# Smoothability of relative stable maps to stacky curves

Kenneth Ascher & Dori Bejleri

###### Abstract.

Using log geometry, we study smoothability of genus zero twisted stable maps
to stacky curves relative to a collection of marked points. One application is
to smoothing semi-log canonical fibered surfaces with marked singular fibers.

##  1\. Introduction

Consider the moduli space $\mathrm{Maps}^{\circ}_{n}$ of degree $n$ maps
$f:C\to\mathbb{P}^{1}$ from a smooth rational curve $C$ such that $f$ is
unramified over infinity. Marking the preimage
$f^{-1}(\infty)=\\{p_{1},\ldots,p_{n}\\}$ of such a map induces a locally
closed embedding

$\widetilde{\mathrm{Maps}^{\circ}_{n}}\hookrightarrow\overline{\mathcal{M}}_{0,n}({\mathbb{P}}^{1},n)$

of the $\mathfrak{S}_{n}$-torsor
$\widetilde{\mathrm{Maps}^{\circ}_{n}}\to\mathrm{Maps}^{\circ}_{n}$
parametrizing such $f:C\to\mathbb{P}^{1}$ as well as a labeling of
$f^{-1}(\infty)$ into the space of $n$-pointed degree $n$ genus zero stable
maps to $\mathbb{P}^{1}$. The image of this embedding is the locus of stable
maps $(f:C\to\mathbb{P}^{1},p_{1},\ldots,p_{n})$ such that $C$ is smooth and
$f(p_{i})=\infty$ for all $i$. Let $\overline{\mathrm{Maps}}_{n}$ denote the
closure of this locus.

######  Question 1.

Is there a combinatorial description of the boundary
$\overline{\mathrm{Maps}}_{n}\setminus\widetilde{\mathrm{Maps}^{\circ}_{n}}$?

There are several natural combinatorial conditions (see Proposition 2.2 and
the discussion preceding it) that are necessary for a stable map
$(f:C\xrightarrow{n:1}\mathbb{P}^{1},p_{1},\ldots,p_{n})$ to lie inside
$\overline{\mathrm{Maps}}_{n}$:

  1. (1)

The evaluations $f(p_{i})=\infty$ for all $i$;

  2. (2)

every point of $f^{-1}(\infty)\subset C$ is either a marked point, or is on a
contracted component;

  3. (3)

for each maximal connected closed subvariety $T\subset C$ contracted to
$\infty$ by $f$, we have

$\\#\textrm{ marked points on $T$ }=\sum\textrm{ ramification index of $f$ at
the nodes $T\cap(\overline{C\setminus T})$.}$

More generally, for any tuple of non-negative integers $(d_{1},\ldots,d_{n})$
with $d=\sum d_{i}$, one can consider the locally closed subset
$\mathcal{M}_{(d_{1},\ldots,d_{n})}\subset\overline{\mathcal{M}}_{0,n}({\mathbb{P}}^{1},d)$
of $n$-pointed degree $d$ genus $0$ maps
$(f:C\to{\mathbb{P}}^{1},p_{1},\ldots,p_{n})$ such that $f(p_{i})=\infty$ and
$f$ is ramified of order $d_{i}$ at $p_{i}$ if $d_{i}>0$. If $d_{i}=0$, there
is no condition imposed on the $f$ at $p_{i}$. Denoting by
$\overline{\mathcal{M}}_{(d_{1},\ldots,d_{n})}$ the closure of
$\mathcal{M}_{(d_{1},\ldots,d_{n})}$, we can ask the variant of Question 1 for
this space. The necessary conditions above naturally generalize to Conditions
($\ast$) ‣ 1.1 below.

######  Definition 1.1.

Let $(f:C\to X,p_{1},\ldots,p_{n})$ be a prestable map to a smooth curve and
fix a tuple of positive integers $(d_{1},\ldots,d_{n})$ and a point $x\in X$.
We say that $f$ is a relative map to $(X,x)$ with tangency
$(d_{1},\ldots,d_{n})$ if it satisfies the following conditions:

######  Combinatorial Conditions ($\ast$).

[Gat02, Definition 1.1 & Remark 1.7]

  1. (1)

The evaluations $f(p_{i})=x$ for all $i$.

  2. (2)

Every point of $f^{-1}(x)\subset C$ is either a marked point, or is on a
contracted component.

  3. (3)

For each maximal connected closed subvariety $T\subset f^{-1}(x)$, we have

$\sum_{p_{i}\in T}d_{i}=\sum_{q\in T\cap(\overline{C\setminus T})}e_{f}(q)$

where the first sum is over marked points contained in $T$ and $e_{f}(q)$
denotes the ramification of $f|_{\overline{C\setminus T}}$ at the point $q$.

This question in much greater generality was studied by Gathmann [Gat02,
Proposition 1.14], building off previous work of Vakil [Vak00, Theorem 6.1].
In particular, Gathmann showed that Conditions ($\ast$) ‣ 1.1 relative to
$\infty$ are both necessary and sufficient for a stable map to lie in
$\overline{\mathcal{M}}_{(d_{1},\ldots,d_{n})}$. As a consequence, the set of
points of $\overline{\mathcal{M}}_{0,n}({\mathbb{P}}^{1},d)$ satisfying
Conditions ($\ast$) ‣ 1.1 relative to $\infty$ are the points of an
irreducible closed substack.

In [AV02], Abramovich and Vistoli introduced moduli spaces of _twisted stable
maps_ , which allows the target to instead be a Deligne-Mumford stack. To form
a compact moduli space, the source curves obtain a stacky structure. Let
$\mathcal{K}_{0,n}(\mathcal{X},d)$ denote the moduli space of $n$-pointed
genus $0$ degree $d$ twisted stable maps $n$ to a Deligne-Mumford stack
$\mathcal{X}$. The goal of this paper is to study the analogue of Question 1
for genus $0$ twisted stable maps to a weighted projective line
$\mathcal{P}(a,b)$, or more generally a rational Deligne-Mumford curve.

When $\mathcal{X}$ is $\mathcal{P}(a,b)$ and $\infty\in\mathcal{P}(a,b)$ is a
fixed point away from $[0:1]$ and $[1:0]$, we have the following (see Theorem
1.5 for a more general result). For this example, the reader can keep in mind
$\overline{\mathcal{M}}_{1,1}=\mathcal{P}(4,6)$ and the point $j=\infty$
parametrizing a nodal elliptic curve.

######  Theorem 1.2.

Let $(f:\mathcal{C}\to\mathcal{P}(a,b),p_{1},\ldots,p_{n})$ be an $n$-pointed
genus $0$ twisted stable map such that the coarse map
$(g:C\to\mathbb{P}^{1},q_{1},\ldots,q_{n})$ satisfies Conditions ($\ast$) ‣
1.1 with respect to $\infty$. Then $f$ is smoothable in a family with generic
fiber satisfying: $f(p_{i})=\infty$ and $f$ is ramified to order $d_{i}$ at
$p_{i}$ for all $i=1,\ldots,n$.

In what follows, we setup notation needed to state our more general results.

Let $(\mathcal{X},x_{1},\ldots,x_{r})$ be a smooth and proper $1$-dimensional
rational Deligne-Mumford curve and suppose $x_{i}\in\mathcal{X}$ are points
with generic stabilizer. For each $j=1,\ldots,r$, let
$\Gamma_{j}=(d_{j1},\ldots,d_{jn_{j}})$ be a tuple of positive integers and
fix $n_{0}\geq 0$. Denote by $n=\sum_{j=0}^{r}n_{j}$ and

$d=\sum_{j=1}^{r}\sum_{k=1}^{n_{j}}d_{jk}$

and let $\Gamma=(n_{0},\\{\Gamma_{1},x_{1}\\},\ldots,\\{\Gamma_{r},x_{r}\\})$
be the tuple of combinatorial data.

######  Definition-Notation 1.3.

For any tuple of combinatorial data $\Gamma$, let
$\bm{\mathcal{M}_{\Gamma}(\mathcal{X})}$ be the locally closed substack of
$\mathcal{K}_{0,n}(\mathcal{X},d)$ parametrizing $n$-pointed genus zero degree
$d$ twisted stable maps
$(f:\mathcal{C}\to\mathcal{X},\\{\\{p_{jk}\\}_{k=1}^{n_{j}}\\}_{j=0}^{r})$
such that

  1. (1)

$p_{jk}$ are marked points with stabilizer of order $a_{jk}$ lying over smooth
points $q_{jk}$ of the coarse space $C$,

  2. (2)

for each $j>0$, the image $f(p_{jk})=x_{j}$,

  3. (3)

for each $j>0$ the coarse map $h:C\to X$ is ramified to order $d_{jk}$ at
$q_{jk}$, and

  4. (4)

$\mathcal{C}$ is smooth.

Similarly, we let $\bm{\mathcal{N}_{\Gamma}(\mathcal{X})}$ be the locally
closed substack of $\mathcal{K}_{0,n}(\mathcal{X},d)$ of maps satisfying
conditions $(1),(2)$ and $(3)$ above as well as the condition

  1. $(4^{\prime})$

$\mathcal{C}$ is smooth in a neighborhood of $f^{-1}(x_{j})$ for all $j$.

We denote by $\bm{\overline{\mathcal{M}}_{\Gamma}(\mathcal{X})}$ (resp.
$\bm{\overline{\mathcal{N}}_{\Gamma}(\mathcal{X})}$) the closure of
$\mathcal{M}_{\Gamma}(\mathcal{X})$ (resp.
$\mathcal{N}_{\Gamma}(\mathcal{X})$) inside
$\mathcal{K}_{0,n}(\mathcal{X},d)$.

Note here and throughout we are fixing a bijection between the set
$\\{1,\ldots,n\\}$ and the set $\\{\\{(j,k)\\}_{k=1}^{n_{j}}\\}_{j=0}^{r}$
indexing the marked points. Note also that the marked points with $j=0$ are
the ones with no tangency conditions, so in the situation of
$\mathcal{M}_{(d_{1},\ldots,d_{n})}$, we may assume for notational convenience
that the first $n_{0}$ entries $(d_{1},\ldots,d_{n_{0}})$ of
$(d_{1},\ldots,d_{n})$ are $0$ so that
$\mathcal{M}_{(d_{1},\ldots,d_{n})}=\mathcal{M}_{\Gamma}(\mathbb{P}^{1})$
where $\Gamma=(n_{0},\\{(d_{n_{0}+1},\ldots,d_{n}),\infty\\})$.

######  Definition-Notation 1.4.

For any tuple of combinatorial data $\Gamma$, let
$\bm{\mathcal{K}_{\Gamma}(\mathcal{X})}$ denote the subset of
$\mathcal{K}_{0,n}(\mathcal{X},d)$ parametrizing those twisted stable maps
$(f:\mathcal{C}\to\mathcal{X},\\{\\{p_{jk}\\}_{k=1}^{n_{j}}\\}_{j=0}^{r})$
such that for each $j=1,\ldots,r$, the coarse moduli map $(h:C\to
X,q_{j1},\ldots,q_{jn_{j}})$ is a relative map to $(X,x_{j})$ with tangency
$\Gamma_{j}$.

######  Theorem 1.5.

Let $\mathcal{X}$ and $\Gamma$ be as above. Then we have an equality
$\overline{\mathcal{N}}_{\Gamma}(\mathcal{X})=\mathcal{K}_{\Gamma}(\mathcal{X})$.
That is, every twisted stable map whose coarse moduli map satisfies the
relative condition for $\\{\Gamma_{j},x_{j}\\}$ and for each $j=1,\dots,r$, is
smoothable to a twisted stable map parametrized by
$\mathcal{N}_{\Gamma}(\mathcal{X})$. In particular,
$\mathcal{K}_{\Gamma}(\mathcal{X})$ is the set of points of a closed substack
of $\mathcal{K}_{0,n}(\mathcal{X},d)$.

