\end{tikzcd}$$

Since $\phi$ is crepant and a local log isomorphism at $S_1$, we can write

where $\coeff_{S_W}\Gamma=0$. Thus by adjunction

\begin{eqnarray*}
(u|_{S^n_W})^*(K_{S_1}+\Diff^*_{S_1}\Delta_1) & = & n^*u^*(K_{Y_1}+\Delta_1) \\
& = & n^*(K_W+S_W+\Gamma) \\
& = & K_{S_W^n}+\Diff_{S_W^n}\Gamma
\end{eqnarray*}

and similarly $(v|_{S^n_W})^*(K_{S_2^n}+\Diff_{S_2^n}\Delta_2)=K_{S_W^n}+\Diff_{S_W^n}\Gamma$. This implies that the birational map $\phi|_{S_1}\colon (S_1,\Diff^*_{S_1}\Delta_1)\dashrightarrow (S_2^n,\Diff_{S_2^n}\Delta_2)$ is crepant.

§.§.§ Geometry of lc lcenters

Let $(Y,\Delta_Y)\to (X,\Delta)$ be as in <ref>, $Z\subset X$ an lc center of $(X,\Delta)$ and $S,S'\subset Y$ lc centers of $(Y,\Delta_Y)$ that are minimal for the property of dominating $Z$. In characteristic $0$, a crucial observation due to Kollár [39] is that $S$ and $S'$ are birational, and that the birational map arises from a special structure which he calls a standard $\bP^1$-link. In this section, we prove some analogous statements in positive characteristic.

We begin by defining the $\bP^1$-links.

A weak standard $\bP^1$-link is a $\bQ$-factorial pair $(T, W_1+W_2+E)$ together with a proper morphism $\pi\colon T\to W$ such that

* $K_T+W_1+W_2+E\sim_{\bQ,\pi} 0$,

* $W_1$ and $W_2$ are disjoint and normal,

* both $\pi\colon W_i\to W$ are isomorphisms, and

* $\red(T_w)\cong \bP^1_{k(w)}$ for every $w\in W$.

In Kollár's definition of standard $\bP^1$-link <cit.>, it is assumed that $(T,W_1+W_2+E)$ is plt. Let us justify our deviation. The use of (weak) $\bP^1$-links is motivated by <ref> below, which is analogous to <cit.>. In both cases, during the proof we find a weak $\bP^1$-link $(T,W_1+W_2+E)\to W$ such that $(T,E)$ is klt. Using cyclic covers, we can see that $(T,W_1+W_2+E)$ is locally the quotient of a product $(\tilde{W}\times\bP^1, \tilde{W}\times [0;1]+\tilde{W}\times[1;0]+E_{\tilde{W}}\times\bP^1)$. By inversion of adjunction, this product is plt. In characteristic $0$ this implies that the quotient is plt, but this is not necessarily the case in positive characteristic.

However, our weaker assumption on standard links is sufficient for our purpose, as we shall see.

Let $\pi\colon (T,W_1+W_2+E)\to W$ be a weak standard $\bP^1$-link over a field of characteristic $>2$. Then $\Supp(E)$ is a union of fibers and there is a log isomorphism $(W_1,\Diff_{W_1}E)\cong (W_2,\Diff_{W_2}E)$ that commutes with the projections onto $W$.

A general fiber $F$ is normal by <ref>, and by assumption $K_F$ is anti-ample. In particular $2=-K_T\cdot F=(W_1+W_2+E)\cdot F$. Since $W_i\cdot F=1$, we deduce that the support of $E$ is a union of fibers. Since $W_i\to W$ are isomorphisms, the projections provide an isomorphism $\phi\colon W_1\cong W_2$ that commutes with the projections $\pi_i\colon W_i\to W$. The morphism $\phi\colon (W_1,\Diff_{W_1}E)\to (W_2, \Diff_{W_2}E)$ is crepant by <ref>.

