# Gluing for stable families of surfaces in mixed characteristic

Quentin Posva  École Polytechnique Fédérale de Lausanne, SB MATH CAG, MA C3
595 (Bâtiment MA), Station 8, CH-1015 Lausanne, Switzerland
<EMAIL_ADDRESS>
> Abstract. Motivated by the question of properness of the moduli space of
> stable surfaces in mixed characteristic, we study the gluing problem for
> stable families of lc surfaces over a mixed characteristic DVR. We prove
> that the quotient exists, and give sufficient conditions for the the fibers
> of the quotient to be the quotients of the fibers.

###### Contents

  1. 1 Introduction
    1. 1.1 Acknowledgements
  2. 2 Preliminaries
    1. 2.1 Conventions and notations
    2. 2.2 Quotients by finite equivalence relations
    3. 2.3 Demi-normal varieties and slc pairs
  3. 3 Gluing for families of surfaces
    1. 3.1 Existence of the quotient
    2. 3.2 Descent of the log canonical sheaf
  4. 4 Fibers of the quotient
    1. 4.1 Commutativity of fibers and quotients
    2. 4.2 Serre properties of the fibers

##  1\. Introduction

Semi-log canonical (slc) singularities play a central role in the moduli
theory of canonically polarized varieties in characteristic zero: they appear
at the boundary of the compact moduli space of stable varieties [Kol21]. It is
expected that slc varieties play a similar role in positive and mixed
characteristics.

Recent breakthroughs in the Minimal Model Program (MMP) for threefolds in
mixed characteristic [BMP+21, TY21] have led to advances in the moduli theory
of stable surfaces in mixed characteristic. The corresponding moduli stack is
known to be a separated Artin stack with finite diagonal and of finite type
[BMP+21, Theorem I]. Properness is not known yet, except for some specific
subspaces [BMP+21, Theorem J].

In characteristic zero, properness of the moduli stack of stable varieties is
proved through a delicate use of semi-stable reductions and MMP methods: see
[Kol21, §2.4] for a presentation of the proof. In addition to semi-stable
reduction and MMP, that proof needs a technical gluing statement. Indeed, we
consider a pointed curve $(t\in T)$ and a stable family $X^{0}\to
T^{0}=T\setminus\\{t\\}$, and we need to complete the family over $T$,
possibly after a finite base-change $T^{\prime}\to T$. The variety $X^{0}$ is
only demi-normal in general, and the MMP might fail for demi-normal varieties
[Fuj14, Example 5.4]. Thus we normalize $X^{0}$ and try to complete the family
$(X^{0})^{n}\to T^{0}$. If we succeed, we need to de-normalize the completed
family. In characteristic zero, this is achieved through Kollár’s gluing
theory [Kol13, §5], that gives a dictionnary between slc stable varieties and
their normalizations.

In [Pos21] I have extended this dictionnary to surfaces and threefolds in
positive characteristic, and given applications to the properness of the
moduli space of stable surfaces in positive characteristic (see also [Pat17]
for related results about this moduli space). In this paper, I study the
gluing statement necessary for the proof of properness of the moduli stack of
stable surfaces in mixed characteristic along the lines sketched above. The
tailor-made statement is our main result:

######  Theorem 1 (see Section 3.1 and Section 3.2).

Let $R$ be a DVR of mixed characteristic with maximal ideal $\pi R$. Then
normalization gives a bijection

$\begin{pmatrix}\text{Threefold pairs }(X,\Delta)\\\ \text{flat and proper
over }R\\\ \text{such that }(X,\Delta+X_{\pi})\text{ is slc}\\\ \text{and
}K_{X}+\Delta\text{ is
ample}\end{pmatrix}\overset{1:1}{\longrightarrow}\begin{pmatrix}\text{Threefold
pairs }(\bar{X},\bar{D}+\bar{\Delta})\\\ \text{flat and proper over }R\\\
\text{such that }(\bar{X},\bar{D}+\bar{\Delta}+\bar{X}_{\pi})\text{ is lc}\\\
\text{and }K_{\bar{X}}+\bar{D}+\bar{\Delta}\text{ is ample}\\\ \text{plus a
generically fixed point free}\\\ R\text{-involution }\tau\text{ of
}(\bar{D}^{n},\operatorname{Diff}_{\bar{D}^{n}}\bar{\Delta}).\end{pmatrix}$

Let us comment on the condition that $(X,\Delta+X_{\pi})$ is slc. Since we
think of the family of surfaces $(X,\Delta)\to\operatorname{{Spec}}R$ as an
element in moduli stack of stable surfaces, it would be natural to ask for
each fiber to be slc. In characteristic zero, this is equivalent to
$(X,\Delta+X_{\pi})$ being log canonical (lc) [Kol21, Theorem 2.4]. The local-
to-global direction relies on inversion of adjunction, which is known in
dimension three in positive and mixed characteristics provided that the non-
zero characteristics are $>5$ ([Pat17, Lemma 3.3] and [BMP+21, Corollary
10.1]). On the other hand, there is a difficult step in the proof of the
global-to-local direction, namely that $(X,\Delta+X_{\pi})$ being slc implies
that $X_{\pi}$ is $S_{2}$: in characteristic zero this follows from a theorem
of Alexeev [Kol13, 7.21]. A similar result is not yet proved in positive and
mixed characteristic, so the slc condition on $(X,\Delta+X_{\pi})$ is a priori
weaker.

We prove our gluing statement for this weaker stability assumption. We also
prove in Section 4.2 that, under the assumption
$\operatorname{char}R/(\pi)\geq 7$, if
$(\bar{X}_{\pi},\operatorname{Diff}_{\bar{X}_{\pi}}(\bar{D}+\bar{\Delta}))$ is
slc then the pair $(X_{\pi},\operatorname{Diff}_{X_{\pi}}\Delta)$ is also slc.
However the converse statement seems more difficult without an Alexeev-type
statement (see Section 4.2). Thus we cannot yet formulate an analog of Theorem
1 for the stronger stability assumption.

Let us say a word about the proof of Theorem 1. We construct an inverse map to
the normalization process: $\bar{X}$ should be the normalization of an slc
quotient $\bar{X}/R(\tau)$. It turns out that the main difficulty is the
existence of the quotient, for then it is easy to show that it has an slc
structure, see Section 3.2.

To show that the quotient exists, thanks to a result of Witaszek [Wit20,
Theorem 1.4] and to Kollár’s theory [Kol13, 5.13], we only need to show that
the equivalence relations induced by $\tau$ on the special fiber $X_{\pi}$ is
finite. To achieve this, we relate $\bar{D}^{n}_{\pi}$ to a divisor contained
in the round-down of
$\operatorname{Diff}_{X_{\pi}^{n}}(\bar{D}+\bar{\Delta})$, and show that the
relation on $\bar{X}_{\pi}$ comes from a log involution on the reduced
boundary of the $k(\pi)$-pair
$(\bar{X}_{\pi}^{n},\operatorname{Diff}_{X_{\pi}^{n}}(\bar{D}+\bar{\Delta})$.
This is done using adjunction and classification of codimension two lc
singularities. Then we are in position to apply the results from [Pos21] and
obtain finiteness. We also obtain a conditional finiteness result in case
$\dim\bar{X}=4$: see Section 3.1 for the precise statement.

We emphasize that all our results are independent of $\dim\bar{X}$ and of the
properties of the positive characteristic residue field $k(\pi)$ — except for
Section 3.1, which shows that the quotient indeed exists in several
situations.

In the second part of the paper, we study the fibers of $X=\bar{X}/R(\tau)$
over $\operatorname{{Spec}}R$. The involution $\tau$ induces an involution on
both fibers of $\bar{D}^{n}\to\operatorname{{Spec}}R$, and therefore we can
take the quotient of both fibers of $\bar{X}\to\operatorname{{Spec}}R$. We ask
whether these quotients are equal to the fibers of $X$. This commutativity
property is always true for the generic fiber (Section 4.1). For the special
fiber, we first study the general case of the normalization of a demi-normal
scheme over $\operatorname{{Spec}}R$ and give some necessary and sufficient
conditions (see Section 4.1 and Section 4.1). For the normalization of an slc
pair, we deduce the following simple condition:

######  Theorem 2 (Section 4.1).

Suppose $(X,\Delta+X_{\pi})$ is slc and that $\operatorname{char}R/(\pi)>2$.
If $\bar{D}_{\pi}$ is reduced, the special fiber of $X$ is the quotient of the
special fiber of $\bar{X}$.

We remark that $\bar{D}_{\pi}$ is the scheme-theoretic intersection of two lc
centers of $(\bar{X},\bar{D}+\bar{\Delta}+\bar{X}_{\pi})$. It is a question of
independent interest whether intersections of lc centers are reduced. This is
true in characteristic zero [Kol13, 7.8], but not much is known in mixed (or
positive) characteristics. Nonetheless, we can prove that the hypothesis of
Theorem 2 holds in relative dimension two and residue characteristic large
enough:

######  Theorem 3 (Section 4.1).

Let $(\bar{X},\bar{D}+\bar{\Delta}+\bar{X}_{\pi})$ be a lc threefold pair that
is projective over $\operatorname{{Spec}}R$. Assume that
$\operatorname{char}R/(\pi)\geq 7$. Then $\bar{D}_{\pi}$ is reduced.

The proof relies on the technique of [Kol13, 7.8] and on the results of
[BK21]. We record the following consequence of Theorem 2 and Theorem 3 for
families of surfaces:

######  Theorem 4.

Let $\mathcal{S}\to\operatorname{{Spec}}R$ be a family of surfaces over a DVR
with residue characteristic $\geq 7$, with normalization $\bar{\mathcal{S}}$.
Assume that $(\mathcal{S},\Delta+\mathcal{S}_{\pi})$ is slc. Then the fibers
of $\mathcal{S}$ are the quotients of the fibers of $\bar{\mathcal{S}}$.

