=a_{1},a_{2},\ldots,a_{k}$ be the
immediate descendants of $v$. Let $\gamma_{a}$ be the tree advancing of $a$ in
$\gamma$. Let $\pi\colon\widetilde{V_{\gamma}}\to V_{\gamma}$ be the blow-up
of $V_{\gamma}$ along the ideal $I_{\gamma}=(z_{a_{1}},\ldots,z_{a_{k}})$. We
view $\widetilde{V_{\gamma}}$ embedded inside
$V_{\gamma}\times\mathbb{P}^{k-1}$. There is a natural way to associate to
each generator $z_{a_{i}}$ of $I_{\gamma}$ one chart of $\mathbb{P}^{k-1}$,
and thus also of $\widetilde{V_{\gamma}}$. We denote such chart by
$\widetilde{V}_{\gamma,a_{i}}$. Let $\pi_{a}\colon\widetilde{V}_{\gamma,a}\to
V_{\gamma}$ be the restriction of the natural projection, where $a=a_{1}$.
Then, by the proof of [HL10, Lemma 5.14], one of the following must hold

  * •

either $\gamma_{a}$ is a path tree, and then the zero locus of
$\pi_{a}^{*}(\Phi_{\gamma})$ has smooth components;

  * •

or $\gamma_{a}$ is not a path tree and then

(46) $\pi_{a}^{*}(\Phi_{\gamma})=\Phi_{\gamma_{a}}.$

The whole blow-up process is summarized as follows. Fix $\gamma$. First blow
up $V_{\gamma}$ along $I_{\gamma}$. The pullback of $\Phi_{\gamma}$ is
controlled by $\mathrm{Mon}(\gamma)$. If $\mathrm{Mon}(\gamma)$ consists only
of path trees, we are done. Otherwise, for every element $\gamma^{\prime}$ in
$\mathrm{Mon}(\gamma)$ which is not a path tree, blow up the chart of
$\widetilde{V_{\gamma}}$ corresponding to $\gamma^{\prime}$ along
$I_{\gamma^{\prime}}$. Continue recursively. The process concludes by [HL10,
Lemma 3.12].

Now we want to describe $\mathrm{Bl}_{\mathfrak{F}}\mathfrak{Pic}$ locally.
Namely, we want to describe which loci inside $\mathfrak{Pic}$ we are blowing
up locally. We have a short exact sequence

$0\to\rho_{*}\mathcal{L}\to\rho_{*}\mathcal{L}(\mathcal{A})\to\rho_{*}\mathcal{L}(\mathcal{A})\mid_{\mathcal{A}}$

by eq. 23. The change of notation is due to the fact that eq. 23 was global in
$\mathfrak{Pic}$, but we now work locally. After a careful study of the second
morphism, [HL10, Theorem 4.16] concludes that $\rho_{\ast}\mathcal{L}$ is the
direct sum of a trivial bundle with the kernel of the morphism

(47)
$\bigoplus_{v\in\mathrm{Ver}(\gamma)^{t}}\varphi_{v}\colon\mathcal{O}_{\mathcal{V}}^{\oplus\ell}\to\mathcal{O}_{\mathcal{V}},$

where $\ell$ is the cardinality of $\mathrm{Ver}(\gamma)^{t}$ and
$\varphi_{v}=\prod_{o\prec v^{\prime}\preceq v}\zeta_{v}$, with $\zeta_{v}$
the smoothing parameter of the disconnecting node corresponding to the vertex
$v$.

By lemma 7.1.1, the sheaf $\mathfrak{F}$ can be described locally as the dual
of eq. 47. In particular, by Remark 8.1.6 we have that
$\mathrm{Bl}_{\mathfrak{F}}\mathfrak{Pic}$ agrees, locally, with the blowup
along the ideal generated by the entries
$(\varphi_{v})_{v\in\mathrm{Ver}(\gamma)^{t}}$. In local coordinates, this
ideal can be described as follows.

######  Definition 8.1.3.

Let $\gamma$ be a semistable terminally weighted rooted tree with
$\mathrm{Ver}(\gamma)^{t}=\\{v_{1},\ldots,v_{t}\\}$. To $\gamma$ we associate
the ideal

$J_{\gamma}=(z_{[v_{1},o]},\ldots,z_{[v_{t},o]}).$

Similarly to eq. 46, in the same setup we have that

$\pi_{a}^{*}(J_{\gamma})=J_{\gamma_{a}},$

independently of whether $\gamma_{a}$ is has a branch vertex. This follows
again from the proof of [HL10, Lemma 5.14].