Let $(S, \Delta)$ be a quasi-projective klt surface pair defined over a field of characteristic $>2$, with a projective morphism $f\colon S\to Z$. Let $D$ be an effective $\bZ$-divisor that dominates $Z$, such that $K_S+\Delta+D\sim_{\bQ,f}0$ and $(S,D+\Delta)$ is dlt. Pick a point $z\in Z$ such that $f^{-1}(z)$ is connected but $f^{-1}(z)\cap D$ is disconnected.

Then there exists an elementary étale morphism $(z'\in Z')\to (z\in Z)$ and a proper morphism $W\to Z'$ such that $(S,\Delta+D)\times_Z Z'=(S', \Delta'+D')$ is birational to a weak standard $\bP^1$-link over $W$.

Given an elementary étale neighborhood $(z'\in Z')\to (z\in Z)$, we form the Cartesian diagram

(S',D'+\Delta')\arrow[r]\arrow[d, "f'"] & (S,D+\Delta)\arrow[d, "f"] \\
Z'\arrow[r] & Z
\end{tikzcd}$$

We can choose $Z'\to Z$ such that the different connected components of $(f')^{-1}(z')\cap D'$ are contained in different connected components of $D'$. Notice that $(S',\Delta')$ is klt, $(S',D'+\Delta')$ is dlt, $K_{S'}+D'+\Delta'\sim_{\bQ,f'} 0$ and $(f')^{-1}(z')$ is connected.

By <cit.>, we may run an MMP for $(S',\Delta')$ over $Z'$. Since $K_{S'}+\Delta'\sim_{\bQ,f'}-D'$ is not pseudo-effective on the generic fiber of $f'$, the MMP terminates with a Fano contraction $p\colon (S^*,\Delta^*)\to W$ over $Z'$. We picture our construction as follows:

(S',D'+\Delta') \arrow[rr,"\varphi", dotted]\arrow[d, "f'"] && (S^*,D^*+\Delta^*)\arrow[d, "p"]\arrow[dll, "g"] \\
Z' && W\arrow[ll]
\end{tikzcd}$$

where $D^*=\varphi_*D$ and $\Delta^*=\varphi_*\Delta$. (Since $S'$ is a surface, $\varphi$ is actually a morphism.)

By construction we have $K_{S^*}+D^*+\Delta^*\sim_{\bQ,g}0$, so in particular $K_{S^*}+D^*+\Delta^*\sim_{\bQ,p}0$. Moreover $\varphi\colon (S',D'+\Delta')\dashrightarrow (S^*,D^*+\Delta^*)$ is crepant, so it follows from <ref> that $g^{-1}(z')\cap D^*$ is still disconnected.

Since $p$ is a Fano contraction, there is a component $D_1^*\subset D^*$ that has positive intersection with the contracted ray inducing $p\colon S^*\to W$. Thus $D_1^*$ is $p$-ample. Now there is another component $D_2^*\subset D^*$ that is disjoint from $D^*_1$. Take a curve $C$ which intersects $D_2^*$ and is contained in a fiber of $p$. The intersection $C\cdot D_2^*$ cannot be negative, otherwise the fiber would be included in $D_2^*$, and $D_1^*$ and $D_2^*$ would not be disjoint. So $D_2^*$ also has positive intersection with the contracted ray, and is also $p$-ample.

So $D_1^*$ and $D_2^*$ are disjoint and have positive intersection with every curve contracted by $p$. Assume that a fiber $F$ of $p$ has dimension at least $2$: then $F\cap D_1^*$ must be at least $1$-dimensional, hence it intersects $D^*_2$, contradiction. Thus the fibers of $p$ have dimension $1$. Since $(S^*,\Delta^*)$ is a klt surface, it is $\bQ$-factorial and $K_{S^*}$ is $\bQ$-Cartier. As $K_{S^*}+\Delta^*$ is $p$-anti-ample, it follows that $K_{S^*}$ is $p$-anti-ample. By <ref>, a general fiber $F$ of $p$ is smooth rational. As

$$F\cdot (\Delta^*+D^*)=-K_{S^*}\cdot F=2,$$

we deduce via <ref> that $D_1^*$ and $D_2^*$ are sections of $p$ and that the other components of $\Delta^*+D^*$ are vertical over $W$. If there was another component of $D^*$ whose image under $g$ contained $z'$, then that component would also contain $g^{-1}(z)$, and thus $g^{-1}(z)\cap D^*$ would not be disconnected. Thus we can shrink $Z'$ and assume that $D^*=D^*_1+D_2^*$.