A corollary of Theorem 4 is the fact, already mentioned above, that
$(\mathcal{S}_{\pi},\operatorname{Diff}_{\mathcal{S}_{\pi}}\Delta)$ is slc if
$(\bar{\mathcal{S}}_{\pi},\operatorname{Diff}_{\bar{\mathcal{S}}_{\pi}}(\bar{D}+\bar{\Delta}))$
is slc. The main difficulty is to show that $\mathcal{S}_{\pi}$ is $S_{2}$,
which follows here from the commutativity of fibers and quotients. We take
this problem as an opportunity to study the Serre properties of a demi-normal
quotient. In view of [Kol13, 10.18], it is not surprising that there is an
interplay between the Serre properties of the quotient, and those of its
normalization and of the conductor subschemes. The precise statement is given
in Section 4.2.

###  1.1. Acknowledgements

Financial support was provided by the European Research Council under grant
number 804334.

##  2\. Preliminaries

###  2.1. Conventions and notations

We will work with schemes that are not necessarily defined over a field, but
we keep most of the usual terminology.

A _variety_ is a connected separated reduced equidimensional excellent
Noetherian scheme. Note that a variety in our sense might be reducible. A
_curve_ (resp. a _surface_ , resp. a _threefold_) is a variety of dimension
one (resp. two, resp. three).

Let $X$ be a variety and $\mathcal{F}$ a coherent $\mathcal{O}_{X}$-module. We
let
$\mathcal{F}^{*}:={\mathchanc{H}\mathchanc{o}\mathchanc{m}}_{X}(\mathcal{F},\mathcal{O}_{X})$
be the dual of $\mathcal{F}$ and write
$\mathcal{F}^{[m]}:=\left(\mathcal{F}^{\otimes m}\right)^{**}$. We say that
$\mathcal{F}$ is $S_{i}$ if
$\operatorname{{depth}}_{\mathcal{O}_{X,x}}\mathcal{F}_{x}\geq\min\\{i,\dim\mathcal{F}_{x}\\}$
for every $x\in X$.

If $X$ is a variety, we define $\mathbb{Q}$-divisors and $\mathbb{Q}$-Cartier
divisors the usual way. We usually use these notions when $X$ is
(demi-)normal.

If $R$ is a ring that is the quotient of a regular one, and if $X$ is a proper
scheme over $\operatorname{{Spec}}R$, then $X$ has a dualizing complex
$\omega_{X}^{\bullet}$ and a dualizing sheaf
$\omega_{X}:=h^{-i}\left(\omega_{X}^{\bullet}\right)$ where $i:=\max\\{j\mid
h^{-j}\left(\omega_{X}^{\bullet}\right)\neq 0\\}$. See [BMP+21, §2.1] for more
details. If $X$ is a variety and $\omega_{X}$ is invertible in codimension
one, it defines a canonical divisor $K_{X}$.

A pair is a data $(X,\Delta)$ where $X$ is a demi-normal variety, $\Delta$ a
$\mathbb{Q}$-divisor with coefficients in $[0,1]$ without component in common
with $\operatorname{{Sing}}(X)$, such that $K_{X}+\Delta$ is
$\mathbb{Q}$-Cartier. If $f\colon Y\to X$ is a proper birational morphism from
a normal variety, then there is a $\mathbb{Q}$-divisor $\Delta_{Y}$ on $Y$
such that

$K_{Y}+\Delta_{Y}=f^{*}(K_{X}+\Delta)$

and $\Delta_{Y}$ is uniquely defined if we assume $f_{*}K_{Y}=K_{X}$, which we
always do. Running through all such $f\colon Y\to X$, we define the
singularity of the pair $(X,\Delta)$ (e.g. lc, klt) the usual way as in
[KM98]. See [BMP+21, §2.5] for details.

Let $R$ be a discrete valuation ring (DVR) with maximal ideal $\pi R$. If $X$
is an $R$-scheme, we denote by $X_{\pi}$ the scheme-theoretic special fiber.
We usually assume that $X$ is flat over $\operatorname{{Spec}}R$: if in
addition $X$ is pure-dimensional, then every irreducible component of
$X_{\pi}$ has dimension $\dim X-1$ by [GW20, 14.97].

We will use many times the coarse classification of codimension two lc
singularities [Kol13, 2.31]: if $(X,\Delta)$ is an lc pair and $x\in X$ a
codimension two point, then $\lfloor\Delta\rfloor$ is either regular or nodal
at $x$. Notice that the proof given in [Kol13, 2.31] holds in the generality
of normal excellent Noetherian schemes.

Let $(X,D+\Delta)$ be a pair where $D$ is a reduced divisor. Then adjunction
theory produces canonically a pair
$(D^{n},\operatorname{Diff}_{D^{n}}\Delta)$: see [Kol13, §4.1]. In some cases
adjunction already produces a different divisor on $D$, which will be
important for us at a few places. See [Kol13, 4.2] for minimalistic
assumptions necessary to perform adjunction.

###  2.2. Quotients by finite equivalence relations

The theory of quotients by finite equivalence relations is developped in
[Kol12] and [Kol13, §9]. For convenience, we recall the basic definitions and
constructions that we will need.

Let $S$ be a base scheme, and $X,R$ two reduced $S$-schemes. An $S$-morphism
$\sigma=(\sigma_{1},\sigma_{2})\colon R\to X\times_{S}X$ is a set theoretic
equivalence relation (over $S$) if, for every geometric point
$\operatorname{{Spec}}K\to S$, the induced map

$\sigma(\operatorname{{Spec}}K)\colon\operatorname{Hom}_{S}(\operatorname{{Spec}}K,R)\to\operatorname{Hom}_{S}(\operatorname{{Spec}}K,X)\times\operatorname{Hom}_{S}(\operatorname{{Spec}}K,X)$

is injective and an equivalence relation of the set
$\operatorname{Hom}_{S}(\operatorname{{Spec}}K,X)$. We say in addition that
$\sigma\colon R\to X\times_{S}X$ is finite if both $\sigma_{i}\colon R\to X$
are finite morphisms.

Suppose that $\sigma\colon R\hookrightarrow X\times_{S}X$ is a reduced closed
subscheme. Then there is a minimal set theoretic equivalence relation
generated by $R$: see [Kol13, 9.3]. Even if both $\sigma_{i}\colon R\to X$ are
finite morphisms, the resulting pro-finite relation may not be finite:
achieving transitivity can create infinite equivalence classes.

Let $f\colon X^{\prime}\to X$ be a finite $S$-morphism, and $\tau$ an
involution of $X^{\prime}$. We denote by $R_{X}(\tau)\rightrightarrows X$ the
set theoretic equivalence relation induced by the closed subscheme $(f\times
f)\left(\Delta_{X^{\prime}/S}\cup\Gamma_{\tau}\right)$ of $X\times_{S}X$,
where $\Delta_{X^{\prime}/S}\subset X^{\prime}\times_{S}X^{\prime}$ is the
diagonal and $\Gamma_{\tau}\subset X^{\prime}\times_{S}X^{\prime}$ is the
graph of $\tau$. If there is no risk of confusion, we drop the subscript and
simply write $R(\tau)$.

Let $(\sigma_{1},\sigma_{2})\colon R\to X\times_{S}X$ be a finite set
theoretic equivalence relation. A geometric quotient of this relation is an
$S$-morphism $q\colon X\to Y$ such that

  1. (a)

$q\circ\sigma_{1}=q\circ\sigma_{2}$,

  2. (b)

$(Y,q\colon X\to Y)$ is initial in the category of algebraic spaces for the
property above, and

  3. (c)

$q$ is finite.

We note that if the quotient $Y$ exists, then it is local for the flat
topology of $Y$ [Kol13, 9.11].

The most important result for us is the following theorem of Witaszek [Wit20,
Theorem 1.4]. Assume $X$ is separated of finite type over an excellent scheme
$S$ (in our applications, $S$ will be the spectrum of a DVR), and $R\to
X\times_{S}X$ a finite set-theoretic equivalence relation over $S$. If the
geometric quotient $X_{\mathbb{Q}}/R_{\mathbb{Q}}$ exists, then the geometric
quotient $X/R$ also exists (as an algebraic space) and is of finite type over
$S$. If $X$ is projective over $S$, then $X$ has the Chevalley–Kleiman
property, so $X/R$ is a scheme by [Kol12, Corollary 48].

######  Lemma 2.2.1.

Let $X$ be a reduced equidimensional scheme that is separated of finite type
over a field of positive characteristic, and $D\subset X$ a reduced scheme of
pure codimension one. Let $\tau$ be an involution of $D^{n}$, such that
$R(\tau)\rightrightarrows X$ is a finite set theoretic equivalence relation,
and let $p\colon X\to Y:=X/R(\tau)$ be the quotient.

If $X$ is $S_{2}$, then so is $Y$.

###### Proof.

The argument is contained in the proof of [Pos21, 3.4.1], although it is not
phrased in our generality. Since $X$ is reduced, the morphism $p\colon X\to Y$
factors through $Y_{\text{red}}$. It follows by the universal property of $p$
that $Y_{\text{red}}\to Y$ is an isomorphism, so $Y$ is reduced and in
particular $S_{1}$. Thus by [Har07, 1.8], it is sufficient to show that for
every $U\subset Y$ open and $Z\subset U$ closed of codimension $\geq 2$, the
restriction map $\mathcal{O}_{Y}(U)\to\mathcal{O}_{Y}(U\setminus Z)$ is
surjective. The quotient is Zariski-local, so we may assume that $U=Y$.