######  Remark 1.6.

Note that Theorem 1.5 is local on the target. Indeed both the definitions of
$\mathcal{K}_{\Gamma}(\mathcal{X})$ and $\mathcal{N}_{\Gamma}(\mathcal{X})$
are in terms of local conditions around the points $x_{i}\in\mathcal{X}$.
Therefore, the theorem has a natural generalization to higher genus maps to
higher genus target $\mathcal{X}$ provided we only consider those maps for
which the $1$-dimensional components of the preimages $f^{-1}(x_{i})$ are
rational curves. In this case, the theorem reads that any such map which is
also contained in $\mathcal{K}_{\Gamma}(\mathcal{X})$ is smoothable in a
family with generic fiber contained in $\mathcal{N}_{\Gamma}(\mathcal{X})$.

When $\mathcal{X}$ has some global positivity, we obtain the following
stronger statement.

######  Theorem 1.7.

Let $\mathcal{X}=\mathcal{P}(a,b)$ be a weighted projective line and let
$\Gamma$ be a tuple of combinatorial data as above. Then we have an equality
$\overline{\mathcal{M}}_{\Gamma}(\mathcal{X})=\mathcal{K}_{\Gamma}(\mathcal{X})$.
That is, every twisted stable map which satisfies the relative condition for
$\\{\Gamma_{j},x_{j}\\}$ and for each $j=1,\ldots,r$ is smoothable to a family
of stable maps from a smooth rational curve satisfying $f(p_{jk})=x_{j}$ and
with ramification $d_{jk}$ at $p_{jk}$.

###  1.1. Applications to moduli of fibered surfaces

Our original motivation for writing this paper came from studying
compactifications of the moduli space of fibered surfaces. Twisted stable maps
are used in [AV00, AB19] to construct a compactification
$\mathcal{F}_{g,n}^{v}(\gamma,\nu)$ of the moduli space of genus $\gamma$
fibrations over a genus $g$ curve with $\nu$ marked sections and $n$ marked
fibers. The objects of the boundary are certain semi-log canonical unions of
birationally fibered surfaces called _twisted surfaces_. This compactification
is closely related to the compactification via stable log varieties from the
minimal model program. In [AB19, Section 1.4] we proposed the problem of using
log geometry to give the main component a moduli theoretic interpretation and
classify the boundary components. The present paper solves this problem for
elliptic fibrations with marked singular fibers (we refer the reader to [AB21]
and [AB20, Section 4]). A key observation is Conditions ($\ast$) ‣ 1.1
relative to $\infty\in\overline{\mathcal{M}}_{1,1}$ as well as the choice of
stabilizers on the marked points translate to conditions on the configuration
of singular fibers on the components of the twisted elliptic surface ([AB20,
Propositions 4.1 & 4.4]).

######  Theorem 1.8.

Theorem 1.7 gives necessary and sufficient combinatorial conditions for a
twisted elliptic surface over a genus $0$ curve with marked singular fibers to
be smoothable to an elliptic surface over $\mathbb{P}^{1}$ with marked
singular fibers.

### Acknowledgements

We thank Dan Abramovich, Qile Chen, Martin Olsson, Dhruv Ranganathan, Ravi
Vakil, Jonathan Wise, and David Yang for helpful discussions. K.A. supported
in part by funds from NSF Grant DMS-2140781 (formerly DMS-2001408). D.B.
supported in part by an NSF postdoctoral fellowship.

##  2\. Genus $0$ relative stable maps to $(\mathbb{P}^{1},\infty)$

In this section, we prove the special case of Theorem 1.2 where the target is
${\mathbb{P}}^{1}$, i.e. $a=b=1$. This special case was originally proved by
Gathmann (see [Gat02]). Our approach differs from that of [Gat02] in that we
give a direct construction of smoothings of comb-type maps (Propositions 2.5
and 2.8) rather than appealing to [Vak00, Theorem 6.1]. This will be a key
step in the proof of the general case of Theorem 1.2. For the remainder of
this section, fix positive integers $\Gamma=(d_{1},\ldots,d_{n})$ with $d=\sum
d_{i}$.

We begin with some examples motivating Conditions ($\ast$) ‣ 1.1 when
$\Gamma=(1,\ldots,1)$. First, it is clear that Condition ($\ast$) ‣ 1.1(1) is
required as the evaluation condition is closed. However, consider a degree $n$
map $f_{1}:(C_{1},q_{1})\to{\mathbb{P}}^{1}$ from a smooth rational curve
$C_{1}$ with $f(q_{1})=\infty$. Let $C=C_{1}\cup_{q_{1},q_{2}}C_{2}$ be a
nodal union of two rational curves and let $p_{1},\ldots,p_{n}\in
C_{2}\setminus q_{2}$ be $n$ marked points. Then there is an $n$-marked degree
$n$ stable map $(f:C\to{\mathbb{P}}^{1},p_{1},\ldots,p_{n})$ given by taking
the map $f_{1}$ on $C_{1}$ and the constant map $\infty$ on $C_{2}$. While
this map satisfies Condition ($\ast$) ‣ 1.1(1), a simple dimension count shows
that the dimension of this locus inside
$\overline{\mathcal{M}}_{0,n}({\mathbb{P}}^{1},n)$ is equal to
$\dim\overline{\mathrm{Maps}}_{n}+n-2$, and so this condition alone is not
enough to cut out the locus $\overline{\mathrm{Maps}}_{n}$ at least for $n\geq
3$.

The above example motivates Condition ($\ast$) ‣ 1.1(2), which in this case
requires that the map $f_{1}$ be totally ramified at $q_{1}$ so that there are
no other points of $C_{1}$ in $f^{-1}(\infty)$. Thus the locus of maps of this
combinatorial type satisfying both Conditions ($\ast$) ‣ 1.1(1) and ($\ast$) ‣
1.1(2) has dimension $\dim\overline{\mathrm{Maps}}_{n}-2$. Note that such maps
automatically satisfy Condition ($\ast$) ‣ 1.1(3) as well. However, the
following example illustrates that Conditions ($\ast$) ‣ 1.1(1) and ($\ast$) ‣
1.1(2) do not imply Condition ($\ast$) ‣ 1.1(3) in general.

Consider a degree $3$ map $f_{1}:(C_{1},q_{1},q_{2})\to{\mathbb{P}}^{1}$ from
a smooth rational curve such that $f(q_{i})=\infty$ and suppose that $f$ has
ramification index $i$ at $q_{i}$. Let $(C_{2},q_{1}^{\prime},p_{1},p_{2})$ be
a smooth pointed rational curve and let
$C=C_{1}\cup_{q_{1},q_{1}^{\prime}}C_{2}$. Then we have a $3$-pointed degree
$3$ stable map $(f:C\to\mathbb{P}^{1},q_{2},p_{1},p_{2})$ where
$f|_{C_{1}}=f_{1}$ and $f|_{C_{2}}$ is constant $\infty$. This map satisfies
Conditions ($\ast$) ‣ 1.1(1) and ($\ast$) ‣ 1.1(2) but not ($\ast$) ‣ 1.1(3).
Note furthermore that this stable map cannot lie in
$\overline{\mathrm{Maps}}_{n}$ as $f_{1}$ is ramified to order $2$ at the
marked point $q_{2}$ but the only way this can happen in the limit of a family
in $\mathrm{Maps}^{\circ}_{n}$ is if two marked points collided at $q_{2}$.

######  Remark 2.1.

Note that a degenerate special case of Condition ($\ast$) ‣ 1.1(3) is when
$T=q$ is itself a point. Since $T$ must be a maximal closed subvariety
contracted by $f$, this means that $q$ cannot lie on a contracted component.
Therefore, by Condition ($\ast$) ‣ 1.1(2), $q$ must be a marked point. Then
Condition ($\ast$) ‣ 1.1(3) reads that $f$ must be unramified at $q$. Indeed
we saw this was necessary in the above example.

We now prove that conditions (1), (2), and (3) of Conditions ($\ast$) ‣ 1.1
are necessary.

######  Proposition 2.2.

Conditions ($\ast$) ‣ 1.1(1), (2), and (3) are necessary for a stable map to
lie in $\overline{\mathcal{M}}_{(d_{1},\ldots,d_{n})}$.

###### Proof.

It is clear that (1) is necessary as evaluation is continuous so the condition
$f(p_{i})=\infty$ is closed, so we proceed to (2) and (3). Consider a
$1$-parameter family of stable maps

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over the spectrum of a DVR $S$ with generic fiber lying in
$\mathrm{Maps}^{\circ}_{n}$ and denote by
$(f_{0}:C_{0}\to{\mathbb{P}}^{1},p_{1},\ldots,p_{n})$ the central fiber.

First we show $(2)$. Let $U\subset C$ denote the open complement of the locus
of components contracted by the map $f$, and consider $D:=f^{-1}(\infty)$. As
the total space $C$ is normal and $f$ is non-constant, $D$ is a Cartier
divisor. Moreover, note that any $\pi$-vertical component of $D$ is contracted
by $f$ and therefore is not in $U$. Thus, restricting to $U$, we see that
$D|_{U}$ is Cartier divisor which is horizontal over $S$, and therefore any
point of $D|_{U}$ lying over $0\in S$ must be in the closure of the marked
points of the generic fiber.

Finally, we show (3). Ramification corresponds to a polynomial having a
multiple root at a point, and this multiplicity takes into account precisely
how many points collided, i.e the number of marked points on the relevant
contracted component. More formally, we can consider the intersection product
$T.D$. By the projection formula $T.D=0$. On the other hand,
$D=\sum\sigma_{i}+E$ where $E$ is the $\pi$-vertical component. Computing
$T.(\sum\sigma_{i}+E)=0$ in terms of local multiplicities gives exactly the
equality in Condition ($\ast$) ‣ 1.1(3). ∎

Our task now is to show that Conditions ($\ast$) ‣ 1.1(1) - ($\ast$) ‣ 1.1(3)
are sufficient for a stable map $(f:C\to{\mathbb{P}}^{1},p_{1},\ldots,p_{n})$
to lie in $\overline{\mathcal{M}}_{(d_{1},\ldots,d_{n})}$. We will do this by
constructing a smoothing of the marked curve and a linear series on the total
space which restricts to $f$ on the cental fiber but whose generic fiber lies
in $\mathcal{M}_{(d_{1},\ldots,d_{n})}$.

We begin with a preliminary lemma that shows that nodal curves can be smoothed
to surfaces admitting $A_{m}$ singularities for any $m$ at the nodes.

######  Lemma 2.3.

Let $C_{0}$ be a genus zero nodal curve with $k$ nodes $q_{1},\ldots,q_{k}$
and let $m_{1},\ldots,m_{k}\geq 1$. Then there exists a smooth $C\to S$ where
$S=\mathrm{Spec}R$ is the spectrum of a DVR such that the total space $C$ has
an $A_{m_{i}-1}$ singularity at $q_{i}$ for each $i=1,\ldots,k$.

###### Proof.

The miniversal deformation space of a pointed prestable curve of genus zero is
$k\llbracket t_{1},\dots,t_{k}\rrbracket$ where $t_{i}$ is the smoothing
parameter of the $i^{th}$ node. Let $R=\llbracket z\rrbracket$ be the DVR in
the statement. To prove the lemma it suffices to note we can construct a map
of rings $k\llbracket t_{1}\dots t_{k}\rrbracket\to k\llbracket z\rrbracket,$
such that $t_{i}\mapsto z^{m_{i}}$. The locally around the $i^{th}$ node, the
family of curves over $\mathrm{Spec}R$ will be isomorphic to $xy=z^{m_{i}}$ as
required. ∎

We now show that it suffices to consider the case of a stable map where every
maximal connected subtree contracted by $f$ (as in Condition ($\ast$) ‣
1.1(3)) is irreducible.