####  8.1.4. Examples

For two concrete trees $\gamma$, we compute the equations $\Phi_{\gamma}$ as
well as the ideals $I_{\gamma}$ and $J_{\gamma}$. We describe the blow up
process of Hu and Li and show that the result indeed desingularizes the main
component of $\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$ locally.
Furthermore, we check that the ideal $J_{\gamma}$ becomes locally principal in
Hu–Li’s blow-up.

######  Example 8.1.5.

Consider the following labelled graph:

$\gamma$ =  [.$o$ $a$ [.$b$ $c$ $d$ ]]

$\displaystyle\Phi_{\gamma}$
$\displaystyle=z_{a}w_{a}+z_{b}(z_{c}w_{c}+z_{d}w_{d}),$ $\displaystyle
I_{\gamma}$ $\displaystyle=(z_{a},z_{b}),$ $\displaystyle J_{\gamma}$
$\displaystyle=(z_{a},z_{b}z_{c},z_{b}z_{d}).$

Let $\widetilde{V_{\gamma}}$ be the blow up along $I_{\gamma}$, that is the
zero locus of $z_{a}z_{b}^{\prime}-z_{b}z_{a}^{\prime}$ inside
$\mathbb{A}^{4}_{z_{a},z_{b},z_{c},z_{d}}\times\mathbb{P}^{1}_{z_{a}^{\prime},z_{b}^{\prime}}$.
The chart associated to $a$ is that where $z_{a}^{\prime}\neq 0$.
Dehomogenizing amounts to the change of variables $z_{b}=z_{b}^{\prime}z_{a}$.
By doing so, we get that

$\pi_{a}^{*}(\Phi_{\gamma})=z_{a}(w_{a}+z_{b}^{\prime}(z_{c}w_{c}+z_{d}w_{d}))$

and that

$\pi_{a}^{*}(J_{\gamma})=(z_{a},z_{a}z_{b}^{\prime}z_{c},z_{a}z_{b}^{\prime}z_{d})=(z_{a}).$

This means that the zero locus of $\pi_{a}^{*}(\Phi_{\gamma})$ already has
smooth components, so no further blowups are needed on this chart, and that
$\pi_{a}^{*}(J_{\gamma})$ is principal on this chart too.

Below are the trees $\gamma_{a}^{\prime}$ obtained by advancing $a$ without
pruning, and $\gamma_{a}$ obtained by advancing $a$. We know that
$\pi_{a}^{*}J_{\gamma}=J_{\gamma_{a}}$, but we check it in this example.

$\gamma_{a}^{\prime}$ =  [.$o$ [.$a$ [.$b$ $c$ $d$ ]]]

$\gamma_{a}$ =  [.$o$ [.$a$ ]]

$\displaystyle J_{\gamma_{a}}$ $\displaystyle=(z_{a}).$

Similarly, we now look at the chart associated to $b$, where
$z_{b}^{\prime}\neq 0$. The change of variables is now
$z_{a}=z_{b}z_{a}^{\prime}$. It follows that

$\pi_{b}^{*}(\Phi_{\gamma})=z_{b}(z_{a}^{\prime}w_{a}+z_{c}w_{c}+z_{d}w_{d})$

and that

$\pi_{b}^{*}(J_{\gamma})=(z_{b}z_{a}^{\prime},z_{b}z_{c},z_{b}z_{d})=z_{b}(z^{\prime}_{a},z_{c},z_{d}).$

This means that we still need to blow up. This time the tree $\gamma_{b}$
obtained by advancing $b$ in $\gamma$ (no pruning is needed) is not a path
tree. We also check the identities $J_{\gamma_{a}}=\pi_{a}^{*}J_{\gamma}$ and
$\pi_{b}^{*}(\Phi_{\gamma})=\Phi_{\gamma_{b}}$.