Take $s\in\mathcal{O}(U\setminus Z)$. Since $X$ is $S_{2}$ and $p^{-1}(Z)$ has
codimension $\geq 2$, the section $p^{*}s$ extends to a global section over
$X$. By construction $p^{*}s|_{D^{n}}$ is $\tau$-invariant on an open dense
subset, hence it is globally $\tau$-invariant. Thus $p^{*}s$ descends to a
global section of $Y$ [Kol13, 9.10] which is an extension of $s$. ∎

###  2.3. Demi-normal varieties and slc pairs

We give the definitions and the basic properties of demi-normal varieties and
of slc pairs over a DVR.

######  Definition 2.3.1.

A one-dimensional Noetherian local ring $(R,\mathfrak{m})$ is called a node if
there exists a ring isomorphism $R\cong S/(f)$, where $(S,\mathfrak{n})$ is a
regular two-dimensional local ring and $f\in\mathfrak{n}^{2}$ is an element
that is not a square in $\mathfrak{n}^{2}/\mathfrak{n}^{3}$.

A locally Noetherian reduced scheme (or ring) is called nodal if its
codimension one local rings are regular or nodal. It is called demi-normal if
it is $S_{2}$ and nodal.

Let $X$ be a reduced scheme with normalization $\pi\colon\bar{X}\to X$. The
conductor ideal of the normalization is defined as

$\mathfrak{I}:={\mathchanc{H}\mathchanc{o}\mathchanc{m}}_{X}(\pi_{*}\mathcal{O}_{\bar{X}},\mathcal{O}_{X}).$

It is an ideal in both $\mathcal{O}_{X}$ and $\mathcal{O}_{\bar{X}}$. We let

$D:=\operatorname{{Spec}}_{X}\mathcal{O}_{X}/\mathfrak{I},\quad\bar{D}:=\operatorname{{Spec}}_{\bar{X}}\mathcal{O}_{\bar{X}}/\mathfrak{I},$

and call them the conductor subschemes.

######  Lemma 2.3.2.

Notations as above. Assume that $X$ is the spectrum of a local ring that is
essentially of finite type over a DVR. Then:

  1. (a)

$D$ and $\bar{D}$ are reduced of pure codimension $1$.

  2. (b)

If $\eta\in D$ is a generic point such that $\operatorname{char}k(\eta)\neq
2$, the morphism $\bar{D}\to D$ is étale of degree $2$ in a neighborhood of
$\eta$.

  3. (c)

$X$ has a dualizing sheaf which is invertible in codimension one. In
particular, it has a well-defined canonical divisor $K_{X}$.

  4. (d)

Let $\Delta$ be a $\mathbb{Q}$-divisor on $X$ with no component supported on
$D$, and such that $K_{X}+\Delta$ is $\mathbb{Q}$-Cartier. If $\bar{\Delta}$
denotes the divisorial part of $\pi^{-1}(\Delta)$, then there is a canonical
isomorphism

$\pi^{*}\omega_{X}^{[m]}(m\Delta)\cong\omega_{\bar{X}}^{[m]}(m\bar{D}+m\bar{\Delta})$

for $m$ divisible enough.

###### Proof.

We note that $X$ has a dualizing sheaf, since its is the spectrum of a
quotient of a regular ring. The rest is proved as in [Kol13, 5.2, 5.7] (notice
that the adjunction property is proved in [Kol13, 4.2, 5.7] in the generality
we need). ∎

######  Lemma 2.3.3.

Notations as above. Assume that $\operatorname{char}k(\eta)\neq 2$ for every
generic point $\eta$ of $D$. Then:

  1. (a)

the induced morphism of normalizations $\bar{D}^{n}\to D^{n}$ is the geometric
quotient by a Galois involution $\tau$;

  2. (b)

if $K_{X}+\Delta$ is $\mathbb{Q}$-Cartier, then $\tau$ is a log involution of
$(\bar{D}^{n},\operatorname{Diff}_{\bar{D}^{n}}\bar{\Delta})$.

###### Proof.

See [Pos21, 2.2.5]. ∎

######  Definition 2.3.4.

We say that $m(X,\Delta)$ is an semi-log canonical (slc) pair over a DVR if:
$X$ is demi-normal of finite type over a DVR, $\Delta$ is a
$\mathbb{Q}$-divisor with no components along $D$, $K_{X}+\Delta$ is
$\mathbb{Q}$-Cartier, and the normalization $(\bar{X},\bar{D}+\bar{\Delta})$
is an lc pair.

##  3\. Gluing for families of surfaces

The goal of this section is the proof of Theorem 1. We consider the following
situation (where, compared to Theorem 1, we modify slightly our notations):

######  Notation 3.0.1.

Let $R$ be a DVR of mixed characteristic $(0,p>0)$, let
$X\to\operatorname{{Spec}}R$ be a flat proper morphism with connected fibers
from an equidimensional normal scheme $X$ of any dimension, let $D$ be an
effective reduced Weil divisor on $X$ and $\Delta$ an effective
$\mathbb{Q}$-Weil divisor, and assume that $(X,D+\Delta+X_{\pi})$ is log
canonical. In particular, $X_{\pi}$ is reduced and has no component contained
in the support of $D+\Delta$.

We denote by $\bar{D}\to D$ the normalization of $D$, by $\bar{D}_{\pi}$ the
special fiber of $\bar{D}\to\operatorname{{Spec}}R$ and by $\bar{D}_{\pi}^{n}$
its normalization.

We also assume that there exists an involution $\tau$ of the log pair
$(\bar{D},\operatorname{Diff}_{\bar{D}}\Delta)$ over $\operatorname{{Spec}}R$.

###  3.1. Existence of the quotient

######  Lemma 3.1.1.

Let $Z\to\operatorname{{Spec}}R$ be a morphism and $\tau$ an involution of $Z$
over $\operatorname{{Spec}}R$. Then $\tau$ restricts to involutions of
$Z_{\pi}$, $\operatorname{red}(Z_{\pi})$ and
$Z\otimes_{R}\operatorname{Frac}(R)$.

###### Proof.

The ideal of $Z_{\pi}$ is $\pi\mathcal{O}_{Z}$, so by looking at the exact
sequence

$0\to\pi\mathcal{O}_{Z}\to\mathcal{O}_{Z}\to\mathcal{O}_{Z_{\pi}}\to 0$

we see that $\tau$ restricts to an involution of $Z_{\pi}$ if and only if
$\tau(\pi)\subseteq\pi\mathcal{O}_{Z}$. But $\tau(\pi)=\pi$, so this is
immediate. So $\tau$ descends to $Z_{\pi}$, and thus to
$\operatorname{red}(Z_{\pi})$. Since $\tau(\pi)=\pi$, it also holds that
$\tau$ descends to an involution of
$\mathcal{O}_{Z}\otimes_{R}R[1/\pi]=\mathcal{O}_{Z\otimes_{R}\operatorname{Frac}(R)}$.
∎

######  Lemma 3.1.2.

Let $(X,D+\Delta)\to\operatorname{{Spec}}R$ be as in Section 3. If $E$ is a
divisor over $X$ whose center $c_{X}(E)$ belongs to the special fiber
$X_{\pi}$, then $a(E;X,D+\Delta)\geq 0$.

###### Proof.

The proof is the same as [Kol21, 2.14]. Let $E$ be a divisor over $X$,
appearing on a proper birational model $\pi\colon Y\to X$. Write
$b_{E}:=\operatorname{{coeff}}_{E}\pi^{*}X_{\pi}$. Since $\pi^{*}X_{\pi}$ is
Cartier and effective, $b_{E}$ is a non-negative integer. If $c_{X}(E)\subset
X_{\pi}$ then $b_{E}$ is actually a positive integer. Then:

$-1\leq a(E;X,D+\Delta+X_{\pi})=a(E;X,D+\Delta)-b_{E}$

so $a(E;X,D+\Delta)\geq 0$. ∎

######  Lemma 3.1.3.

In the situation of Section 3, every irreducible component of $D$ dominates
$\operatorname{{Spec}}R$, and the irreducible components of $D\cap X_{\pi}$
have dimension $\dim D-1$.

###### Proof.

Since $X$ is flat over $\operatorname{{Spec}}R$, every component of $X_{\pi}$
is a divisor by [GW20, 14.97]. Since $(X,D+\Delta+X_{\pi})$ is lc, no
component of the boundary has coefficient $>1$. Thus $D$ does not contain any
component of $X_{\pi}$, and so every component of $D$ dominates
$\operatorname{{Spec}}R$ and so $D\to\operatorname{{Spec}}R$ is flat [Har77,
III.9.7]. Applying [GW20, 14.97] again yields the result. ∎

######  Lemma 3.1.4.

The special fiber $\bar{D}_{\pi}$ of $\bar{D}\to\operatorname{{Spec}}R$ is
reduced.

###### Proof.

Since $\bar{D}$ is normal and $\bar{D}_{\pi}$ is an hypersurface,
$\bar{D}_{\pi}$ is $S_{1}$. Thus we only need to show that $\bar{D}_{\pi}$ is
generically reduced.

The generic points of $\bar{D}_{\pi}$ dominates the generic points of the
intersection $D\cap X_{\pi}$, and by Section 3.1 these points have codimension
two in $X$. Since $(X,D+\Delta+X_{\pi})$ is lc, the classification of
codimension two lc singularities [Kol13, 2.32] shows that around the generic
points of $D\cap X_{\pi}$, the divisors $D$ and $X_{\pi}$ are regular and meet
transversally. Hence $\bar{D}\to D$ is an isomorphism around those generic
points, with $\bar{D}_{\pi}$ isomorphic to the regular $D\cap X_{\pi}$. This
shows that $\bar{D}_{\pi}$ is generically reduced. ∎

######  Lemma 3.1.5.

The involution $\tau$ on $(\bar{D},\operatorname{Diff}_{\bar{D}}\Delta)$
induces an involution $\sigma$ of the lc pair $(\bar{D}_{\pi}^{n},\Gamma)$,
where $\Gamma$ is defined by the adjunction formula
$(K_{\bar{D}}+\operatorname{Diff}_{\bar{D}}(\Delta)+\bar{D}_{\pi})|_{\bar{D}_{\pi}^{n}}=K_{\bar{D}_{\pi}^{n}}+\Gamma$.