######  Lemma 2.4.

Let $(f_{0}:C_{0}\to{\mathbb{P}}^{1},p_{1},\ldots,p_{n})$ be a stable map
satisfying Conditions ($\ast$) ‣ 1.1(1), (2), and (3). Then there exists a
deformation to a family of stable maps
$(f:C\to{\mathbb{P}}^{1},\sigma_{i})\to\mathrm{Spec}R$ over the spectrum of a
DVR such that

  1. (1)

the generic fiber $(f_{\eta},C_{\eta}\to{\mathbb{P}}^{1},(\sigma_{i})_{\eta})$
satisfies Conditions ($\ast$) ‣ 1.1(1), (2), and (3), and

  2. (2)

every connected component of $f_{\eta}^{-1}(\infty)\subset C_{\eta}$ is
irreducible.

###### Proof.

Let $E^{1},\ldots,E^{k}$ be the $1$-dimensional connected components of
$f_{0}^{-1}(\infty)$ and write $C^{1},\ldots,C^{r}$ for the connected
components of the closure of the complement
$C_{0}\setminus\\{E^{i}\\}_{i=1}^{k}$. Each $E^{i}$ is pointed by
$(q_{ij},p_{il})$ where the $p_{il}$ are the subset of marked points that lie
on $E^{i}$ and $q_{ij}$ are the points of $E^{i}$ along which $E^{i}$ is glued
to the non-contracted components of $C$. Then each $E_{i}$ is a pointed genus
$0$ prestable curve and there exists a smoothing
$\widetilde{E}^{i}\to\mathrm{Spec}R$ with sections $\tau_{ij}$ smoothing the
$q_{ij}$ and $\sigma_{il}$ smoothing the $p_{il}$. Now consider the constant
family $\bigsqcup_{l}C^{l}_{R}$ with constant sections corresponding to the
marked points and the points $q^{\prime}_{ij}$ along which $q_{ij}$ are glued.

We can glue $\bigsqcup_{l}C^{l}_{R}$ with $\bigsqcup_{t}\widetilde{E}^{t}$ by
identifying the sections $q^{\prime}_{ij}\times\mathrm{Spec}R$ with
$\tau_{ij}$ for all $(i,j)$ and call the result $C\to\mathrm{Spec}R$ which is
now a family of pointed curves. By construction this is a partial smoothing of
the pointed curve $(C_{0},p_{1},\ldots,p_{n})$. Moreover, there is a stable
map $f:C\to{\mathbb{P}}^{1}$ constructed by taking
$f_{R}|_{C^{l}_{R}}:C^{l}_{R}\to{\mathbb{P}}^{1}$ for each $l$ and taking the
constant map $\infty$ on each $\widetilde{E}^{t}$. This descends to a map $f$
as desired since $f(q^{\prime}_{ij})=\infty$. Conditions ($\ast$) ‣ 1.1(1),
(2) are satisfied by $f_{\eta}$ by construction and ($\ast$) ‣ 1.1(3) is
satisfied since the maximal irreducible components $\tilde{E}^{i}_{\eta}$ of
$f_{\eta}^{-1}(\infty)$ have the same number of marked points as $E^{i}$ and
$f_{\eta}$ has the same ramification at $(q^{\prime}_{ij})_{\eta}$ as $f_{0}$
at $q^{\prime}_{ij}$. ∎

We now restrict ourselves to the case where the connected components of
$f^{-1}(\infty)$ are irreducible. Let
$(f:C_{0}\to{\mathbb{P}}^{1},p_{1},\ldots,p_{n})$ be such a map and suppose
that $E_{1},\ldots,E_{k}$ are the $1$-dimensional connected component of
$f^{-1}(\infty)$ and write $E=\sqcup_{i}E_{i}$. By the above reduction, each
$E_{i}$ is a smooth rational curve.

We write the closure of the complement of $E$ in $C_{0}$ as a union of
connected components $C_{1},\ldots,C_{r}$. Thus each $C_{i}$ is at tree of
rational curves such that $(f_{0})|_{C_{i}}$ is non-constant and such that the
preimage of $\infty$ is zero dimensional. For each $i$, let
$I_{i}\subset\\{1,\ldots,n\\}$ be the subset of indices $k$ such that $p_{k}$
lies on $E_{i}$. If $E_{i}$ and $C_{j}$ intersect, we let $q_{ij}\in E_{i}$
and $t_{ij}\in C_{j}$ be the points of $E_{i}$ and $C_{j}$ respectively at
which they are glued and we let $e_{ij}$ be the ramification index of
$(f_{0})|_{C_{j}}$ at $t_{ij}$. Note that Condition ($\ast$) ‣ 1.1(3) then
reads that for each $i$,

$n_{i}:=\sum_{k\in I_{i}}d_{k}=\sum_{j}e_{ij}$

where the left side is a definition and the right sum is over $j$ such that
$E_{i}$ meets $C_{j}$.

Consider a smoothing $(C\to S,\sigma_{1},\ldots,\sigma_{n})$ of
$(C_{0},p_{1},\ldots,p_{n})$ over the spectrum of a DVR and let $D=\sum
d_{i}\sigma_{i}$ denote the divisor of $\Gamma$-weighted marked sections on
this surface. We wish to construct a rank one linear series on such a
smoothing so that the central fiber agrees with $f_{0}$ and the generic fiber
satisfies that $D|_{\eta}$ is the preimage of $\infty$. Our strategy is to
consider the map $\pi:C\to C^{\prime}$ which contracts $E$ to a point, and
instead construct the appropriate linear series on $C^{\prime}$.

######  Proposition 2.5.

In the setting above, there exists a smoothing $(C\to
S,\sigma_{1},\ldots,\sigma_{n})$ and a Cartier divisor $D^{\prime}$ on
$C^{\prime}$ where $\pi:C\to C^{\prime}$ is the contraction of the $E$ such
that

  1. (1)

$\pi^{*}D^{\prime}=D+\sum a_{i}E_{i}$ for some $a_{i}$ where $D$ is the
divisor of marked sections, and

  2. (2)

$a_{i}E_{i}.C_{j}=e_{ij}t_{ij}$ for all $i$ and $j$ such that $E_{i}$ meets
$C_{j}$.

###### Proof.

For each $i$, we let $a_{i}=\prod_{j=1}^{r}e_{ij}$ and let
$m_{ij}=\prod_{k\neq j}e_{ik}$. By Lemma 2.4, there exists a smoothing $(C\to
S,\sigma_{1},\ldots,\sigma_{n})$ of $(C_{0},p_{1},\ldots,p_{n})$ such that $C$
has an $A_{m_{ij}-1}$ singularity at the node $t_{ij}$. Computing intersection
products in the surface $C$, we have $E_{i}.C_{0}=0$ since $C_{0}$ is a fiber
containing $E_{i}$. Moreover, if $E_{i}$ and $C_{j}$ intersect, then

$E_{i}.C_{j}=\frac{1}{m_{ij}}t_{ij}$

since locally around the node $t_{i}$, the curves $E_{i}$ and $C_{j}$ are
distinct lines through an $A_{m_{ij}-1}$ singularity. Therefore,

$E_{i}^{2}=\sum_{j}-\frac{1}{m_{ij}}$

where the sum is over those $j$ for which $C_{i}$ and $E_{j}$ meet. Here, we
go back and forth between viewing these intersection products as numbers or as
divisors on the curves depending on whether its convenient to emphasize that
the particular intersection points.

Now consider the divisor $D+\sum a_{i}E_{i}$. We can compute that
$D.E_{i}=\sum_{k\in I_{i}}d_{k}=n_{i}$ since $D$ is the divisor of weighted
marked sections and $E_{i}$ contains $p_{k}$ for $k\in I_{i}$. On the other
hand,

$aE_{i}.C_{j}=\frac{a_{i}}{m_{ij}}=\frac{\prod_{j}e_{ij}}{\prod_{k\neq
j}e_{ik}}=e_{ij}.$

Finally,

$(D+\sum
a_{k}E_{k}).E_{i}=n_{i}-\sum_{j}\frac{a_{i}}{m_{ij}}=n_{i}-\sum_{j}\frac{\prod_{i}e_{ij}}{\prod_{k\neq
j}e_{ik}}=n_{i}-\sum_{j}e_{ij}=0$

by Condition ($\ast$) ‣ 1.1(3). Here we have used that $E_{k}.E_{i}=0$ if
$i\neq k$.

Therefore, we need to show that $D+\sum a_{i}E_{i}$ descends to a Cartier
divisor $D^{\prime}$ along the contraction $\pi:C\to C^{\prime}$. Note that
$\pi$ exists, $C^{\prime}$ is a normal quasiprojective surface, and
$\pi_{*}\mathcal{O}_{C}=\mathcal{O}_{C^{\prime}}$ by [KM98, Theorem 3.7].
Moreover, we have an exact sequence

(1)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{Pic}(C^{\prime})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi^{*}}$$\textstyle{\mathrm{Pic}(C)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathbb{Z}}$

where the map $\mathrm{Pic}(C)\to\mathbb{Z}$ is the restriction $L\mapsto
L|_{E}$ [KM98, Corollary 3.17]. In particular, $\mathcal{O}_{C}(D+\sum
a_{i}E_{i})$ is in the kernel of this map and thus there exists a line bundle
$L$ on $C^{\prime}$ such that $\pi^{*}L=\mathcal{O}_{C}(D+aE)$. On the other
hand, $\pi_{*}\mathcal{O}_{C}=\mathcal{O}_{C^{\prime}}$ and so by projection
formula

$\pi_{*}\mathcal{O}_{C}(D+\sum a_{i}E_{i})=L.$

Therefore, there exists a section $s\in H^{0}(L)$ which pulls back to the
section cutting out $D+\sum a_{i}E_{i}$ on $C$ and so
$D^{\prime}=\mathrm{div}(s)$ is the required Cartier divisor. ∎

######  Remark 2.6.

Fix a list of positive integers $r_{ij}$ for $i$ and $j$ such that $E_{i}$
meets $C_{j}$. Then in the construction of of the smoothing in Proposition
2.5, we can replace $e_{ij}$ by $r_{ij}e_{ij}$ in the definitions of $a_{i}$
and $m_{ij}$. In this way we get a smoothing satisfying the properties of
Proposition 2.5 where the total space has $A_{m_{ij}-1}$-singularities with
$m_{ij}$ divisible by any fixed choice of integers.

Let $0\in{\mathbb{P}}^{1}$ be a general point of the target and consider the
divisor $B_{0}\subset C_{0}$ given by $f_{0}^{-1}(0)$. Then $B_{0}$ consists
of a union of points on $C_{0}$ which are disjoint from $E$ and contained in
the locus where $f_{0}$ is étale. Since $C\to S$ is a family of genus $0$
curves birational to ${\mathbb{P}}^{1}_{S}$, we can extend the collection of
points $B_{0}$ into a Cartier divisor $B$ on $C$ which is horizontal and
satisfies $B|_{C_{0}}=B_{0}$.

######  Proposition 2.7.

With notation as above, $D+\sum a_{i}E_{i}\sim B$ are linearly equivalent
Cartier divisors on $C$.