[.$o$ [.$b$ [.$a$ ] [.$c$ ] [.$d$ ]]]

$\displaystyle\Phi_{\gamma_{b}}$
$\displaystyle=z_{b}(z_{a}w_{a}+z_{c}w_{c}+z_{d}w_{d}),$ $\displaystyle
I_{\gamma_{b}}$ $\displaystyle=(z_{a},z_{c},z_{d}),$ $\displaystyle
J_{\gamma_{b}}$ $\displaystyle=z_{b}(z_{a},z_{c},z_{d}).$

To conclude the example, we need to blow up along the ideal
$(z_{a},z_{c},z_{d})$. We collect the result below.

Advancing $a$, or equivalently looking at the chart $z_{a}^{\prime}\neq 0$, we
have

$\gamma_{b,a}^{\prime}$ =  [.$o$ [.$b$ [.$a$ [.$c$ ] [.$d$ ]]]]

$\gamma_{b,a}^{\prime}$ =  [.$o$ [.$b$ [.$a$ ]]]

$\displaystyle\pi_{a}^{*}\pi_{b}^{*}(\Phi_{\gamma})=\pi_{a}^{*}(\Phi_{\gamma_{b}})$
$\displaystyle=z_{a}z_{b}(w_{a}+z_{c}w_{c}+z_{d}w_{d}),$ $\displaystyle
J_{\gamma_{b,a}}$ $\displaystyle=(z_{a}z_{b}).$

Advancing $c$, or equivalently looking at the chart $z_{c}^{\prime}\neq 0$, we
have

$\gamma_{b,c}^{\prime}$ =  [.$o$ [.$b$ [.$c$ [.$a$ ] [.$d$ ]]]]

$\gamma_{b,c}^{\prime}$ =  [.$o$ [.$b$ [.$c$ ]]]

$\displaystyle\pi_{c}^{*}\pi_{b}^{*}(\Phi_{\gamma})=\pi_{c}^{*}(\Phi_{\gamma_{b}})$
$\displaystyle=z_{b}z_{c}(z_{a}w_{a}+w_{c}+z_{d}w_{d}),$ $\displaystyle
J_{\gamma_{b,c}}$ $\displaystyle=(z_{b}z_{c}).$

And finally, advancing $d$, or equivalently looking at the chart
$z_{d}^{\prime}\neq 0$, we have

$\gamma_{b,d}^{\prime}$ =  [.$o$ [.$b$ [.$d$ [.$a$ ] [.$c$ ]]]]

$\gamma_{b,d}^{\prime}$ =  [.$o$ [.$b$ [.$d$ ]]]

$\displaystyle\pi_{d}^{*}\pi_{b}^{*}(\Phi_{\gamma})=\pi_{d}^{*}(\Phi_{\gamma_{b}})$
$\displaystyle=z_{b}z_{d}(z_{a}w_{a}+z_{c}w_{c}+w_{d}),$ $\displaystyle
J_{\gamma_{b,d}}$ $\displaystyle=(z_{b}z_{d}).$

######  Remark 8.1.6.

Example 8.1.5 shows that the Rossi–Villamayor blow-up process of
$\overline{\mathcal{M}}_{g,n}(\mathbb{P}^{r},d)$ is not equal to the
Vakil–Zinger blow-up. This is compatible with Remark 4.4. in [HN19]. Indeed,
the Rossi–Villamayor blowup around $\gamma$ is given by
$\mathrm{Bl}_{J_{\gamma}}V_{\gamma}$ and the Vakil–Zinger one is the iterated
blow-up $\mathrm{Bl}_{I_{\gamma_{b}}}\mathrm{Bl}_{I_{\gamma}}V_{\gamma}$. We
know there is a natural morphism

$\mathrm{Bl}_{I_{\gamma_{b}}}\mathrm{Bl}_{I_{\gamma}}V_{\gamma}\to\mathrm{Bl}_{J_{\gamma}}V_{\gamma}$

over $V_{\gamma}$, either by proposition 4.2.6 or because we checked that
$J_{\gamma}$ pulls back to a principal ideal in
$\mathrm{Bl}_{I_{\gamma_{b}}}\mathrm{Bl}_{I_{\gamma}}V_{\gamma}$. By
contradiction, if there is a morphism