Moreover, the equivalence relation $R_{X_{\pi}}(\sigma)\rightrightarrows
X_{\pi}$ is equal to the restriction of the equivalence relation
$R_{X}(\tau)\rightrightarrows X$ to $X_{\pi}$.

###### Proof.

By Section 3.1 and Section 3.1 the involution $\tau$ preserves the Cartier
divisor $\bar{D}_{\pi}^{n}$ and descends to an involution $\sigma^{\prime}$ on
$\bar{D}_{\pi}$. By the universal property of normalization, we obtain an
involution $\sigma$ of the normalization $\bar{D}_{\pi}^{n}$ that makes the
diagram

(3.1.5.a)
${\bar{D}^{n}_{\pi}}$${\bar{D}^{n}_{\pi}}$${\bar{D}_{\pi}}$${\bar{D}_{\pi}}$$\scriptstyle{\sigma}$$\scriptstyle{\sigma^{\prime}}$

commutative.

Since the $\mathbb{Q}$-Cartier divisors
$K_{\bar{D}}+\operatorname{Diff}_{\bar{D}}\Delta$ and $\bar{D}_{\pi}$ are
$\tau$-invariant, so is their sum. Hence the pullback of
$K_{\bar{D}}+\operatorname{Diff}_{\bar{D}}+\bar{D}^{n}_{\pi}$ to
$\bar{D}_{\pi}^{n}$ is $\sigma$-invariant.

The pair $(\bar{D}_{\pi}^{n},\Gamma)$ is lc by adjunction.

It is clear from the construction that $R(\sigma^{\prime})\rightrightarrows
X_{\pi}$ is equal to the restriction of $R_{X}(\tau)$ to $X_{\pi}$, so we only
need to compare $R_{X_{\pi}}(\sigma)$ and $R_{X_{\pi}}(\sigma^{\prime})$.
Since (missing) is commutative, we see that the two equivalence relations are
the same. ∎

######  Lemma 3.1.6.

The special fiber $X_{\pi}$ is reduced, regular at the generic points of
$D\cap X_{\pi}$ and at worst nodal at other codimension one points. Moreover
if $X_{\pi}^{n}\to X_{\pi}$ is the normalization morphism then:

  1. (a)

it is an isomorphism over the generic points of $D\cap X_{\pi}$,

  2. (b)

the pair $(X_{\pi}^{n},\operatorname{Diff}_{X_{\pi}^{n}}(D+\Delta))$ is lc,
and

  3. (c)

the strict transform of $D\cap X_{\pi}$ is contained in
$\lfloor\operatorname{Diff}_{X_{\pi}^{n}}(D+\Delta)\rfloor$.

###### Proof.

The hypersurface $X_{\pi}$ of $X$ is $S_{1}$. It is regular at the generic
points of $D\cap X_{\pi}$ and at worst nodal at other codimension one points
by [Kol13, 2.32]. To decide whether the strict transform of $D\cap X_{\pi}$
appears in $\lfloor\operatorname{Diff}_{X_{\pi}^{n}}(D+\Delta)\rfloor$ is a
local question around the generic points of $D\cap X_{\pi}$. So the first and
third points hold. The pair
$(X_{\pi}^{n},\operatorname{Diff}_{X_{\pi}^{n}}(D+\Delta))$ is lc by
adjunction. ∎

######  Lemma 3.1.7.

Denote by $E\subset\lfloor\operatorname{Diff}_{X_{\pi}^{n}}(D+\Delta)\rfloor$
the strict transform of $D\cap X_{\pi}$, and by $(E^{n},\Theta)$ the lc pair
obtained from $(X_{\pi}^{n},\operatorname{Diff}_{X_{\pi}^{n}}(D+\Delta))$ by
adjunction. Then there exists a log isomorphism
$f\colon(E^{n},\Theta)\cong(\bar{D}_{\pi}^{n},\Gamma)$ such that the diagram

(3.1.7.b) ${E^{n}}$${\bar{D}^{n}_{\pi}}$${X}$$\scriptstyle{f}$

commutes.

###### Proof.

As observed in Section 3.1 and Section 3.1, the two morphisms

$E\to D\cap X_{\pi}\leftarrow\bar{D}_{\pi}$

are birational and finite. This induces finite birational morphisms

$E^{n}\to(D\cap X_{\pi})^{n}\leftarrow\bar{D}_{\pi}^{n}$

of normal schemes, so these must be isomorphisms. Thus we obtain an
isomorphism $f\colon E^{n}\cong\bar{D}_{\pi}^{n}$ that commutes with the
morphisms to $X$. Since the divisors $\Theta$ and $\Gamma$ are defined by
adjunction, we see that $f$ is a log isomorphism. ∎

######  Proposition 3.1.8.

In the situation of Section 3, assume that $X$ is proper over
$\operatorname{{Spec}}R$ and that $K_{X}+D+\Delta$ is ample over
$\operatorname{{Spec}}R$. Assume also that

  1. (a)

$\dim X=3$, or

  2. (b)

$\dim X=4$, and $X_{\pi}$ is $S_{2}$ in a neighborhood of $D\cap X_{\pi}$ and
the residue field of $R$ is perfect of characteristic $>5$.

Then the quotient $X/R(\tau)$ exists as a demi-normal scheme that is flat and
proper over $\operatorname{{Spec}}R$.

###### Proof.

As explained in Section 2.2, the geometric quotient $X/R(\tau)$ exists as a
scheme as soon as $R(\tau)$ is finite and the geometric quotient
$X_{\mathbb{Q}}/R(\tau)_{\mathbb{Q}}$ exists.

Section 3.1 implies that $\tau$ induces an involution $\tau_{\mathbb{Q}}$ on
the generic fiber
$(\bar{D}_{\mathbb{Q}},\operatorname{Diff}_{\bar{D}_{\mathbb{Q}}}(\Delta_{\mathbb{Q}}))$
and $R(\tau_{\mathbb{Q}})=R(\tau)|_{X_{\mathbb{Q}}}$. Since $K_{X}+D+\Delta$
is ample over $\operatorname{{Spec}}R$, we see that
$(X_{\mathbb{Q}},D_{\mathbb{Q}}+\Delta_{\mathbb{Q}})$ is a projective lc pair
over a field of characteristic zero with ample log canonical divisor a log
involution $\tau_{\mathbb{Q}}$ on
$(\bar{D}_{\mathbb{Q}},\operatorname{Diff}_{\bar{D}_{\mathbb{Q}}}(\Delta_{\mathbb{Q}}))$.
By [Kol13, 5.13] the quotient $X_{\mathbb{Q}}/R(\tau)_{\mathbb{Q}}$ exists.

We still have to show that $R(\tau)$ is finite. Since it respects the
fibration to $\operatorname{{Spec}}R$, we only need to show that
$R(\tau)|_{X_{\pi}}$ is finite. By Section 3.1 it is equivalent to show that
$R_{X_{\pi}}(\sigma)\rightrightarrows X_{\pi}$ is finite. By Section 3.1 and
the commutativity of (missing), we may transport $\sigma$ to an involution of
$(E^{n},\Theta)$. We are now in situation to apply [Pos21, 4.1.1, 4.1.3,
5.3.2] which shows that $R_{X_{\pi}^{n}}(\sigma)$ is finite.

To go from $X_{\pi}^{n}$ to $X_{\pi}$, observe that the equivalence generated
by $\sigma$ on $X_{\pi}^{n}$, respectively on $X_{\pi}$, is trivial away from
the support of $E$, respectively away from the support of $D\cap X_{\pi}$. Now
$X_{\pi}$ is $R_{1}$ is a neigborhood of $D\cap X_{\pi}$ by Section 3.1. If it
is also $S_{2}$ then $X_{\pi}^{n}\to X_{\pi}$ is an isomorphism in a
neighborhood of $E$, and we deduce that $R_{X_{\pi}}(\sigma)$ is finite.

If $\dim X=3$, then we do not need the fact that $X_{\pi}$ is $S_{2}$. In this
case $X_{\pi}$ is a reduced surface that is regular in codimension one and
such that $\omega_{X_{\pi}}^{[m]}(m(D+\Delta)|_{X_{\pi}}))$ is invertible for
$m$ sufficiently divisible (this follows from adjunction along $X_{\pi}\subset
X$), hence we can perform adjunction along $E^{n}\to X_{\pi}$ even if
$X_{\pi}$ is not normal (see [Kol13, 4.2]). The crucial point is that $E^{n}$
is a curve, so the points of $E^{n}$ where $E^{n}\to D\cap X_{\pi}$ is not an
isomorphism, are contained in $\operatorname{{Supp}}\Theta$ by [Kol13, 4.5.1].
Thus the the proof of [Pos21, 4.1.1] is also valid in this situation: the
finite closed subset $\Sigma:=\operatorname{{Supp}}\Theta$ is
$\sigma$-invariant and $E^{n}\setminus\Sigma\to E\setminus n(\Sigma)$ is an
isomorphism. Thus we obtain finiteness.

Flatness of $X/R(\tau)$ over $\operatorname{{Spec}}R$ follows from [Har77,
III.9.7].

It remains to show that $X/R(\tau)$ is demi-normal and proper over
$\operatorname{{Spec}}R$. This is shown as in [Pos21, 3.4.1] (which is
formulated for schemes over a field, but the proof also works over a DVR). ∎

######  Remark 3.1.9.

More generally, the proof of Section 3.1 applies to $(X,D+\Delta,\tau)$ as
soon as we have a gluing theorems for stable lc varieties of dimension $\dim
X-1$ above the residue field $k(\pi)$ of $R$, and that the special fiber
$X_{\pi}$ is $S_{2}$.