###### Proof.

Let $\psi:C\to S$ denote the morphism to the base, and consider $\psi_{*}N$
where $N=\mathcal{O}_{C}(D+\sum a_{i}E_{i}-B)$. First we claim that
$H^{1}(C_{0},N|_{C_{0}})=0$. Indeed since $C_{0}$ is a nodal genus $0$ curve,
it suffices to show that the degree of $N$ restricted to each component of
$C_{0}$ is $\geq 0$. Now $N|_{E}=\mathcal{O}_{E}$ by construction since
$(D+\sum a_{i}E_{i}).E=0$ and $B$ avoids $E$. On the other hand, for each $i$,
we have $(D+\sum a_{i}E_{i}).C_{j}=f_{0}^{-1}(\infty)|_{C_{j}}$ and
$B.C_{j}=f_{0}^{-1}(0)|_{C_{j}}$ so these divisors are linearly equivalent.
Therefore, we in fact have that $N$ is trivial on each component of $C_{0}$.
Therefore, by cohomology and base change, $\psi_{*}N$ is a vector bundle whose
formation commutes with base change. Moreover, over the generic fiber,
$C_{\eta}\cong{\mathbb{P}}^{1}_{\eta}$ and $B$ and $D+\sum a_{i}E_{i}$ are
divisors of the same degree $n$ by construction. Therefore,
$H^{0}(C_{\eta},N|_{C_{\eta}})=1$. Since any line bundle on the spectrum of a
DVR is trivial, we conclude that $\psi_{*}N\cong\mathcal{O}_{S}$. Therefore,
$N$ has a non-vanishing section which exhibits the required linear equivalence
$D+\sum a_{i}E_{i}\sim B$. ∎

Now let $B^{\prime}$ be the image of $B$ under $\pi$ which is also a Cartier
divisor since $B$ is contained in the locus where $\pi$ is an isomorphism. By
the exact sequence 1, the linear equivalence in Proposition 2.7 is equivalent
to a linear equivalence $D^{\prime}\sim B^{\prime}$. In particular,
$D^{\prime}$ and $B^{\prime}$ form a basepoint free rank one linear subseries
of $H^{0}(C^{\prime},L)$ where $L=\mathcal{O}_{C^{\prime}}(D^{\prime})$.

######  Proposition 2.8.

With notation as above, let $g:C^{\prime}\to{\mathbb{P}}^{1}$ be the morphism
induced by the basepoint free linear series $\langle
B^{\prime},D^{\prime}\rangle$. Then the composition
$C\xrightarrow{\pi}C^{\prime}\xrightarrow{g}{\mathbb{P}}^{1}$ satisfies
$(g\circ\pi)|_{C_{0}}=f_{0}$.

###### Proof.

The map $(g\circ\pi)$ contracts $E$ to $\infty$ by construction. Moreover, on
each $C_{j}$, the restriction

$B^{\prime}|_{C_{j}}=f_{0}^{-1}(0)|_{C_{j}}$

and

$D^{\prime}|_{C_{j}}=f_{0}^{-1}(\infty)|_{C_{j}}$

by Proposition 2.5. Therefore $(g\circ\pi)|_{C_{j}}=(f_{0})_{C_{j}}$ for all
$j=1,\ldots,k$. We conclude that $(g\circ\pi)|_{C_{0}}=f_{0}$ as it agrees
with $f_{0}$ on each component of $C_{0}$. ∎

Putting these together, we conclude the following.

######  Theorem 2.9.

Let $(f_{0}:C_{0}\to{\mathbb{P}}^{1},p_{1},\ldots,p_{n})$ be a stable map
satisfying Conditions ($\ast$) ‣ 1.1(1), (2) and (3). Then there exists a
smoothing $(C\to S,\sigma_{1},\ldots,\sigma_{n})$ over $S=\mathrm{Spec}R$ the
spectrum of a DVR and a stable map $f:C\to{\mathbb{P}}^{1}$ such that the
generic fiber
$(f_{\eta},C_{\eta}\to{\mathbb{P}}^{1}_{\eta},(\sigma_{i})_{\eta})$ is
contained in $\mathcal{M}_{(d_{1},\ldots,d_{n})}$ and $f|_{C_{0}}=f_{0}$. In
particular, Conditions ($\ast$) ‣ 1.1(1), (2) and (3) are both necessary and
sufficient for a stable map to be contained in
$\overline{\mathcal{M}}_{(d_{1},\ldots,d_{n})}$.

###### Proof.

By Propositions 2.5 & 2.8, there exists a smoothing $(C\to S,\sigma_{i})$ and
a map $f:C\to{\mathbb{P}}^{1}$ such that $f|_{C_{0}}=f_{0}$ and
$f^{-1}(\infty)=D+\sum a_{i}E_{i}$ where $D$ is the divisor of weighted marked
sections and $E_{i}$ are contained on $C_{0}$. Therefore, $f|_{C_{\eta}}$ is
in $\mathcal{M}_{(d_{1},\ldots,d_{n})}$. ∎

##  3\. Log twisted curves and maps

We refer the reader to [ACG+13] for the standard definitions in log geometry
and to [CJRS18, Section 2] for log twisted stable maps. We will use $M_{X}$ to
denote the log structure of $X$, which will be implicit if no confusion
arises. The characteristic of the log structure will be denoted by
$\overline{M}_{X}$.

###  3.1. Log geometry

We will need the following results regarding log smooth morphisms.

######  Remark 3.1.

  * •

If a morphism of log schemes $f:X\to pt$ is log smooth (where the point has
the trivial log structure), then $U\subset X$ is a toroidal embedding where
$U$ is the locus with trivial log structure and $M_{X}$ is the divisorial log
structure.

  * •

If $C$ is a curve, then log smooth is equivalent to nodal [ACG+13, Theorem 5].

  * •

Given a family $f:C\to S$ of nodal curves, there is a _minimal_ log structure
$M_{S}$ on $S$ such that any other log structure that makes $f$ log smooth is
pulled back from it [ACG+13, Section 7.3]. We call the corresponding log
structure and structure morphism $f^{\flat}:f^{*}M_{S}\to\tilde{M}_{C}$ the
_canonical_ log structure $f:C\to S$. Note here $\tilde{M}_{C}$ does not
include marked points. If $f$ is equipped with marked points then we denote
the natural log structure

$M_{C}:=\tilde{M}_{C}\bigoplus_{\mathcal{O}_{C}^{*}}\left(\oplus_{\mathcal{O}_{C}^{*},i}N_{i}\right)$

where the sum is over marked points and $N_{i}$ is the divisorial log
structure associated to the $i^{th}$ marked point and call $M_{C}$ the
_canonical log structure_ associated to a pointed curve if there is no
confusion.

There exists a log cotangent complex $\mathbb{L}^{\mathrm{log}}_{X/Y}$ (see
[Ols05] or [ACG+13, Section 7]) for morphisms of log schemes $f:X\to Y$, and
deformation theory of log schemes is controlled by the log cotangent complex
([Ols05, Theorem 5.2]).

######  Remark 3.2.

If $f$ is a log smooth morphism, then the log cotangent complex
$\mathbb{L}^{\mathrm{log}}_{X/Y}$ is represented by the sheaf of log
differentials ([Ols05, 1.1 (iii)]). There does not exist a distinguished
triangle in general, however, Olsson constructs a distinguished triangle for
log flat or integral morphisms (see [Ols05, 1.3]).

###  3.2. (log) Twisted curves

Our main reference is [Ols07]. To compactify moduli spaces of maps
$f:C\to\mathcal{M}$, where $\mathcal{M}$ is a Deligne-Mumford stack, one needs
to allow $C$ to be a stack as well, known as a _twisted curve_ (see e.g.
[AV02, AB19]).

######  Definition 3.3.

A _twisted curve_ is a purely 1-dimensional Deligne-Mumford stack
$\mathcal{C}$, with at most nodes as singularities, satisfying the following
conditions:

  1. (1)

If $\pi:\mathcal{C}\to C$ denotes the coarse space morphism, then
$\mathcal{C}^{sm}=\pi^{-1}C^{sm}$, and $\pi$ is an isomorphism over a dense
open subset of $C$.

  2. (2)

If $\bar{x}\to C$ is a node such that the strictly henselian local ring
$\mathcal{O}_{C,\bar{x}}$ is the strict henselization of $k[x,y]/(xy)$. Then

$\mathcal{C}\times_{C}\mathrm{Spec}\mathcal{O}_{C,\bar{x}}\cong\bigg{[}\mathrm{Spec}\mathcal{O}_{C,\bar{x}}[z,w]/(zw,z^{m}-x,w^{m}-y)/\mu_{m}\bigg{]},$

where $\xi\in\mu_{m}$ acts by $(z,w)\mapsto(\xi z,\xi^{-1}w)$.

######  Definition 3.4.

An _$n$ -pointed twisted curve_ $\mathcal{C}/S$ marked by disjoint closed
substacks $\\{\Sigma_{i}\\}_{i=1}^{n}$ in $\mathcal{C}$ is assumed to satisfy
the following:

  1. (1)

the $\Sigma_{i}$ are contained in $\mathcal{C}^{sm}$,

  2. (2)

each $\Sigma_{i}$ is a tame étale gerbe over $S$, and

  3. (3)

$\mathcal{C}^{sm}\setminus\cup\Sigma_{i}\to C$ is an open embedding.

Let $\mathcal{X}$ be a Deligne–Mumford stack. We say that a fine log structure
$M_{\mathcal{X}}$ is said to be locally free if for every geometric point
$\bar{x}\to\mathcal{X}$, the characteristic sheaf
$\overline{M}_{\mathcal{X}}\cong\mathbb{N}^{r}$ for some $r$.

######  Definition 3.5.

In the above situation, we say that a morphism of sheaves of monoids $M\to
M^{\prime}$ is _simple_ if for every geometric point $\bar{x}\to\mathcal{X}$
we have the following

${\overline{M}_{\mathcal{X}}}$${\overline{M}^{\prime}_{\mathcal{X}}}$${\mathbb{N}^{r}}$${\mathbb{N}^{r}}$$\scriptstyle{\phi}$

where $\phi$ is given by $(m_{1},\dots,m_{r})$.

######  Definition 3.6.

An _$n$ -pointed twisted log curve over $S$_ is the data

$(C/S,\\{\sigma_{i},a_{i}\\},l:M_{S}\to M^{\prime}_{S})$

where

  * •

$(C,\\{\sigma_{i}\\})/S$ is an $n$-pointed nodal curve,

  * •

$M_{S}$ is the minimal log structure for the family $C\to S$,

  * •

$a_{i}:S\to\mathbb{Z}_{>0}$ are locally constant, and

  * •

$l$ is a simple morphism.

######  Theorem 3.7.

[Ols07, Theorem 1.8] The fibered category of $n$-pointed twisted curves is
naturally equivalent to the stack of $n$-pointed log twisted curves.

######  Example 3.8.

Consider a smoothing of a nodal curve as in Lemma 2.3 and let
$(C,M_{C})\to(S,M_{S})$ be the minimal log structure. The appearance of
$A_{m_{i}-1}$ singularities on the total space of the smoothing at the nodes
of $C_{0}$ is equivalent to the existence of a simple extension
$M_{S}\hookrightarrow M_{S}^{\prime}$ with $\phi=(m_{1},\ldots,m_{k})$. Thus,
by Olsson’s Theorem 3.7, such a smoothing is the coarse space of a smoothing
of the twisted curve with stabilizer $\mu_{m_{i}}$ at the $i^{th}$ node of
$C_{0}$. Note that we can see this directly by taking the canonical stack of
the total space $C$ which introduces a $\mu_{m_{i}}$ stabilizer at the
$A_{m_{i}-1}$ singularity.

Let us briefly explicate the construction of Theorem 3.7 as we will use it in
the sequel. If $\pi:\mathcal{C}\to C$ is the coarse map of a twisted curve
over $S$, then the simple extension $l:M_{S}\hookrightarrow M_{S}^{\prime}$ is
the map between the minimal log structures of $h:C\to S$ and $f:\mathcal{C}\to
S$ respectively. The pushforward log structure
$\pi_{*}\tilde{M}_{\mathcal{C}}$ sits in a pushout square

$\textstyle{h^{*}M_{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h^{*}l}$$\scriptstyle{h^{\flat}}$$\textstyle{h^{*}M_{S}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{*}f^{\flat}}$$\textstyle{\tilde{M}_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\pi_{*}\tilde{M}_{\mathcal{C}}}$

The $a_{i}$ keeps track of the stabilizer of $\mathcal{C}/C$ along
$\sigma_{i}$ and we have simple extensions
$N_{C,i}\hookrightarrow\pi_{*}N_{\mathcal{C},i}$ corresponding to
multiplication by $a_{i}$ on characteristic monoids.