$f\colon\mathrm{Bl}_{J_{\gamma}}V_{\gamma}\to\mathrm{Bl}_{I_{\gamma_{b}}}\mathrm{Bl}_{I_{\gamma}}V_{\gamma}$

over $V_{\gamma}$, then we get a morphism

$\tilde{f}\colon\mathrm{Bl}_{J_{\gamma}}V_{\gamma}\to\mathrm{Bl}_{I_{\gamma}}V_{\gamma}$

over $V_{\gamma}$. By [Moo01], there is a fractional ideal $K$ in $V_{\gamma}$
and a positive integer $\alpha$ such that

$I_{\gamma}\cdot K=J_{\gamma}^{\alpha}.$

This is not true for $I_{\gamma}=(z_{a},z_{b})$ and
$J_{\gamma}=(z_{a},z_{b}z_{c},z_{b}z_{d})$ in
$V_{\gamma}=\mathbb{A}^{4}_{z_{a},z_{b},z_{c},z_{d}}$.

######  Example 8.1.7.

We do a similar study for the following labelled graph $\gamma$:

[.$o$ [.$a$ $c$ $d$ ] [.$b$ $e$ $f$ ]]

$\displaystyle\Phi_{\gamma}$
$\displaystyle=z_{a}(z_{c}w_{c}+z_{d}w_{d})+z_{b}(z_{e}w_{e}+z_{f}w_{f}),$
$\displaystyle I_{\gamma}$ $\displaystyle=(z_{a},z_{b}),$ $\displaystyle
J_{\gamma}$ $\displaystyle=(z_{a}z_{c},z_{a}z_{d},z_{b}z_{e},z_{b}z_{f}).$

After blowing up, there are two charts, corresponding to the advacings of $a$
and $b$ respectively.

$\gamma_{a}$ =  [.$o$ [.$a$ $c$ $d$ [.$b$ $e$ $f$ ]]]

$\displaystyle\Phi_{\gamma_{a}}$
$\displaystyle=z_{a}(z_{c}w_{c}+z_{d}w_{d}+z_{b}(z_{e}w_{e}+z_{f}w_{f})),$
$\displaystyle I_{\gamma_{a}}$ $\displaystyle=(z_{b},z_{c},z_{d}),$
$\displaystyle J_{\gamma_{a}}$
$\displaystyle=z_{a}(z_{c},z_{d},z_{b}z_{e},z_{b}z_{f}).$

$\gamma_{b}$ =  [.$o$ [.$b$ [.$a$ $c$ $d$ ] $e$ $f$ ]]

$\displaystyle\Phi_{\gamma_{b}}$
$\displaystyle=z_{b}(z_{a}(z_{c}w_{c}+z_{d}w_{d})+z_{e}w_{e}+z_{f}w_{f}),$
$\displaystyle I_{\gamma_{b}}$ $\displaystyle=(z_{a},z_{e},z_{f}),$
$\displaystyle J_{\gamma_{b}}$
$\displaystyle=z_{b}(z_{a}z_{c},z_{a}z_{d},z_{e},z_{f}).$

By symmetry, it is enough to understand how to proceed in one of the charts.
We choose the one corresponding to $a$. We get three new charts corresponding
to the vertices $c$, $d$ and $b$.

$\gamma_{a,c}^{\prime}$ =  [.$o$ [.$a$ [.$c$ $d$ [.$b$ $e$ $f$ ]]]]

$\gamma_{a,c}$ =  [.$o$ [.$a$ [.$c$ ]]]

$\displaystyle\pi_{c}^{*}\Phi_{\gamma_{a}}$
$\displaystyle=z_{a}z_{c}(w_{c}+z_{d}w_{d}+z_{b}(z_{e}w_{e}+z_{f}w_{f})),$
$\displaystyle J_{\gamma_{a,c}}$ $\displaystyle=(z_{a}z_{c}).$

$\gamma_{a,d}^{\prime}$ =  [.$o$ [.$a$ [.$d$ $c$ [.$b$ $e$ $f$ ]]]]

$\gamma_{a,d}$ =  [.$o$ [.$a$ [.$d$ ]]]

$\displaystyle\pi_{d}^{*}\Phi_{\gamma_{a}}$
$\displaystyle=z_{a}z_{d}(z_{c}w_{c}+w_{d}+z_{b}(z_{e}w_{e}+z_{f}w_{f})),$
$\displaystyle J_{\gamma_{a,d}}$ $\displaystyle=(z_{a}z_{d}).$