###  3.2. Descent of the log canonical sheaf

We show that in the situation of Section 3.1, the log canonical
$\mathbb{Q}$-Cartier divisor descends to the quotient. We can actually show it
in any dimension, as soon as the quotient exists:

######  Proposition 3.2.1.

Let $(Y,\Delta_{Y})\to\operatorname{{Spec}}R$ be a demi-normal flat
$\operatorname{{Spec}}R$-scheme, with induced normalization
$(X,\Delta+D,\tau)$. If $(X,D+\Delta+X_{\pi})$ is lc and
$\operatorname{Diff}_{D^{n}}(\Delta)$ is $\tau$-invariant, then
$K_{Y}+\Delta_{Y}$ is $\mathbb{Q}$-Cartier.

###### Proof.

First base-change over the generic point of $\operatorname{{Spec}}R$. Then
[Kol13, 5.38] shows that $K_{Y}+\Delta_{Y}$ is $\mathbb{Q}$-Cartier on the
generic fiber.

So the closed locus where $K_{Y}+\Delta_{Y}$ is not $\mathbb{Q}$-Cartier, if
not empty, is contained in the special fiber $Y_{\pi}$. By Section 3.1 and the
fact that $(X,D+\Delta)\to(Y,\Delta_{Y})$ is crepant, we see that no lc center
of $(Y,\Delta_{Y})$ is contained in $Y_{\pi}$. Thus we may apply the first
part of the proof of [Kol13, 5.38], which is valid in our setting, and
conclude that $K_{Y}+\Delta_{Y}$ is also $\mathbb{Q}$-Cartier along the
special fiber. ∎

###### Proof of Theorem 1.

If $(X,\Delta)$ is as in the left-hand side of Theorem 1, we claim its
normalization is a triplet $(\bar{X},\bar{D}+\bar{\Delta},\tau)$ as on the
right-hand side. We indeed have a generically fixed point free involution
$\tau$: this follows from Section 2.3 since by Section 3.1 the generic points
of $\bar{D}$ have characteristic zero residue fields. The other properties
clearly hold.

Let $(\bar{X},\bar{D}+\bar{\Delta},\tau)$ be as in the right-hand side of
Theorem 1. By Section 3.1 the quotient $X:=\bar{X}/R(\tau)$ exists, it is a
demi-normal scheme flat and proper over $\operatorname{{Spec}}R$. By Section
3.2 we obtain that $(X,\Delta)$ is slc.

This defines a map in the opposite direction as the normalization. They are
inverse to each other by [Kol13, 5.3] and the fact that $\bar{X}$ is the
normalization of $X$. ∎

##  4\. Fibers of the quotient

In the previous section, we have shown that the quotient of the family exists
in several situations. In this section, we study the fibers of the quotient
whenever it exists. We will use the following notations:

######  Notation 4.0.1.

Let $R$ be a DVR of mixed characteristic $(0,p)$, let
$Y\to\operatorname{{Spec}}R$ be a flat morphism of finite type from a demi-
normal scheme $Y$. (We do not assume any properness property relatively to
$\operatorname{{Spec}}R$.) Let $p\colon(X,D,\tau)\to Y$ be the normalization
morphism. Then $D$ is reduced of pure codimension one and we let $\bar{D}\to
D$ be its normalization.

Since $\tau$ is an $R$-involution on $\bar{D}$, by Section 3.1 it restricts to
an involution $\tau_{\pi}$ of $\bar{D}_{\pi}$ and $\tau_{\mathbb{Q}}$ of
$\bar{D}_{\mathbb{Q}}$. It is clear that
$R_{X_{\pi}}(\tau_{\pi})=R_{X}(\tau)|_{X_{\pi}}$ and that
$R_{X_{\mathbb{Q}}}(\tau_{\mathbb{Q}})=R_{X}(\tau)|_{X_{\mathbb{Q}}}$. Since
$R(\tau)\rightrightarrows X$ is finite, it follows that
$R(\tau_{\pi})\rightrightarrows X\pi$ and
$R(\tau_{\mathbb{Q}})\rightrightarrows X_{\mathbb{Q}}$ are finite.

We do not assume systematically that $Y$ has an slc structure. When we do, we
let $\Delta_{Y}$ be the boundary on $Y$, and $\Delta$ its strict transform on
$X$.

###  4.1. Commutativity of fibers and quotients

We let $p\colon(X,D,\tau)\to Y\to\operatorname{{Spec}}R$ be as in Section 4.
We investigate to which extent the quotients
$X_{\mathbb{Q}}/R(\tau_{\mathbb{Q}})$ and $X_{\pi}/R(\tau_{\pi})$ are
comparable to the fibers $Y_{\mathbb{Q}}$ and $Y_{\pi}$.

######  Lemma 4.1.1.

$Y_{\mathbb{Q}}=X_{\mathbb{Q}}/R(\tau_{\mathbb{Q}})$.

###### Proof.

This follows immediately from [Kol13, 9.11] since $Y_{\mathbb{Q}}\to Y$ is
flat. ∎

######  Lemma 4.1.2.

The quotient $Z:=X_{\pi}/R(\tau_{\pi})$ exists, and $X_{\pi}\to Y_{\pi}$
factorizes through a finite birational universal homeomorphism $Z\to Y_{\pi}$.

###### Proof.

The quotient exists as a scheme by [Kol12, Theorem 6, Corollary 48]. By the
universal property of the quotient, $X_{\pi}\to Y_{\pi}$ factors through a
morphism $q\colon Z\to Y_{\pi}$.

(4.1.2.c)
${X_{\pi}}$${X}$${Z}$${Y_{\pi}}$${Y}$$\scriptstyle{p}$$\scriptstyle{q}$

Using [Kol13, 9.2] for $X_{\pi}\to Z$ and $X\to Y$, we see that
$q(\operatorname{{Spec}}K)\colon Z(\operatorname{{Spec}}K)\to
Y_{\pi}(\operatorname{{Spec}}Z)$ is a bijection for every geometric point
$\operatorname{{Spec}}K\to\operatorname{{Spec}}k(\pi)$. Thus $q$ is a
universal homeomorphism by [Gro60, 3.5.3-5]. It is finite birational since
$X_{\pi}\to Y_{\pi}$ is so. ∎

Let us study more precisely the morphism $Z\to Y_{\pi}$. The question is flat-
local on $Y$ by [Kol13, 9.11], so in particular we may assume that every
scheme appearing in (missing) is affine, and work with sections of structural
sheaves as if they were global sections.

We introduce the following sub-sheaves:

$\mathcal{O}_{X}^{+}:=\\{s\in\mathcal{O}_{X}\mid s|_{\bar{D}}\text{ is
}\tau\text{-invariant}\\},\quad\mathcal{O}_{D}^{+}:=\\{s\in\mathcal{O}_{D}\mid
s|_{\bar{D}}\text{ is }\tau\text{-invariant}\\}$

and

$\mathcal{O}_{X_{\pi}}^{+}:=\\{s\in\mathcal{O}_{X}\mid
s|_{\bar{D}_{\pi}}\text{ is
}\tau_{\pi}\text{-invariant}\\},\quad\mathcal{O}_{D_{\pi}}^{+}:=\\{s\in\mathcal{O}_{D_{\pi}}\mid
s|_{\bar{D}_{\pi}}\text{ is }\tau_{\pi}\text{-invariant}\\}.$

They fit into the following commutative diagram:

(4.1.2.d)
${\mathcal{O}_{D_{\pi}}^{+}}$${\mathcal{O}_{D}^{+}}$${\mathcal{O}_{X_{\pi}}^{+}}$${\mathcal{O}_{X}^{+}}$

######  Claim 4.1.3.

The quotient $W:=D_{\pi}/R_{D_{\pi}}(\tau_{\pi})$ exists as a scheme, and
$\mathcal{O}_{W}=\mathcal{O}_{D_{\pi}}^{+}$.

###### Proof.

The quotient exists as a scheme by [Kol13, 9.10] applied to $D_{\pi}\to
Y_{\pi}$, and the second assertion follows from the same reference. ∎

######  Claim 4.1.4.

$\mathcal{O}_{Z}=\mathcal{O}_{X_{\pi}}^{+}$.

###### Proof.

Indeed, by [Kol12, Proposition 25] the diagram

${D_{\pi}}$${X_{\pi}}$${W=D_{\pi}/R_{D_{\pi}}(\tau_{\pi})}$${Z}$

is a universal push-out. This implies that

$\mathcal{O}_{Z}=\mathcal{O}_{X_{\pi}}\times_{\mathcal{O}_{D_{\pi}}}\mathcal{O}_{W}=\mathcal{O}_{X_{\pi}}\times_{\mathcal{O}_{D_{\pi}}}\mathcal{O}_{D_{\pi}}^{+}=\mathcal{O}_{X_{\pi}}^{+}$

as claimed. ∎

######  Proposition 4.1.5.

$Z\to Y_{\pi}$ is an isomorphism if and only if the restriction map
$\mathcal{O}_{X}^{+}\to\mathcal{O}_{X_{\pi}}^{+}$ appearing in (missing) is
surjective.

###### Proof.

By [Kol13, 9.10] we have $\mathcal{O}_{Y}=\mathcal{O}_{X}^{+}$. Combining this
with the previous claim and the commutative diagram (missing) we obtain

${\mathcal{O}_{X_{\pi}}^{+}}$${\mathcal{O}_{X}^{+}}$${\mathcal{O}_{Z}}$${\mathcal{O}_{Y_{\pi}}}$${\mathcal{O}_{Y}}$$\scriptstyle{\alpha}$$\scriptstyle{=}$$\scriptstyle{=}$

It is easy to see that $\alpha$ is surjective if and only if
$\mathcal{O}_{Y_{\pi}}\hookrightarrow\mathcal{O}_{Z}$ is bijective. ∎

######  Proposition 4.1.6.

Assume that:

  1. (a)

$\tau(\mathcal{O}_{D})\subseteq\mathcal{O}_{D}$ holds in a neighbourhood of
$D_{\pi}$,

  2. (b)

$\mathcal{O}_{D_{\pi}}\to\mathcal{O}_{\bar{D}_{\pi}}$ is injective,

  3. (c)

$D\to\operatorname{{Spec}}R$ is flat, and

  4. (d)

$p=\operatorname{char}R/(\pi)\neq 2$.