######  Definition 3.9.

Let $(\mathcal{X},M_{\mathcal{X}})$ be a separated Deligne-Mumford stack wtih
fs log structure. A _log twisted prestable map_ is a diagram

$\leavevmode\hbox to110.22pt{\vbox
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where $\mathcal{C}\to S$ is a twisted curve and
$(\mathcal{C},M_{\mathcal{C}})\to(S,M_{S})$ is log smooth. A log twisted
prestable map is _stable_ if the underlying map $f$ is representable and the
coarse map $h:C/S\to X$ is a stable map.

We will also make use of the _root stack_ construction of Borne-Vistoli [BV12,
Section 4.2] in the case of a locally free log structure. Suppose $(X,M_{X})$
is a locally free log scheme and
$\phi:\overline{M}_{X}\to\overline{M}^{\prime}$ is a simple extension. The
root stack $\sqrt[\phi]{(X,M_{X})}$ is a log algebraic stack over $Sch_{X}$
which associates to $s:T\to X$ the groupoid of pairs $(l,\varphi)$ where
$l:s^{*}M_{X}\to M_{T}^{\prime}$ is a simple extension of log structures and
$\varphi$ is an isomorphism of maps of characteristic monoids
$\overline{l}\cong s^{*}\phi$.

We conclude this section by comparing the sheaves of log differentials between
$\mathcal{C}$ and $C$. This simple comparison is one of the advantages of
using log geometry to study twisted curves. Let
$(f,f^{\flat}):(\mathcal{C},M_{\mathcal{C}})\to(S,M_{S}^{\prime})$ and
$(h,h^{\flat}):(C,M_{C})\to(S,M_{S})$ denote the natural log smooth maps where
$M_{S}\hookrightarrow M_{S}^{\prime}$ is a simple extension.

######  Notation 3.10.

We will denote by $\Omega^{log}_{\mathcal{C}/S}:=\Omega^{log}_{(f,f^{\flat})}$
and $\Omega^{log}_{C/S}:=\Omega^{log}_{(h,h^{\flat})}$ and similarly for the
cotangent complex.

######  Proposition 3.11.

Let $\pi:\mathcal{C}\to C$ be the coarse space of a twisted curve. Then

$\mathbb{L}^{\mathrm{log}}_{\mathcal{C}/S}\simeq_{qis}\Omega^{log}_{\mathcal{C}/S}[0]\cong\pi^{*}\Omega^{log}_{C/S}[0]\simeq_{qis}L\pi^{*}\mathbb{L}^{\mathrm{log}}_{C/S}.$

###### Proof.

Since both $(f,f^{\flat})$ and $(h,h^{\flat})$ are log smooth, then
$\mathbb{L}^{\mathrm{log}}_{\mathcal{C}/S}\simeq_{qis}\Omega^{log}_{\mathcal{C}/S}$
and $\mathbb{L}^{\mathrm{log}}_{C/S}\simeq_{qis}\Omega^{log}_{C/S}$. Since
$\Omega^{log}_{C/S}$ is locally free,
$L\pi^{*}\Omega^{log}_{C/S}[0]\simeq_{qis}\pi^{*}\Omega^{log}_{C/S}[0]$.
Therefore, it suffices to show that the natural map

$\rho:\pi^{*}\Omega^{log}_{C/S}\to\Omega^{log}_{\mathcal{C}/S}$

is an isomorphism which we can compute étale locally. Toward that end, suppose
$S=\mathrm{Spec}R$. In an étale neighborhood of a marked gerbe with with
stabilizer $\mu_{a}$, the map $\pi$ can be written as $R[u]\mapsto R[x]$ with
$u\mapsto x^{a}$ so $\rho:d\log u\mapsto ad\log x$. In a neighborhood of a
node with stabilizer $\mu_{a}$, we have $R[u,v]/(uv-t^{a})\mapsto
R[x,y]/(xy-t)$ with $(u,v)\mapsto(x^{a},y^{a})$ so $\rho$ is given by $d\log
u\mapsto ad\log x$, $d\log v\mapsto ad\log y$. In either case, $\rho$ is an
isomorphism since $a\in R$ is invertible. ∎

##  4\. Log maps to $1$-dimensional targets

In this section we collect some results on log twisted maps to stacky curves
which we will use to lift the smoothings from Section 2 to the twisted case.

###  4.1. Introducing log structures

We begin by giving a criterion to lift a prestable map with $1$-dimensional
target to a log map. More precisely, suppose $g:C\to X$ is a prestable map to
a smooth curve $X$ with log structure $M_{X}$ such that $(X,M_{X})$ is log
smooth over the trivial log point. We wish to find a log structure $M_{C}$ on
$C$ such that $(C,M_{C})$ is a log smooth curve and $g$ lifts to log map. For
the purpose of constructing smoothings, it suffices by Lemma 2.4 to consider
the case where the connected components of $C$ which are contracted to the
marked points of $(X,M_{X})$ are smooth (see also Remark 4.2).

######  Proposition 4.1.

Let $g:C\to X$ be a prestable map with $(X,M_{X})$ as above. Suppose that the
$1$-dimensional connected components of the support of
$f^{*}(\overline{M}_{X})$ are smooth genus $0$ curves. Then $g$ lifts to a
logarithmic prestable map. That is, there exist log structures $M_{C}$ on $C$
and $M$ on $S=\mathrm{Spec}k$ and a map $g^{\flat}:g^{*}M_{X}\to M_{C}$ such
that $(C,M_{C})\to(S,M)$ is log smooth.

###### Proof.

The question is étale local on the target so we may pull back to a chart for
the log structure $(X,M_{X})$ so without loss of generality, we may assume
that the target is $(\mathbb{A}^{1},0)=(X,D)$ with its toric log structure and
that $g:C\to\mathbb{A}^{1}$ is a proper map satisfying the assumptions of the
proposition. Moreover, we write
$g^{-1}(0)=\\{p_{1},\ldots,p_{k},E_{1},\ldots,E_{r}\\}$ as a disjoint union of
points and smooth rational curves. Let us denote by $q_{ij}$ the nodes of $C$
along $E_{i}$ and by $e_{ij}$ the ramification of $g$ at these points. First
we claim that $g$ factors through an expansion of the target

$\textstyle{C\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{\widetilde{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\varphi}$$\textstyle{X.}$

Indeed consider $\widetilde{X}=E\cup_{0}\mathbb{A}^{1}$ where
$E=\mathbb{P}(N_{D/X}+\mathcal{O}_{D})$. To construct such a factorization it
suffices to construct nonconstant maps $h_{i}:E_{i}\to E$ with
$h_{i}(q_{ij})=0$ and such that the order of tangency of $h_{i}$ at $q_{ij}$
is equal to $e_{ij}$. Since $E_{i}$ are smooth rational curves, such maps
exist.

Therefore, we have a factorization $h:C\to\widetilde{X}$ and it suffices to
show that $h$ can be lifted to a log map where

$M_{\widetilde{X}}=\widetilde{M}_{\widetilde{X}}\oplus_{\mathcal{O}_{\widetilde{X}}^{*}}\varphi^{*}M_{X}$

since $\varphi$ underlies a log map
$(\widetilde{X},M_{\widetilde{X}})\to(X,M_{X})$ and the composition of log
maps is a log map. Now we have reduced to the case where the support of
$h^{*}(\overline{M}_{\widetilde{X}}/\mathbb{N})=\\{p_{1},\ldots,p_{k},p_{k+1},\ldots,p_{n}\\}$
is a union of isolated points. Here the copy of $\mathbb{N}$ in the quotient
is the pullback of the minimal log structure on $\mathrm{Spec}k$ along
$\tilde{X}\to\mathrm{Spec}k$. In a neighborhood of the smooth $p_{i}$ there is
a unique divisorial log structure such that the map lifts encoding the order
of tangency. In a neighborhood of the nodal $p_{i}$, the map lifts to a log
map by the tangency condition above which is precisely the predeformability
condition for a map to an expansion (see for example [ACG+13, Section 12]). ∎

######  Remark 4.2.

The proposition may be generalized to the case when the $1$-dimensional
connected components of the support of $f^{*}\overline{M}_{X}$ are trees of
rational curves at the expense of allowing $C$ to be blown up at the nodes. To
do this, one needs to consider the map $g^{trop}$ of tropical curves and
subdivide the source and target to make $g^{trop}$ tropically transverse (see
[Ran20, Sections 2.5 & 2.6] for more details). Such a subdivision corresponds
to choosing a possibly larger expansion $\widetilde{X}\to X$ and a sequence of
log blowups $\widetilde{C}\to C$ after which we may lift $g$ to map
$h:\widetilde{C}\to\widetilde{X}$ which underlies a log map as in the
proposition. Since this is not strictly necessary for our stated goal of
constructing smoothings, we leave the details to the reader.

Next we address the question of lifting the log structure to a twisted map
from its coarse space. More precisely, consider
$f:\mathcal{C}/S\to\mathcal{X}$ a proper morphism from a twisted curve to a
smooth Deligne-Mumford curve $\mathcal{X}$. Let $M_{\mathcal{X}}$ be a log
structure making $(\mathcal{X},M_{\mathcal{X}})$ a log twisted curve with
coarse space $(X,M_{X}\hookrightarrow M^{\prime}_{X})$. Here $(X,M_{X})$ is
log smooth and $M_{X}\hookrightarrow M^{\prime}_{X}$ is the simple extension
encoding the stabilizers of $\mathcal{X}$. Suppose furthermore that there
exist log structures $M_{C}$ and $M_{S}$ on the coarse space $C$ of
$\mathcal{C}$ and $S$ respectively such that

$\textstyle{(C,M_{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{(X,M_{X})}$$\textstyle{(S,M_{S})}$

is a log prestable map.

######  Proposition 4.3.

Let $f:\mathcal{C}/S\to\mathcal{X}$ and $f:(C,M_{C})/(S,M_{S})\to(X,M_{X})$ as
above. Then there exists a root stack $\pi:\tilde{\mathcal{C}}\to\mathcal{C}$
and log structures $M_{\tilde{\mathcal{C}}}$ on $\tilde{\mathcal{C}}$ and
$M_{S}^{\prime}$ on $S$ such that
$(\tilde{\mathcal{C}},M_{\tilde{\mathcal{C}}})\to(S,M_{S}^{\prime})$ is a log
smooth curve and there exists $f^{\flat}:(f\circ\pi)^{*}M_{\mathcal{X}}\to
M_{\tilde{\mathcal{C}}}$ making $f$ into a log prestable map.

###### Proof.

Let $g:C\to X$ be the coarse map. By Theorem 3.7 we have simple extensions of
log structures $M_{C}\hookrightarrow M^{\prime}_{C}$, $M_{S}\hookrightarrow
M^{\prime}_{S}$, and $M_{X}\hookrightarrow M_{X}^{\prime}$ on $C$, $S$ and $X$
respectively such that $(C\to S,\\{a_{i},\sigma_{i}\\},M_{S}\hookrightarrow
M^{\prime}_{S})$ and $(X\to\mathrm{Spec}k,\\{b_{i},x_{i}\\},k^{*}\to k^{*})$
are log twisted curves. Here $\sigma_{i}:S\to C$ correspond to the marked
gerbes of $\mathcal{C}\to S$ with order of stabilizer given by $a_{i}$.
Similarly, $x_{i}\in X$ correspond to the marked gerbes of $\mathcal{X}$ with
stabilizer order $b_{i}$.

The question is local on $X$ so we can restrict to a neighborhood of one of
the marked points $x_{i}$ with stabilizer $\mu_{b_{i}}$ on $X$ and suppose
without loss of generality that there is a single marked point $x\in X$ with
stabilizer $\mu_{b}$. This point is exactly the support of
$\overline{M}_{X}=\mathbb{N}$ and $\overline{M}_{X}^{\prime}=\mathbb{N}$ and
the simple extension is simply the map $1\mapsto b$.