$\gamma_{a,b}$ =  [.$o$ [.$a$ [.$b$ $c$ $d$ $e$ $f$ ]]]

$\displaystyle\Phi_{\gamma_{a,b}}$
$\displaystyle=z_{a}z_{b}(z_{c}w_{c}+z_{d}w_{d}+z_{e}w_{e}+z_{f}w_{f}),$
$\displaystyle I_{\gamma_{a,b}}$ $\displaystyle=(z_{c},z_{d},z_{e},z_{f}),$
$\displaystyle J_{\gamma_{a,b}}$
$\displaystyle=z_{a}z_{b}(z_{c},z_{d},z_{e},z_{f}).$

To conclude, we need to blow up the last chart along $I_{\gamma_{a,b}}$. This
produces four new charts corresponding to $c,d,e$ and $f$. We will only write
down one of them since the rest are very similar.

$\gamma_{a,b,c}^{\prime}$ =  [.$o$ [.$a$ [.$b$ [.$c$ $d$ $e$ $f$ ]]]]

$\gamma_{a,b,c}$ =  [.$o$ [.$a$ [.$b$ [.$c$ ]]]]

$\displaystyle\pi_{c}^{*}\Phi_{\gamma_{a,b}}$
$\displaystyle=z_{a}z_{b}z_{c}(w_{c}+z_{d}w_{d}+z_{e}w_{e}+z_{f}w_{f}),$
$\displaystyle J_{\gamma_{a,b,c}}$ $\displaystyle=(z_{a}z_{b}z_{c}).$

####  8.1.8. Smoothness

In genus one,
$\overline{\mathcal{M}}_{1,n}(\mathbb{P}^{r},d)\times_{\mathfrak{Pic}}\mathrm{Bl}_{\mathfrak{F}}\mathfrak{Pic}$
has simple normal crossings following the same argument as in [HL10, Theorem
5.24]. It is enough to show that the zero locus of the ideal $\Phi_{\gamma}$
becomes a simple normal crossing in the blow-up $\widetilde{V_{\gamma}}$ of
$V_{\gamma}$ along the ideal $J_{\gamma}$.

Remember that $\Phi_{\gamma}=(\Phi_{\gamma,1},\ldots,\Phi_{\gamma,r})$ with

$\Phi_{\gamma,i}=\sum_{v\in\mathrm{Ver}(\gamma)^{t}}z_{\left[v,o\right]}w_{b,i}\quad
1\leq i\leq r,$

where

$z_{\left[v,o\right]}=\prod_{o\prec a\preceq v}z_{a},$

and that

$J_{\gamma}=(z_{[v,o]})_{v\in\mathrm{Ver}(\gamma)^{t}}.$

For given $v^{\prime}\in\mathrm{Ver}(\gamma)^{t}$, the pullback of the
equation $\Phi_{\gamma,i}$ on the chart corresponding to $v^{\prime}$ is equal
to

$z_{[v^{\prime},o]}\left(w_{v^{\prime},i}+\sum_{v^{\prime}\neq
v\in\mathrm{Ver}(\gamma)^{t}}z_{\left[v,o\right]}w_{b,i}\right)$

by [HL10, Lemma 5.14]. This proves the claim.

####  8.1.9. Maps between blowups

By proposition 4.2.6 there is a morphism from Vakil–Zinger’s blowup to Rossi-
Villamayor’s blowup. In genus one, we can check it locally: it is equivalent
to the fact that the pullback of the ideal $J_{\gamma}$ to each chart
$\widetilde{V_{\gamma}}$ of the Hu–Li blowup of $V_{\gamma}$ is principal. We
have checked this in example 8.1.5 and example 8.1.7, More generally, we can
give a proof for every $\gamma$ as follows.

By eq. 46 if $z_{a}$ is any of the generators of $I_{\gamma}$, then
$\pi_{a}^{*}(J_{\gamma})=J_{\gamma_{a}}$ where $\gamma_{a}$ is the advancing
of $a$ in $\gamma$. In particular, it is enough to show that all the (natural)
charts of $\widetilde{V_{\gamma}}$ correspond to path trees, which is proven
in [HL10, Lemma 3.14].

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