Then $Z=Y_{\pi}$.

###### Proof.

We prove that $\mathcal{O}_{X}^{+}\to\mathcal{O}_{X_{\pi}}^{+}$ is surjective.
The only question is along $D_{\pi}$, so we may assume that
$\tau(\mathcal{O}_{D})\subseteq\mathcal{O}_{D}$ holds globally. Take
$s\in\mathcal{O}_{X_{\pi}}^{+}$ and any lift $t\in\mathcal{O}_{X}$. Write
$v=t|_{D}$. By hypothesis $\tau(v)\in\mathcal{O}_{D}$ and by construction
$\tau(v)-v$ vanishes when restricted to $\bar{D}_{\pi}$. Since
$\mathcal{O}_{D_{\pi}}\to\mathcal{O}_{\bar{D}_{\pi}}$ is injective, we see
that $v-\tau(v)$ belongs to the ideal $I_{D_{\pi}}=\pi\cdot\mathcal{O}_{D}$.
Thus we can write $\tau(v)=v+a\pi$ for some $a\in\mathcal{O}_{D}$. Since
$\tau$ is an $R$-involution,

$v=\tau^{\circ 2}(v)=v+(a+\tau(a))\pi.$

By flatness $\pi$ is not a zero-divisor in $\mathcal{O}_{D}$, so we have
$a=-\tau(a)$. Let $b\in\mathcal{O}_{X}$ be any lift of $a$. Since
$2\in\mathcal{O}_{X}$ is invertible we can form the element
$t^{\prime}:=t+\frac{b}{2}\pi$. Then $t^{\prime}\in\mathcal{O}_{X}^{+}$ and
$t^{\prime}|_{X_{\pi}}=t|_{X_{\pi}}=s$, as desired. ∎

######  Remark 4.1.7.

The conditions of Section 4.1 are sufficient but not necessary for $Z=Y_{\pi}$
to hold. More precisely, it can happen that $Z=Y_{\pi}$ while $\tau$ does not
descend to an involution of a neighbourhood of $D_{\pi}\subset D$. To wit, let
$Y=\operatorname{{Spec}}R[x,y,z]/(xyz)$. Then its normalization $X$ is the
union of three copies of $\mathbb{A}^{2}_{R}$, with the morphism $X\to Y$
given by:

$\begin{matrix}R[x,y,z]/(xyz)&\longrightarrow&R[u_{1},u_{2}]&\oplus&R[v_{1},v_{2}]&\oplus&R[w_{1},w_{2}]\\\
x&\mapsto&u_{1}&\oplus&v_{2}&\oplus&0\\\
y&\mapsto&u_{2}&\oplus&0&\oplus&w_{1}\\\
z&\mapsto&0&\oplus&v_{1}&\oplus&v_{2}\end{matrix}$

The conductor $D$ is given by $V(u_{1}u_{2})\sqcup V(v_{1}v_{2})\sqcup
V(w_{1}w_{2})\subset X$. The involution $\tau$ on $\bar{D}$ is given by three
isomorphisms of lines, namely

$\tau=\left[(\mathbb{A}^{1}_{u_{1}}\cong\mathbb{A}^{1}_{v_{2}},\ u_{1}\mapsto
v_{2}),\quad(\mathbb{A}^{1}_{v_{1}}\cong\mathbb{A}^{1}_{w_{2}},\ v_{1}\mapsto
w_{2}),\quad(\mathbb{A}^{1}_{u_{2}}\cong\mathbb{A}^{1}_{w_{1}},\ u_{2}\mapsto
w_{1})\right].$

The picture on the special fiber is exactly the same, except that $R$ is
replaced with its residue field $k(\pi)$.

The involution $\tau$ does not descend to $D$, since the three origins would
belong to the same equivalence class.

On the other hand, $\mathcal{O}_{X}^{+}$ is the set of
$f(s(u_{1},v_{2}),s^{\prime}(v_{1},w_{2}),s^{\prime\prime}(u_{2},w_{1}))$
where $f\in R[X,Y,Z]$ and $s,s^{\prime},s^{\prime\prime}$ run through the
symmetric polynomials in two variables. Similarly for
$\mathcal{O}_{X_{\pi}}^{+}$, except that we take $f\in k(\pi)[X,Y,Z]$. In
particular we see that $\mathcal{O}_{X}^{+}\to\mathcal{O}_{X_{\pi}}^{+}$ is
surjective.

We are mostly interested in the slc cases, which we study in the following
theorem.

######  Theorem 4.1.8.

Suppose that $p\neq 2$ and that $(Y,\Delta_{Y}+Y_{\pi})$ is slc. Then:

  1. (a)

$Z\to Y_{\pi}$ is an isomorphism in codimension one.

  2. (b)

If $Y_{\pi}$ is $S_{2}$, then $Z\to Y_{\pi}$ is an isomorphism.

  3. (c)

If $D_{\pi}$ is reduced, then $Z\to Y_{\pi}$ is an isomorphism.

###### Proof.

We verify the conditions of Section 4.1. In general, since
$(X,\Delta+D+X_{\pi})$ is lc, every component of $D$ dominates
$\operatorname{{Spec}}R$, so $D\to\operatorname{{Spec}}R$ is flat. By [Kol13,
2.32] we see that $D$ is normal in codimension one along $D_{\pi}$, thus
$\tau(\mathcal{O}_{D})\subseteq\mathcal{O}_{D}$ holds in a neighborhood of the
generic points of $D_{\pi}$.

We consider the pullback morphism
$\mathcal{O}_{D_{\pi}}\to\mathcal{O}_{\bar{D}_{\pi}}$. Recall that
$\bar{D}_{\pi}$ is reduced Section 3.1 and $\bar{D}_{\pi}\to D_{\pi}$ is
finite. Thus $\mathcal{O}_{D_{\pi}}\to\mathcal{O}_{\bar{D}_{\pi}}$ is
injective if and only if $D_{\pi}$ is reduced. By [Kol13, 2.32], we know that
$D_{\pi}$ is generically reduced.

Hence we deduce that there is an open subset $U\subset Y$ with
$\operatorname{codim}(Y\setminus U)\geq 2$ and such that the conditions of
Section 4.1 are satisfied for the normalization $(X_{U},D_{U},\tau_{U})\to U$.
It follows that $Z_{U}\to Y_{\pi}\cap U$ is an isomorphism. This proves the
first point. The second point follows easily.

Now assume that $D_{\pi}$ is reduced. In particular it is $S_{1}$. Since it is
a hypersurface of $D$ and that $\pi$ is not a zero-divisor in
$\mathcal{O}_{D}$, we deduce that $D$ is $S_{2}$ in a neighbourhood of
$D_{\pi}$. Therefore $\tau(\mathcal{O}_{D})\subseteq\mathcal{O}_{D}$ holds not
only around the generic points of $D_{\pi}$, but in a neighbourhood of
$D_{\pi}$, and Section 4.1 applies. ∎

We see that for slc pairs, the commutativity of the quotient and the fibers
depends on the reducedness of the scheme-theoretic intersection $D_{\pi}=D\cap
X_{\pi}$. Note that $D_{\pi}$ is the intersection of two lc centers of
$(X,\Delta+D+X_{\pi})$. In characteristic zero, intersections of lc centers
are reduced [Kol13, 7.8]. In mixed characteristic, we can prove the following.

######  Proposition 4.1.9.

If $(Y,\Delta_{Y}+Y_{\pi})$ is slc projective over $\operatorname{{Spec}}R$,
$\dim Y=3$ and $p\geq 7$, then $D_{\pi}$ is reduced.

###### Proof.

First assume that $(X,\Delta+D+X_{\pi})$ is dlt. Then the result holds by
[BK21, Theorem 19 and subsequent paragraph].

In the general case, let
$\varphi\colon(X^{\prime},\Delta^{\prime}+D^{\prime}+E^{\prime}+X_{\pi}^{\prime})\to(X,\Delta+D+X_{\pi})$
be a $\mathbb{Q}$-factorial crepant dlt model. Such a model exists: log
resolutions exist for pairs of dimension three over the spectrum of $R$
[BMP+21, 2.12], and combining [BMP+21, Theorem F] with the arguments of
[Kol13, 1.35-1.36] we can run a MMP to produce a model with the desired
properties. We claim that
$R^{1}\varphi_{*}\mathcal{O}(-D^{\prime}-X_{\pi}^{\prime})=R^{1}\varphi_{*}\mathcal{O}(-D^{\prime})\otimes\mathcal{O}(-X_{\pi})=0$
along $\operatorname{{Supp}}(X_{\pi})$. This follows from [BK21, Proposition
7] applied to $\varphi\colon X^{\prime}\to X$, the dlt pair
$(X^{\prime},X_{\pi}^{\prime}+(\Delta^{\prime}+E^{\prime}))$ and the
$\mathbb{Z}$-divisor $-D^{\prime}$, since:

  1. (a)

$-D^{\prime}\sim_{\varphi,\mathbb{Q}}K_{X^{\prime}}+X^{\prime}_{\pi}+E^{\prime}+\Delta^{\prime}$,

  2. (b)

$\mathcal{O}(-D^{\prime}-mX^{\prime}_{\pi})$ is $S_{3}$ for every $m\geq 1$ by
[BK21, Theorem 19],

  3. (c)

$-X^{\prime}_{\pi}$ is relatively nef over $X$, since it is relatively
trivial,

  4. (d)

strong Grauert–Riemenschneider vanishing holds for $(X_{\pi}^{\prime})^{n}\to
X_{\pi}$ since this is a birational morphism of excellent surfaces (this can
be deduced from [Kol13, 10.4]).