By assumption, there exists $g^{\flat}:g^{*}M_{X}\to M_{C}$ making
$(g,g^{\flat})$ into a log map. By Theorem 3.7, it suffices to show that there
is a further simple extension $M_{C}^{\prime}\hookrightarrow
M_{C}^{\prime\prime}$ which is an isomorphism away from finitely many smooth
points and a map of log structures $f^{\flat}:g^{*}M_{X}^{\prime}\to
M_{C}^{\prime}$ such that the square

$\textstyle{g^{*}M_{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g^{\flat}}$$\textstyle{M_{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{g^{*}M_{X}^{\prime}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f^{\flat}}$$\textstyle{M_{C}^{\prime\prime}}$

commutes. Note that $M^{\prime}_{X}=M_{X}\oplus\mathbb{N}/(e=be^{\prime})$
where $e$ is the element corresponding to a local parameter of $x\in X$ and
$e^{\prime}$ is the generator of the copy of $\mathbb{N}$. By the universal
property of this pushout, to construct $f^{\flat}$, it suffices to show that
$s(g^{\flat}(e))$ is divisible by $b$. However, we can guarantee this by
taking a large enough simple extension $s$. ∎

###  4.2. Deformations of log twisted maps and their coarse spaces

Let

---
$\textstyle{(\mathcal{C},M_{\mathcal{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(f,f^{\flat})}$$\textstyle{(\mathcal{X},M_{\mathcal{X}})}$$\textstyle{(S,M_{S})}$

be a log stable map from a log twisted curve and suppose that
$(\mathcal{X},M_{\mathcal{X}})$ is a log smooth curve. We define the _critical
locus_ of $f$ to be the support of the cokernel of the natural map

$f^{*}\Omega^{log}_{\mathcal{X}}\to\Omega^{\mathrm{log}}_{\mathcal{C}/S}.$

Note that this map, viewed as a two term complex, is a presentation of the log
cotangent complex $\mathbb{L}^{\mathrm{log}}_{f}$ relative to $S$ so we may
equivalently define the critical locus to be the support of
$\mathcal{H}^{0}(\mathbb{L}^{\mathrm{log}}_{f})$.

The critical locus is a union of nodes of $\mathcal{C}$, branch points of $f$
and contracted components of $f$. By contrast, the kernel of the above map, or
equivalently $\mathcal{H}^{-1}(\mathbb{L}^{\mathrm{log}}_{f})$, is supported
solely on the contracted components of $f$.

Let $A_{1},\ldots,A_{k}$ be the connected compponents of the critical locus
and $U_{1},\ldots,U_{k}$ be affine étale neighborhoods of the $A_{i}$ with log
structure pulled back from $\mathcal{C}$. Suppose each $U_{i}$ avoids $A_{j}$
for $j\neq i$. Let $\mathrm{Def}_{(}f,f^{\flat})$ be the versal deformation
space of the log map $(f,f^{\flat})$. Since the moduli space is a Deligne-
Mumford stack, $\mathrm{Def}_{(f,f^{\flat})}$ is pro-representable. Let
$\mathrm{Def}_{i}$ be the miniversal deformation space of the restriction
$(f_{i}:=f|_{U_{i}}:U_{i}\to\mathcal{X},f^{\flat}|_{U_{i}})$. We have the
following proposition (see also [Vak00, Proposition 4.3]).

######  Proposition 4.4.

In the setting above, the natural map

$\operatorname{Def}_{(f,f^{\flat})}\to\operatorname{Def}_{1}\times\ldots\times\operatorname{Def}_{k}$

is an isomorphism.

###### Proof.

The deformation space $\operatorname{Def}_{(f,f^{\flat})}$ (respectively
$\operatorname{Def}_{i}$) is constructed via the complex

$R\operatorname{Hom}(\mathbb{L}^{\mathrm{log}}_{f},\mathcal{O}_{\mathcal{C}})\
(\text{respectively
}R\operatorname{Hom}(\mathbb{L}^{\mathrm{log}}_{f_{i}},\mathcal{O}_{U_{i}}))$

Since $g_{i}:U_{i}\to\mathcal{C}$ are strict étale, then
$Lg_{i}^{*}\mathbb{L}^{\mathrm{log}}_{f}=\mathbb{L}^{\mathrm{log}}_{f_{i}}$.
Let $K_{i}$ be the component of
$K=\mathcal{H}^{-1}(\mathbb{L}^{\mathrm{log}}_{f})$ supported on $A_{i}\subset
U_{i}$ and $Q_{i}$ the component of
$Q=\mathcal{H}^{0}(\mathbb{L}^{\mathrm{log}}_{f})$ supported on $A_{i}\subset
U_{i}$. Since the $A_{i}$ are disjoint, we have $K=\oplus K_{i}$ and $Q=\oplus
Q_{i}$ and $K_{i}=\mathcal{H}^{-1}(\mathbb{L}^{\mathrm{log}}_{f_{i}})$ and
$Q_{i}=\mathcal{H}^{0}(\mathbb{L}^{\mathrm{log}}_{f_{i}})$.

Since both $\mathbb{L}^{\mathrm{log}}_{f}$ and
$\mathbb{L}^{\mathrm{log}}_{f_{i}}$ are two term complexes, we have
distinguished triangles

$K[1]\to\mathbb{L}^{\mathrm{log}}_{f}\to Q\to$

and

$K_{i}[1]\to\mathbb{L}^{\mathrm{log}}_{f_{i}}\to Q_{i}\to.$

There is a natural map
$R\operatorname{Hom}(\mathbb{L}^{\mathrm{log}}_{f},\mathcal{O}_{\mathcal{C}})\to
R\operatorname{Hom}(\mathbb{L}^{\mathrm{log}}_{f_{i}},\mathcal{O}_{U_{i}})$
via restriction. Taking direct sums yields the following diagram of
distinguished triangles

$\textstyle{R\operatorname{Hom}(Q,\mathcal{O}_{\mathcal{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R\operatorname{Hom}(\mathbb{L}^{\mathrm{log}}_{f},\mathcal{O}_{\mathcal{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{R\operatorname{Hom}(K[1],\mathcal{O}_{\mathcal{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus
R\operatorname{Hom}(Q_{i},\mathcal{O}_{U_{i}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus
R\operatorname{Hom}(\mathbb{L}^{\mathrm{log}}_{f_{i}},\mathcal{O}_{U_{i}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\bigoplus
R\operatorname{Hom}(K_{i}[1],\mathcal{O}_{U_{i}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\
}$

where the vertical maps are equalities via the identification $K=\oplus K_{i}$
and $Q=\oplus Q_{i}$. Therefore, the middle map is an isomorphism by the
properties of distinguished triangles or equivalently the five lemma from
which we conclude. ∎

######  Remark 4.5.

Via the big diagram in the above proof, we see that in fact the deformation
spaces $\operatorname{Def}_{i}$ of an étale neighborhood $U_{i}$ of the
component $A_{i}$ of the critical locus is independent of the choice of
$U_{i}$ since the $Q_{i}$ and $K_{i}$ terms are independent of $U_{i}$.

Now we address the question of lifting log smooth deformations along the
coarse space map of a log twisted curve. Let

$\textstyle{(C_{0},M_{C_{0}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{0}}$$\textstyle{(X,M_{X})}$$\textstyle{(S_{0},M_{S_{0}})}$

be a log prestable map and suppose that there exists a simple extension of log
structures $M_{S}\to M^{\prime}_{S}$ such that
$(C_{0}/S_{0},(\sigma_{i},a_{i}),l_{0}:M_{S_{0}}\to M^{\prime}_{S_{0}})$ is a
log twisted curve. Let $(\mathcal{C}_{0},\Sigma_{i})\to S_{0}$ be the
corresponding twisted curve with log map
$(\mathcal{C}_{0},M_{\mathcal{C}_{0}})\to(S_{0},M^{\prime}_{S_{0}})$ and let
$g_{0}:(\mathcal{C}_{0},M_{\mathcal{C}_{0}})\to(X,M_{X})$ be the composition
of $h_{0}$ with the coarse space map $\pi_{0}:\mathcal{C}_{0}\to C$. Let
$(S_{0},l_{0}:M_{S_{0}}\to M^{\prime}_{S_{0}})\hookrightarrow(S,l:M_{S}\to
M^{\prime}_{S})$ be a strict closed immersion of log schemes with simple
extensions defined by a square zero ideal $I\subset\mathcal{O}_{S}$.

######  Proposition 4.6.

With notation as above, suppose we have a log flat deformation

(2)
$\textstyle{(C,M_{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{(X,M_{X})}$$\textstyle{(S,M_{S})}$

of $h_{0}$ over $(S,M_{S})$. Then $h$ lifts uniquely to a log flat deformation

$\textstyle{(\mathcal{C},M_{\mathcal{C}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{g}$$\textstyle{(X,M_{X})}$$\textstyle{(S,M_{S^{\prime}})}$

of $g_{0}$ factoring through $h$.

###### Proof.

We can summarize the situation with the commutative diagram below:

$\textstyle{(\mathcal{C}_{0},M_{\mathcal{C}_{0}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{0}}$$\scriptstyle{g_{0}}$$\textstyle{(C_{0},(\pi_{0})_{*}M_{\mathcal{C}_{0}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathrm{id},s_{0})}$$\textstyle{(C_{0},M_{C_{0}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{0}}$$\textstyle{(X,M_{X})}$$\textstyle{(S_{0},M^{\prime}_{S_{0}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathrm{id},l_{0})}$$\textstyle{(S_{0},M_{S_{0}})}$

where the middle square is a pullback in the category of fs log schemes by
[Ols07, Lemma 4.6, Diagram 4.6.2, and Lemma 4.7]. Given a strict square zero
thickening $(S,l:M_{S}\to M^{\prime}_{S})$ of $(S_{0},l_{0}:M_{S_{0}}\to
M^{\prime}_{S_{0}})$ with ideal $I$ and a log flat deformation $h$ as in
Diagram 2, we can form the pullback

$\textstyle{(C,M^{\prime}_{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(C,M_{C})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{(X,M_{X})}$$\textstyle{(S,M^{\prime}_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(S,M_{S})}$

in the category of fs log schemes to obtain a log flat deformation of
$h_{0}\circ(\mathrm{id},s_{0})$. Then $M_{C}\hookrightarrow M^{\prime}_{C}$ is
a simple extension and by Theorem 3.7, this is equivalent to a log flat
deformation $g$. ∎

##  5\. Smoothing twisted stable maps

Our goal now is to lift the smoothings constructed in Section 2 to the case of
twisted maps. More precisely, suppose we have a genus $0$ twisted stable map
$(f_{0}:\mathcal{C}_{0}\to\mathcal{X},p_{1},\ldots,p_{n})$ such that

  * •

$\mathcal{X}$ is a smooth stacky curve with coarse map $\tau:\mathcal{X}\to
X=\mathbb{P}^{1}$,

  * •

the coarse map $(h_{0}:C_{0}\to X,q_{1},\ldots,q_{n})$ satisfies Conditions
($\ast$) ‣ 1.1 relative to $\infty\in X$, and

  * •

$\tau$ is étale over $\infty$.

The situation can be summarized in the following diagram where
$S_{0}=\mathrm{Spec}k$.

$\textstyle{(\mathcal{C}_{0},p_{1},\ldots,p_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{0}}$$\scriptstyle{\pi_{0}}$$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{(C_{0},q_{1},\ldots,q_{n})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h_{0}}$$\textstyle{X}$$\textstyle{S_{0}}$

By Theorem 2.9, there is a smoothing $h:(C,\sigma_{i})/S\to X$ of $h_{0}$ over
the $S=\mathrm{Spec}R$ the spectrum of a DVR with generic fiber contained in
$\mathcal{M}_{(d_{1},\ldots,d_{n})}$ and we wish to lift this to a smoothing
$f:(\mathcal{C},\Sigma_{i})/S\to\mathcal{X}$ of $f_{0}$ with generic fiber
contained in $\mathcal{M}_{\Gamma}(\mathcal{X})$. That is, we wish to fill in
the dotted arrows in the diagram below.