Therefore pushing forward along $\varphi$ the exact sequence

$0\to\mathcal{O}(-D^{\prime}-X^{\prime}_{\pi})\to\mathcal{O}_{X^{\prime}}\to\mathcal{O}_{D^{\prime}\cup
X^{\prime}_{\pi}}\to 0,$

we obtain that
$\varphi_{*}\mathcal{O}_{X^{\prime}}=\mathcal{O}_{X}\to\varphi_{*}\mathcal{O}_{D^{\prime}\cup
X^{\prime}_{\pi}}$ is surjective in a neighbourhood of $X_{\pi}$. This map
factors through $\mathcal{O}_{D\cup X_{\pi}}$, so we deduce that

$\varphi_{*}\mathcal{O}_{D^{\prime}\cup X^{\prime}_{\pi}}=\mathcal{O}_{D\cup
X_{\pi}}.$

We are now in position to apply the argument of [Kol13, 7.8] to conclude that
$D\cap X_{\pi}$ is reduced. ∎

###  4.2. Serre properties of the fibers

In this subsection we consider the Serre conditions $S_{r}$ on the special
fiber $Y_{\pi}$. Since we always assume $Y\to\operatorname{{Spec}}R$ to be
flat, $\pi$ is not a zero-divisor in $\mathcal{O}_{Y}$ and so

$Y_{\pi}\text{ is }S_{r}\quad\Longleftrightarrow\quad Y\text{ is
}S_{r+1}\text{ along }Y_{\pi}.$

The analog equivalence holds for $X_{\pi}\subset X$, since $X$ is
automatically flat over $\operatorname{{Spec}}R$, by [Har77, III.9.7] and the
fact that $p\colon X\to Y$ is finite. Similarly, an analog equivalence holds
for $D$ and $D/R_{D}(\tau)$ as soon as one of them (equivalently both of them)
is flat over $\operatorname{{Spec}}R$.

######  Remark 4.2.1.

By [Gro65, 7.8.6.iii] the $S_{i}$ loci of excellent schemes are open. If
$Y\to\operatorname{{Spec}}R$ is proper (or simply a closed morphism), a closed
subset that is disjoint from $Y_{\pi}$ must be empty. In this case the
previous equivalence becomes: $Y_{\pi}$ is $S_{r}$ if and only if $Y$ is
$S_{r+1}$, and similarly for $X,D$ and $D/R_{D}(\tau)$ under suitable
hypothesis.

To approach the Serre properties of $Y$, we rely on the fact that it can be
described as a universal push-out [Kol12, Proposition 25]. The main technical
tool is the following lemma.

######  Lemma 4.2.2.

Consider a commutative square of complexes of abelian groups

${A^{\bullet}}$${B^{\bullet}}$${C^{\bullet}}$${D^{\bullet}}$$\scriptstyle{\alpha\bullet}$$\scriptstyle{\beta\bullet}$$\scriptstyle{q\bullet}$$\scriptstyle{p\bullet}$

which is a pullback diagram at every degree. Then for every $j$ there exists a
natural map

$\xi^{j}\colon h^{j}(A^{\bullet})\to
h^{j}(B^{\bullet})\times_{h^{j}(D^{\bullet})}h^{j}(C^{\bullet})$

which is surjective. If $h^{j-1}(D^{\bullet})=0$ and $p^{j-2}$ is surjective,
then $\xi^{j}$ is bijective.

###### Proof.

Let us embark for some diagram-chasing in

${A^{j-1}}$${A^{j}}$${B^{j-1}}$${B^{j}}$${C^{j}}$${C^{j-1}}$${C^{j-2}}$${D^{j}}$${D^{j-1}}$${D^{j-2}}$$\scriptstyle{\alpha^{j}}$$\scriptstyle{\beta^{j}}$$\scriptstyle{q^{j-1}}$$\scriptstyle{p^{j-2}}$

We denote the differential maps of $A^{\bullet}$ by $d_{A}^{j}\colon A^{j}\to
A^{j+1}$, and similarly for the other complexes.

If $x\in\ker(d^{j}_{A})$, then clearly $\alpha^{j}(x)\in\ker(d^{j}_{B})$ and
$\beta^{j}(x)\in\ker(d^{j}_{C})$. This defines a map

$\xi^{j}_{0}\colon\ker(d^{j}_{A})\to\ker(d^{j}_{B})\times_{\ker(d^{j}_{D})}\ker(d^{j}_{C}).$

We claim that $\xi^{j}_{0}$ is surjective. Indeed, a pair
$(s,t)\in\ker(d^{j}_{B})\times_{\ker(d^{j}_{D})}\ker(d^{j}_{C})$ gives, by the
pullback property, a unique element $x\in A^{j}$ such that $\alpha^{j}(x)=s$
and $\beta^{j}(x)=t$. Since

$\alpha^{j+1}(d_{A}^{j}x)=d^{j}_{B}(\alpha^{j}(x))=0,\quad\beta^{j+1}(d_{A}^{j}x)=d_{B}^{j}(\beta^{j}(x))=0$

the pullback property ensures that $d^{j}_{A}(x)=0$. Thus $\xi^{j}_{0}$ is
surjective.

It is easy to see that $\xi_{j}^{\prime}$ descends to a surjective map

$\xi^{j}\colon h^{j}(A^{\bullet})\to
h^{j}(B^{\bullet})\times_{h^{j}(D^{\bullet})}h^{j}(C^{\bullet}).$

Now let us discuss the injectivity of $\xi^{j}$. Let $x\in\ker(d^{j}_{A})$ be
such that $\alpha^{j}(x)=d^{j-1}_{B}(y)$ and $\beta^{j}(x)=d^{j-1}_{C}(z)$ for
some $y\in B^{j-1}$ and $z\in C^{j-1}$. We investigate whether
$x\in\operatorname{{im}}(d^{j-1}_{A})$. The element
$\partial:=q^{j-1}(y)-p^{j-1}(z)$ is non-zero in general, but it belongs to
$\ker(d^{j-1}_{D})$. Thus if $h^{j-1}(D^{\bullet})=0$, there exists
$\partial^{\prime}\in D^{j-2}$ such that
$d^{j-2}_{D}(\partial^{\prime})=\partial$. If $p^{j-2}$ is surjective, there
exists $\delta^{\prime}\in C^{j-2}$ such that
$p^{j-2}(\delta^{\prime})=\partial^{\prime}$. Set
$z_{1}:=z+d^{j-2}_{C}(\delta^{\prime})$. We have

$p^{j-1}(z_{1})=p^{j-1}(z)+\delta=q^{j-1}(y),$

thus by the pullback property the pair $(y,z_{1})$ corresponds to a unique
$x^{\prime}\in A^{j-1}$. Since
$\alpha^{j}(d^{j-1}_{A}(x^{\prime}))=d^{j-1}_{B}(y)=\alpha^{j}(x)$ and
$\beta^{j}(d^{j-1}_{A}(x^{\prime}))=d_{C}^{j-1}(z_{1})=d_{C}^{j-1}(z)=\beta^{j}(x)$,
the pullback property shows that $x=d_{A}^{j-1}(x^{\prime})$. This concludes
the proof. ∎

######  Lemma 4.2.3.

Let $A$ be a Noetherian semi-local ring with Jacobson ideal
$I=\mathfrak{m}_{1}\cap\dots\cap\mathfrak{m}_{n}$. Let $M$ be an $A$-module.
Then $H^{r}_{I}(M)=0$ if and only if
$H^{r}_{\mathfrak{m}_{i}A_{\mathfrak{m}_{i}}}(M_{\mathfrak{m}_{i}})=0$ for
every $\mathfrak{m}_{i}$.

###### Proof.

Since the localization maps $A\to A_{\mathfrak{m}_{i}}$ are flat, we have
isomorphism s

$H^{r}_{I}(M)\otimes_{A}A_{\mathfrak{m}_{i}}\cong
H^{r}_{IA_{\mathfrak{m}_{i}}}(M_{\mathfrak{m}_{i}})\quad\text{for every }i.$

Since the $\mathfrak{m}_{i}$ are pairwise coprime, we have
$I=\mathfrak{m}_{1}\cdots\mathfrak{m}_{n}$ and therefore
$IA_{\mathfrak{m}_{i}}=\mathfrak{m}_{i}A_{\mathfrak{m}_{i}}$. Thus the result
follows from the fact that a module is trivial if and only if its localization
at every maximal ideal is trivial. ∎

######  Proposition 4.2.4.

Let $(X,D,\tau)\to Y$ be as in Section 4 (flatness of
$Y\to\operatorname{{Spec}}R$ is not necessary). Assume in addition that $D$ is
$S_{r}$.

  1. (a)

If $X$ and $D/R_{D}(\tau)$ are $S_{r+1}$, then $Y$ is $S_{r+1}$.

  2. (b)

If $D$ is $S_{r+1}$, then the converse holds.

######  Remark 4.2.5.

As the proof will show, the statement of Section 4.2 is local on $Y$. That is,
the statement holds if we replace adequately being $S_{i}$ with being $S_{i}$
at $y\in Y$, respectively with being $S_{i}$ along $p^{-1}(y)\subset X$.

###### Proof.

For simplicity, we write $\mathcal{W}=D/R_{D}(\tau)$. Then by [Kol12,
Proposition 25] the diagram

${D}$${X}$${\mathcal{W}}$${Y}$$\scriptstyle{p}$

is a universal push-out and $\mathcal{W}\hookrightarrow Y$ is a closed
embedding. Thus if $y\in Y$ is a point with
$\mathfrak{m}_{y}=(f_{1},\dots,f_{n})\subset\mathcal{O}_{y}:=\mathcal{O}_{Y,y}$,
then the diagrams

(4.2.5.e) ${(\mathcal{O}_{D\times\mathcal{O}_{y}})_{f_{i_{0}}\cdots
f_{i_{s}}}}$${(\mathcal{O}_{X\times\mathcal{O}_{y}})_{f_{i_{0}}\cdots
f_{i_{s}}}}$${(\mathcal{O}_{\mathcal{W},y})_{f_{i_{0}}\cdots
f_{i_{s}}}}$${(\mathcal{O}_{y})_{f_{i_{0}}\cdots
f_{i_{s}}}}$$\scriptstyle{p^{i_{0},\dots,i_{s}}}$

are pullback diagrams of semi-local rings. Notice that $p^{i_{0},\dots,i_{s}}$
is always surjective, since $\mathcal{W}\hookrightarrow Y$ is a closed
embedding.

For
$\mathcal{A}\in\\{\mathcal{O}_{y},\mathcal{O}_{\mathcal{W},y},\mathcal{O}_{X\times\mathcal{O}_{y}},\mathcal{O}_{D\times\mathcal{O}_{y}}\\}$
we denote by $\check{C}^{\bullet}(\mathbb{f};\mathcal{A})$ the alternating
Čech complex associated to $\mathcal{A}$ and to
$\mathbb{f}=\\{f_{1},\dots,f_{n}\\}$. Recall that

$\check{C}^{\bullet}(\mathbb{f};\mathcal{A})=\left[0\to\mathcal{A}\to\bigoplus_{i}\mathcal{A}_{f_{i}}\to\bigoplus_{i_{0}<i_{1}}\mathcal{A}_{f_{i_{0}}f_{i_{1}}}\to\dots\to\mathcal{A}_{f_{1}\cdots
f_{n}}\to 0\right]$

with differential defined as alternated sums of the natural localization maps.
Thus it is easy to see that the diagrams (missing) induce a commutative square
of complexes

${\check{C}^{\bullet}(\mathbb{f};\mathcal{O}_{D\times\mathcal{O}_{y}})}$${\check{C}^{\bullet}(\mathbb{f};\mathcal{O}_{X\times\mathcal{O}_{y}})}$${\check{C}^{\bullet}(\mathbb{f};\mathcal{O}_{W,y})}$${\check{C}^{\bullet}(\mathbb{f};\mathcal{O}_{y})}$$\scriptstyle{p^{\bullet}}$

where $p^{\bullet}$ is surjective and the square is a pullback at every
degree. Now the cohomology of $\check{C}(\mathbb{f};\mathcal{A})$ is equal to
the local cohomology of $\mathcal{A}$ along $\mathfrak{m}_{y}\mathcal{A}$
[ILL+07, Theorem 7.13]. By the first part of Section 4.2, we get a surjective
map

$H^{j}_{y}(\mathcal{O}_{y})\twoheadrightarrow
H^{j}_{X_{y}}(\mathcal{O}_{X\times\mathcal{O}_{y}})\times_{H^{j}_{D_{y}}(\mathcal{O}_{D\times\mathcal{O}_{y}})}H^{j}_{y}(\mathcal{O}_{W,y}).$

Now assume that $D$ is $S_{r}$. Then $H^{j}_{z}(\mathcal{O}_{D,z})=0$ for
every $j<r$ and $z\in D$. If $z$ maps to $y\in Y$, then $\mathcal{O}_{D,z}$ is
the localization of $\mathcal{O}_{D\times\mathcal{O}_{y}}$ at some maximal
ideal. Moreover $\mathfrak{m}_{y}\mathcal{O}_{D,z}$ is equal (up to taking its
radical, to which the local cohomology is insensitive) to the Jacobson radical
of $\mathcal{O}_{D\times\mathcal{O}_{y}}$. Thus
$H^{j}_{D_{y}}(\mathcal{O}_{D\times\mathcal{O}_{y}})=0$ for every $j<r$ by
Section 4.2. Therefore by the second part of Section 4.2, we have an
isomorphism

(4.2.5.f) $H^{j}_{y}(\mathcal{O}_{y})\cong
H^{j}_{X_{y}}(\mathcal{O}_{X\times\mathcal{O}_{y}})\times
H^{j}_{y}(\mathcal{O}_{W,y})\quad\forall j<r.$

Similarly to above the group
$H^{j}_{X_{y}}(\mathcal{O}_{X\times\mathcal{O}_{y}})$ is equal to the
$j^{\text{th}}$ local cohomology group of the semi-local ring
$\mathcal{O}_{X\times\mathcal{O}_{y}}$ along its Jacobson radical.

Thus we see that if $X$ and $\mathcal{W}$ are $S_{r+1}$, then by (missing) we
have $H^{j}_{y}(\mathcal{O}_{y})=0$ for every $j<r$. This proves the first
point. Conversely, if $D$ and $Y$ are $S_{r+1}$, then the isomorphisms
(missing) hold for all $j\leq r$, and thus $X$ and $W$ are $S_{r+1}$. ∎

For the $S_{2}$ property, we have a stronger statement in the slc case:

######  Proposition 4.2.6.

Assume that $(Y,\Delta_{Y}+Y_{\pi})$ is slc, $D_{\pi}$ is reduced and
$p=\operatorname{char}R/(\pi)\neq 2$. If
$(X_{\pi},\operatorname{Diff}_{X_{\pi}}(\Delta+D))$ if slc then
$(Y_{\pi},\operatorname{Diff}_{Y_{\pi}}\Delta_{Y})$ is also slc.

###### Proof.

Since $D_{\pi}$ is reduced, $D$ is $S_{2}$ along $D_{\pi}$. By [Kol13, 2.31]
it is also $R_{1}$ in a neighbourhood of $D_{\pi}$. Restricting over an open
subset of $Y$ that contains its special fiber, we may assume that $D$ is
normal.

By Theorem 4.1.8 the special fiber $Y_{\pi}$ is the quotient of $X_{\pi}$ by
the equivalence relation generated by the involution $\tau_{\pi}$ on the
divisor $D_{\pi}$. If $X_{\pi}$ is $S_{2}$, then Section 2.2 shows that
$X_{\pi}/R_{X_{\pi}}(\tau_{\pi})$ is also $S_{2}$.

We show that $Y_{\pi}$ is at worst nodal in codimension one. By [Kol13, 2.31]
we know that $X_{\pi}$ is at worst nodal in codimension one, so it is
sufficient to consider the codimesion one points of $Y_{\pi}$ that belong to
the image of $D_{\pi}$. These are precisely the images of the generic points
of $D_{\pi}$. By [Kol13, 2.31] again, we know that $X_{\pi}$ is normal in a
neighbourhood of the generic points of $D_{\pi}$. Thus we may assume that
$X_{\pi}$ is normal, and the nodal property follows from [Pos21, 3.4.1].

Therefore $Y_{\pi}$ is demi-normal, and we can perform adjunction along
$Y_{\pi}\subset Y$ (see [Kol13, 4.2]). We obtain a pair
$(Y_{\pi},\operatorname{Diff}_{Y_{\pi}}\Delta_{Y})$. From the diagram
(missing) we see that its normalization is
$(X_{\pi}^{n},\operatorname{Diff}_{X_{\pi}^{n}}(\Delta+D))$, which is lc by
assumption. ∎

######  Remark 4.2.7.

The converse direction of Section 4.2 seems more difficult. The question is
whether $Y_{\pi}$ being $S_{2}$ implies that $X_{\pi}$ is also $S_{2}$. The
only question is along $D_{\pi}$, thus we may localize over a point of
$p(D_{\pi})$.

Let $W$ be the quotient of $X_{\pi}^{n}$ by $R(\tau_{\pi})$. We have a
commutative diagram

${X_{\pi}^{n}}$${X_{\pi}}$${W}$${Y_{\pi}}$$\scriptstyle{n}$$\scriptstyle{q}$$\scriptstyle{p}$$\scriptstyle{m}$

where $m$ is a finite morphism. Since $X_{\pi}$ is reduced it is $S_{2}$ in
codimension one. So there is a big open subset $U\subset Y_{\pi}$ such that
$n_{U}$, and therefore $m_{U}$, are isomorphisms. Hence
$\mathcal{O}_{Y_{\pi}}\subset m_{*}\mathcal{O}_{W}$ is an equality over $U$.
Since $\mathcal{O}_{Y_{\pi}}$ is $S_{2}$ and $m_{*}\mathcal{O}_{W}$ is
torsion-free, we obtain by [Sta, 0AV9] that
$\mathcal{O}_{Y_{\pi}}=m_{*}\mathcal{O}_{W}$. In other words $W=Y_{\pi}$ in a
neighbourhood of $p(D_{\pi})$.

However, this equality alone does not imply that $X_{\pi}$ is $S_{2}$ in a
neighbourhood of $D\cap X_{\pi}$. Here is a counterexample. Consider
$\mathcal{S}^{n}=\mathbb{A}^{2}_{x,y}$ over a field $k$, the two lines
$L_{0}=(y=0),L_{1}=(y=1)$ and the two points $s_{0}=(0,0)\in
L_{0},s_{1}=(0,1)\in L_{1}$. Let $\mathcal{S}$ be the surface obtained from
$\mathcal{S}^{n}$ by gluing $s_{0}$ and $s_{1}$ together, and $\mathcal{T}$ be
the surface obtained from $\mathcal{S}^{n}$ by gluing $L_{0}$ and $L_{1}$
along the involution $L_{0}\ni(x,0)\longleftrightarrow(x,1)\in L_{1}$. Then
$\mathcal{T}$ is demi-normal, it is also a quotient of $\mathcal{S}$ and we
have a commutative diagram

${\mathcal{S}^{n}}$${\mathcal{S}}$${\mathcal{T}}$

However $\mathcal{S}$ is $R_{1}$ but not $S_{2}$.

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