$\textstyle{(\mathcal{C}_{0},p_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi_{0}}$$\textstyle{(\mathcal{C},\Sigma_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\pi}$$\textstyle{\mathcal{X}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tau}$$\textstyle{(C_{0},q_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(C,\sigma_{i})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{h}$$\textstyle{X}$$\textstyle{S_{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{S}$

The difficulty is that there are global obstructions to lifting from a map to
$X$ to a map to $\mathcal{X}$. For example the $j$-invariant of a map to
$\overline{\mathcal{M}}_{1,1}$ must satisfy that the discriminant is a sum of
a square and a cube which cuts outs a high codimension locus inside the space
of maps to $\mathbb{P}^{1}$. The key observation is that since $\tau$ is
étale, we can lift such a smoothing locally around $\infty$ (Theorem 5.1) to
obtain a partial smoothing of $f_{0}$ construct a further global smoothing
when $\mathcal{X}$ has some extra positivity (Theorem 5.5).

######  Theorem 5.1.

In the situation above, there exists a partial smoothing
$f:(\mathcal{C},\Sigma_{i})/S\to\mathcal{X}$ such that

  * •

$f(\Sigma_{i})=\infty$,

  * •

$\mathcal{C}_{\eta}$ is smooth in a neighborhood of $f_{\eta}^{-1}(\infty)$,
and

  * •

the coarse map on the generic fiber $h_{\eta}$ is ramified to order $d_{i}$ at
$(\sigma_{i})_{\eta}$.

###### Proof.

We wish to use the log deformation theory results of Section 4.2. To do this
we begin by showing that the smoothing of the coarse map
$h:(C,\sigma_{i})/S\to X$ can be endowed with the structure of a log stable
map.

######  Lemma 5.2.

Let $(h_{0}:C_{0}\to{\mathbb{P}}^{1},q_{1},\ldots,q_{n})$ as above and let

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be a smoothing constructed in Theorem 2.9. Denote by $M$ the divisorial log
structure on ${\mathbb{P}}^{1}$ corresponding to the point $\infty$. Then
there exist log structures $M_{C}$ and $M_{S}$ on $C$ and $S$ and a map
$h^{\flat}:h^{*}M\to M_{C}$ such that $h$ is a log stable map over $S$.

###### Proof.

First note that $C\to S$ is log smooth where we equip $S=\mathrm{Spec}R$ with
the standard log structure and $C$ with the divisorial log structure induced
by the union of the central fiber and marked sections. Indeed this follows
since $C\to S$ is a toroidal morphism of the corresponding toroidal
embeddings. Finally, by construction, $h^{*}[\infty]$ is supported on the
toroidal boundary so $h$ is a log map where $h^{\flat}$ encodes the tangency
at toroidal strata. ∎

Suppose $(R,\mathfrak{m})$ is complete and let
$h_{n}:(C_{n},\sigma_{i,n})/S_{n}\to X$ be the compatible system of
deformations over $S_{n}=\mathrm{Spec}R/\mathfrak{m}^{n}$ obtained by
truncating the smoothing over $S$. We will lift these to a compatible system
of deformations of $h_{0}$ in two steps.

First let
$(g_{0},g_{0}^{\flat}):(\mathcal{C}_{0},M_{\mathcal{C}_{0}})\to(X,M)$ be the
composition $h_{0}\circ\pi_{0}$ where we note that $h_{0}$ is a log map by
Lemma 5.2 and $\pi_{0}$ is a log map by the definition of log twisted curves.
By Lemma 4.6 and induction on $n$, the map $h_{n}$ extends to a log flat
deformation $g_{n}:(\mathcal{C}_{n},M_{\mathcal{C}_{n}})\to(X,M_{X})$ over
$(S_{n},M^{\prime}_{S_{n}})$ where $M_{S}\hookrightarrow M_{S}^{\prime}$ a
simple extension and
$(S_{n},M^{\prime}_{S_{n}})\hookrightarrow(S,M_{S}^{\prime})$ is a strict
closed embedding. Moreover, the $g_{i}$ are compatible by construction. In
this way, we get an $R$-point of the formal deformation space
$\mathrm{Spf}R\to\operatorname{Def}_{(g_{0},g_{0}^{\flat})}$.

Next we will use Proposition 4.4 to lift this $R$-point of
$\operatorname{Def}_{(g_{0},g_{0}^{\flat})}$, at least in a neighborhood of
$\infty$, to a smoothing of $f_{0}$. Note that $f_{0}$ lifts to a log map
$(f_{0},f_{0}^{\flat})$ by Proposition 4.3 where we endow $\mathcal{X}$ with
the pullback log structure $\tau^{*}M$. Equivalently this is the divisorial
log structure for $\infty$ but is different from the canonical log structure
on a log twisted curve. Let $U\subset\mathbb{P}^{1}$ be an étale neighborhood
of $\infty$ such that $\tau|_{V}:V\to U$ is étale where $V=\tau^{-1}(U)$ and
such that $U$ avoids all the components of the critical loci of both $f_{0}$
and $g_{0}$ away from $\infty$. Let $\operatorname{Def}_{f_{0},\infty}$ be the
miniversal deformation space of the restriction of $(f_{0},f_{0}^{\flat})$ to
an étale neighborhood of $f_{0}^{-1}(\infty)$ and similarly for
$\operatorname{Def}_{g_{0},\infty}$. Note that by the independence of the
chosen étale neighborhood (see Remark 4.5), we have that these are the
miniversal deformation spaces of $(f_{0})|_{U}:f_{0}^{-1}(U)\to
U\subset\mathbb{P}^{1}$ and $(g_{0})|_{V}:g_{0}^{-1}(V)\to
V\subset\mathcal{X}$. We will denote by
$U^{\prime}=f_{0}^{-1}(U)=g_{0}^{-1}(V)$, and by $U^{\prime\prime}$ a
neighborhood of the critical loci of both $f_{0}$ and $g_{0}$ away from
$\infty$.

By Proposition 4.4, we have

$\operatorname{Def}_{(f_{0},f_{0}^{\flat})}\cong\operatorname{Def}_{f_{0},\infty}\times\operatorname{Def}_{f_{0},\neq\infty}$

and

$\operatorname{Def}_{(g_{0},g_{0}^{\flat})}\cong\operatorname{Def}_{g_{0},\infty}\times\operatorname{Def}_{g_{0},\neq\infty}$

where $\operatorname{Def}_{f_{0},\neq\infty}$ (resp.
$\operatorname{Def}_{g_{0},\neq\infty}$) is the miniversal deformation space
of a neighborhood of the critical loci of $(f_{0},f_{0}^{\flat})$ (resp.
$(g_{0}.g_{0}^{\flat})$) which are away from $\infty$.

######  Lemma 5.3.

The natural map
$\operatorname{Def}_{f_{0},\infty}\to\operatorname{Def}_{g_{0},\infty}$
induced by composition with $\tau$ is an isomorphism.

###### Proof.

We proceed as in Proposition 4.4 and see that there is a natural isomorphism

$R\operatorname{Hom}(\mathbb{L}^{\mathrm{log}}_{f_{0}},\mathcal{O}_{\mathcal{C}})=R\operatorname{Hom}(\mathbb{L}^{\mathrm{log}}_{f_{0}^{\prime}},\mathcal{O}_{U^{\prime}})\oplus
R\operatorname{Hom}(\mathbb{L}^{\mathrm{log}}_{f_{0}^{\prime\prime}},\mathcal{O}_{U^{\prime\prime}})$

and similarly

$R\operatorname{Hom}(\mathbb{L}^{\mathrm{log}}_{g_{0}},\mathcal{O}_{\mathcal{C}})=R\operatorname{Hom}(\mathbb{L}^{\mathrm{log}}_{g_{0}^{\prime}},\mathcal{O}_{U^{\prime}})\oplus
R\operatorname{Hom}(\mathbb{L}^{\mathrm{log}}_{g_{0}^{\prime\prime}},\mathcal{O}_{U^{\prime\prime}}).$

We now compute

$\mathbb{L}^{\mathrm{log}}_{f_{0}^{\prime}}=[f_{0}^{*}\Omega_{U}\to\Omega^{log}_{U^{\prime}}]$
$\mathbb{L}^{\mathrm{log}}_{g_{0}^{\prime}}=[g_{0}^{*}\Omega_{V}\to\Omega^{log}_{U^{\prime}}].$

Since $\tau|_{V}$ is étale, we have that $g_{0}^{*}\Omega_{V}\cong
g_{0}^{*}\tau^{*}\Omega_{U}=f_{0}^{*}\Omega_{U}$ by functorality of pullbacks.
Therefore the natural map

$\mathbb{L}^{\mathrm{log}}_{g_{0}^{\prime}}\to\mathbb{L}^{\mathrm{log}}_{f_{0}^{\prime}}$

is an isomorphism so we conclude that the natural map

$R\operatorname{Hom}(\mathbb{L}^{\mathrm{log}}_{g_{0}^{\prime}},\mathcal{O}_{U^{\prime}})\to
R\operatorname{Hom}(\mathbb{L}^{\mathrm{log}}_{f_{0}^{\prime}},\mathcal{O}_{U^{\prime}})$

is an isomorphism. ∎

Now let $\mathrm{Spf}R\to\operatorname{Def}_{f_{0},\infty}$ be the composition
of the formal $R$-point of $\operatorname{Def}_{(g_{0},g_{0}^{\flat})}$
constructed above with composition

$\operatorname{Def}_{(g_{0},g_{0}^{\flat})}\to\operatorname{Def}_{g_{0},\infty}\to\operatorname{Def}_{f_{0},\infty}$

where the first map is the projection from Proposition 4.4 and the second map
is the inverse of the isomorphism in Lemma 5.3. There is a canonical splitting
of the projection
$\operatorname{Def}_{(f_{0},f_{0}^{\flat})}\to\operatorname{Def}_{f_{0},\infty}$
via the isomorphism of Proposition 4.4 given by picking the constant
deformation in $\operatorname{Def}_{f_{0},\neq\infty}$. Further composing with
this splitting gives us a formal deformation
$\mathrm{Spf}R\to\operatorname{Def}_{(f_{0},f_{0}^{\flat})}$. Since
$\operatorname{Def}_{(f_{0},f_{0}^{\flat})}$ is the formal neighborhood of a
point of a locally of finite type algebraic stack, this formal deformation
algebraizes to a family
$(f,f^{\flat}):(\mathcal{C},M_{\mathcal{C}})/(S,M_{S}^{\prime})\to(\mathcal{X},\tau^{*}M)$
of log twisted stable maps (see for example [Bha16, Theorem 1.1] and [BHL17,
Corollary 1.5]). Let us denote by $f:(\mathcal{C},\Sigma_{i})/S\to\mathcal{X}$
the underlying twisted stable map. By construction, the coarse map of $f$
agrees with $h$ over an étale neighborhood of $\infty$ so we conclude $f$
satisfies the required conditions. ∎

######  Remark 5.4.

By construction, the smoothing of Theorem 5.1 induces the constant
infinitesimal deformation of the restriction of $f_{0}$ away from $\infty$.
More precisely, suppose $A$ is a component of the critical locus of $f_{0}$
which is disjoint from $f_{0}^{-1}(\infty)$ and let $U\to C$ be an étale
neighborhood of $A$ which is disjoint from all other critical loci. Then
$f|_{U}\mod\mathfrak{m}^{n}$ is the constant infinitesimal deformation of
$f_{0}|_{U\times_{C}C_{0}}$.

Next we consider the problem of smoothing twisted stable maps to a weighted
projective line $\mathcal{P}=\mathcal{P}(a,b)$ without any tangency
conditions.

######  Theorem 5.5.

Let $(f_{0}:\mathcal{C}_{0}\to\mathcal{P},\Sigma_{i,0})$ be a genus $0$
twisted stable map to a weighted projective line
$\mathcal{P}=\mathcal{P}(a,b)$. Then there exists a smoothing

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over $S=\mathrm{Spec}R$ the spectrum of a DVR.

###### Proof.

Suppose $(R,\mathfrak{m})$ is complete. We will construct such a smoothing by
building a compatible system of deformations over $R/\mathfrak{m}^{n}$. Since
the stack of not necessarily representable twisted stable maps to
$\mathcal{P}$ is a locally of finite type algebraic stack, such compatible
system automatically algebraizes as in the proof of Theorem 5.1. First note
that if $a=ka^{\prime}$ and $b=kb^{\prime}$ with $a^{\prime},b^{\prime}$
coprime, then the canonical map
$\rho:\mathcal{P}(a,b)\to\mathcal{P}(a^{\prime},b^{\prime})$ is étale and so
infinitesimal deformations lift uniquely along $\rho$. Thus without loss of
generality we may assume that $a$ and $b$ are coprime and $\mathcal{P}$ is a
twisted curve.

First we reduce to the case where every connected component in the fibers of
$f_{0}:\mathcal{C}_{0}\to\mathcal{P}$ is irreducible. We can argue as in Lemma
2.4. If $E$ is a connected component of the preimage $f_{0}^{-1}(p)$ for some
$p$, then we can view $f_{0}|_{E}$ as a map to the residual gerbe
$\mathcal{G}_{p}$ where $E$ is pointed by the nodes. Then since
$\mathcal{G}_{p}$ is étale over $\mathrm{Spec}k$, it suffices to smooth the
composition $E\to\mathrm{Spec}k$ as a marked twisted curve and then glue
together the smoothing of $E$ with constant family of maps
$f|_{\overline{C_{0}\setminus E}}$ along the markings corresponding to nodes
exactly as in the proof of Lemma 2.4 to obtain a partial smoothing where each
connected component of the contracted locus is irreducible. Therefore, without
loss of generality we may suppose that every connected component of the
contracted locus of $f_{0}$ is irreducible.

By Propositions 4.1 and 4.3, there exists a twisted curve
$\tilde{\mathcal{C}}_{0}$ with a partial coarse space map
$\pi_{0}:\tilde{\mathcal{C}}_{0}\to\mathcal{C}_{0}$ such that
$g_{0}:=f_{0}\circ\pi_{0}:\tilde{\mathcal{C}}_{0}\to\mathcal{P}$ can be
endowed with a log structure where $\mathcal{P}$ has the toric log structure
and
$(\tilde{\mathcal{C}}_{0},M_{\tilde{\mathcal{C}}_{0}})\to(\mathrm{Spec}k,M)$
is log smooth. Since $(\mathcal{P},M_{\mathcal{P}})$ is toric, the log
cotangent complex
$\mathbb{L}^{\mathrm{log}}_{\mathcal{P}}\cong\mathcal{O}_{\mathcal{P}}$ is
trivial. Moreover, since $\tilde{\mathcal{C}}_{0}$ is genus $0$,

(3)
$\mathrm{Ext}^{1}(g_{0}^{*}\mathbb{L}^{\mathrm{log}}_{\mathcal{P}},\mathcal{O}_{\tilde{\mathcal{C}}_{0}})=H^{1}(\tilde{\mathcal{C}}_{0},\mathcal{O}_{\tilde{\mathcal{C}}_{0}})=0.$

Therefore, the log deformation space of $(g_{0},g_{0}^{\flat})$ is log smooth
over the log deformation space of the curve
$(\tilde{\mathcal{C}}_{0},M_{\tilde{\mathcal{C}}_{0}})$. By Theorem 3.7,
$\operatorname{Def}(\tilde{\mathcal{C}}_{0},M_{\tilde{\mathcal{C}}_{0}})$ is
root stack over $\operatorname{Def}(C_{0},p_{1},\ldots,p_{n})$ where $p_{i}$
are the smooth marked lying under $\Sigma_{i,0}$. In particular we can build a
smoothing $(C/S,\sigma_{i})$ of $(C_{0},p_{i})$ whose truncations
$(C_{n}/\mathrm{Spec}R/\mathfrak{m}^{n},\sigma_{i,n})$ lift to a formal
smoothing of $(\tilde{\mathcal{C}}_{0},M_{\tilde{\mathcal{C}}_{0}})$, after
possibly extending the log structure on $S$. By the vanishing obstruction
group (3), the map $(g_{0},g_{0}^{\flat})$ can then be lifted to a compatible
family $(g_{n}:\tilde{C}_{n}\to\mathcal{P},g_{n}^{\flat})$ of log flat
deformations over $R/\mathfrak{m}^{n}$.

Thus, after algebraizing, we have a smoothing
$(g:\tilde{\mathcal{C}}/S\to\mathcal{P},\Sigma_{i})$ of $g_{0}$. Taking the
relative coarse space of $g$ gives a flat family of maps
$f:\mathcal{C}/\mathrm{Spec}R\to\mathcal{P}$ whose formation commutes with
base change (see [AOV11, Proposition 3.4] and [AOV08, Corollary 3.3]).
Therefore we conclude that the central fiber of $f$ agrees with $f_{0}$ and
that the generic fiber is smooth since it is a partial coarse space of the
smooth generic fiber of $\tilde{C}/S$. ∎

######  Remark 5.6.

For the proof of Theorem 5.5, it is necessary to use log deformation theory
even though the statement does not make any reference to log geometry. The
reason is that the stack $\mathcal{P}$ is _not_ convex in the classical sense.
The tangent bundle is not necessarily effective even though it has positive
degree and there are examples of genus $0$ twisted stable maps where the
vanishing (3) does not hold for the usual tangent bundle.

##  6\. Proofs of the main theorems

We use the notation from the introduction. Let
$(\mathcal{X},x_{1},\ldots,x_{r})$ be a smooth and proper pointed stacky curve
and suppose that points $x_{i}$ have generic stabilizers. Fix
$\Gamma=(n_{0},\\{\Gamma_{1},x_{1}\\},\ldots,\\{\Gamma_{r},x_{r}\\})$ a tuple
of discrete data.

######  Proposition 6.1.

The loci $\mathcal{M}_{\Gamma}(\mathcal{X})$ and
$\mathcal{N}_{\Gamma}(\mathcal{X})$ (Definition 1.3) are locally closed
substacks of $\mathcal{K}_{0,n}(\mathcal{X},d)$.

###### Proof.

We run through the conditions of Definition 1.3. Condition (1) is both open
and closed by definition of a twisted curve and condition (2) is closed.
Conditions (4) and ($4\prime$) are open by openness of the smooth locus. For
condition (3), we consider the cohomology sheaves of the cotangent complex
$\mathbb{L}^{\mathrm{log}}_{f}$. The locus where $f$ is constant is given by
the support of $\mathcal{H}^{-1}(\mathbb{L}^{\mathrm{log}}_{f})$. Since the
universal family of curves over the moduli space is proper, the locus where
the marked points avoid
$\mathrm{Supp}\mathcal{H}^{-1}(\mathbb{L}^{\mathrm{log}}_{f})$ and so the
condition of $f$ being non-constant along the marked points is open. Finally,
under this condition, the order of tangency of $f$ at $x_{i}$ is measured by
the length of
$\mathrm{coker}(f^{*}\Omega^{1}_{\mathcal{X}}\to\Omega^{1}_{\mathcal{C}})_{x}$
and by semicontinuity, there is a locally closed subset where this length is
constant $d_{i}$. ∎

###### Proof of Theorem 1.5.

The marked points $p_{0k}$ for $k=1,\ldots,n_{0}$ with no tangency conditions
do not change the outcome of the problem. Indeed if we can construct such a
smoothing with $n_{0}>0$, then by forgetting and stabilizing we obtain a
smoothing with $n_{0}=0$, and conversely, if we have a partial 1-parameter
smoothing over a base $S$ with $n_{0}=0$, then up to a finite base
$S^{\prime}\to S$, we can pick generic sections passing through $p_{0k}$ to
obtain a smoothing with $n_{0}>0$. So without loss of generality, suppose
$n_{0}=0$.

We proceed by induction on $r$. The base case is precisely Theorem 5.1. In
general, let
$(f:\mathcal{C}\to\mathcal{X},\\{\\{p_{jk}\\}_{k=1}^{n_{j}}\\}_{j=1}^{r})$ be
a map in $\mathcal{K}_{\Gamma}(\mathcal{X})$. Applying Theorem 5.1 to
$(f:\mathcal{C}\to\mathcal{X},p_{r1},\ldots,p_{rn_{r}})$ produces a
1-parameter deformation whose generic fiber is a relative map to
$(\mathcal{X},x_{r})$ with tangency $\Gamma_{r}$ which also satisfies that the
curve is smooth along the preimage of $x_{r}$. Moreover, by Remark 5.4, the
generic fiber is contained in $\mathcal{K}_{\Gamma}(\mathcal{X})$. Therefore,
without loss of generality, we may assume that $\mathcal{C}$ is smooth in a
neighborhood of $f^{-1}(x_{r})$. By inductive hypothesis, there exists a
1-parameter deformation of $f$ with generic fiber contained in
$\mathcal{N}_{\Gamma^{\prime}}(\mathcal{X})$ where
$\Gamma^{\prime}=(\\{\Gamma_{1},x_{1}\\},\ldots,\\{\Gamma_{r-1},x_{r-1}\\})$.
Moreover, by Remark 5.4, this partial smoothing induces the constant
deformation in an étale neighborhood of $f^{-1}(x_{r})$ and so the generic
fiber is contained in $\mathcal{N}_{\Gamma}(\mathcal{X})$. ∎

###### Proof of Theorem 1.7.

Let $(f:\mathcal{C}\to\mathcal{X},p_{jk})$ be a map contained in
$\mathcal{K}_{\Gamma}(\mathcal{X})$. By Theorem 1.5, there exists a partial
smoothing of $(f:\mathcal{C}\to\mathcal{X},p_{jk})$ with generic fiber
contained in $\mathcal{N}_{\Gamma}(\mathcal{X})$ so without loss of
generality, suppose that $(f:\mathcal{C}\to\mathcal{X},p_{jk})$ is contained
in $\mathcal{N}_{\Gamma}(\mathcal{X})$. By Theorem 5.5, there exists a
smoothing of the stable map into the interior of
$\mathcal{K}_{0,n}(\mathcal{X},d)$ since $\mathcal{X}=\mathcal{P}$ is a
weighted projective line. Let $\mathrm{Spf}R\to\operatorname{Def}_{f}$ be the
corresponding formal deformation. By Proposition 4.4, we can write

$\operatorname{Def}_{f}=\operatorname{Def}_{f,x_{1}}\times\ldots\operatorname{Def}_{f,x_{r}}\times\operatorname{Def}_{f,\neq}$

where $\operatorname{Def}_{f,x_{i}}$ is a miniversal deformation space of a
small étale neighborhood of the fiber $f^{-1}(x_{i})$ and
$\operatorname{Def}_{f,\neq}$ is a miniversal deformation space of the
critical loci of $f$ which are disjoint from the fibers $f^{-1}(x_{i})$ for
each $i$. Then projecting onto $\operatorname{Def}_{f,\neq}$ and then
composing with the section
$\operatorname{Def}_{f,\neq}\to\operatorname{Def}_{f}$ which picks the
constant deformation on each of the $\operatorname{Def}_{f,x_{i}}$ factors
produces a new formal deformation $\mathrm{Spf}R\to\operatorname{Def}_{f}$
which agrees with the smoothing from Theorem 5.5 away from the the fibers
$f^{-1}(x_{i})$ but is the constant deformation in a neighborhood of
$f^{-1}(x_{i})$ for all $i$. After algebraizing this formal deformation as in
the proof of Theorem 5.1, we obtain a 1-parameter family over $\mathrm{Spec}R$
such that the generic fiber is contained in
$\mathcal{N}_{\Gamma}(\mathcal{X})$ but is also smooth away from the union of
fibers $f^{-1}(x_{i})$. Therefore, the generic fiber is smooth everywhere and
thus contained in $\mathcal{M}_{\Gamma}(\mathcal{X})$. ∎

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