By <cit.>, we have $R^1p_*\sO_{S^*}=0$, and thus every reduced fiber is a tree of $\bP^1$. By intersecting with the $D_i^*$ we see these trees must be irreducible.

Since $S^*$ is a normal surface, $W$ is a normal curve, and as $D^*_i$ are sections of $p$ we deduce that they are normal.

Thus $p\colon (S^*,D_1^*+D_2^*+\Delta^*)\to W$ is a weak standard $\bP^1$-link. This proves the proposition.

Let $f\colon (Y,\Delta_Y)\to (X,\Delta)$ be as in <ref>, and let $Z_1,Z_2\subset Y$ be two lc centers. We say that $Z_1$ and $Z_2$ are directly weakly $\bP^1$-linked if there exists an lc center $W\subseteq Y$ (we allow $W=Y$) such that $f(Z_1)=f(Z_2)=f(W)$ and $(W,\Diff^*_W\Delta_Y)$ is crepant birational to a $\bP^1$-link with the $Z_i$ as the two sections, and the base of the $\bP^1$-link factorizes the morphism to $X$.

We say that $Z_1$ and $Z_2$ are weakly $\bP^1$-linked if there is a sequence of lc centers $Z_1',\dots,Z_m'$ with $Z_1=Z_1'$ and $Z_2=Z_m'$, such that $Z_i'$ and $Z_{i+1}'$ are directly weakly $\bP^1$-linked.

Let $f\colon (Y,\Delta_Y)\to (X,\Delta)$ be as in <ref>, and let $S_1,S_2\subset Y$ be two weakly $\bP^1$-linked lc centers. Then:

* $(S_1,\Diff^*_{S_1}\Delta_Y)$ and $(S_2,\Diff^*_{S_2}\Delta_Y)$ are crepant birational over $X$, and

* $(S_1,\Diff^*_{S_1}\Delta_Y)$ is klt if and only if $(S_2,\Diff^*_{S_2}\Delta_Y)$ is klt.

We may assume that $S_1$ and $S_2$ are directly weakly $\bP^1$-linked. Then there is a lc center $T\subset Y$ containing both $S_i$'s, a weak standard $\bP^1$-link $(T', W_1+W_2+E)\to W$ and a crepant birational map $\phi\colon (T,\Diff^*_T\Delta_Y)\dashrightarrow (T',W_1+W_2+E)$ such that the map $T\dashrightarrow T'\to W$ factors the morphism $T\to X$, and such that $\phi$ maps birationally $S_i$ to $W_i$.

By <ref>, $\phi$ induces crepant birational $X$-maps $(S_i,\Diff^*_{S_i}\Delta_Y)\dashrightarrow (W_i,\Diff_{W_i}E)$. Composing these maps with the log isomorphism $(W_1,\Diff_{W_1}E)\cong (W_2,\Diff_{W_2}E)$ over $W$ given by <ref>, we obtain that the $(S_i,\Diff_{S_i}^*\Delta_Y)$ are crepant birational over $X$.

The second assertion follows immediately from the first one.

Let $f\colon (Y,\Delta_Y)\to (X,\Delta)$ be as in <ref>. Fix $T\subset X$ an lc center of $(X,\Delta)$. Then the lc centers of $(Y,\Delta_Y)$ that are minimal for dominating $T$, are weakly $\bP^1$-linked to each other.

We prove the following more precise result:

Given a point $x\in X$, an lc center $Z\subset Y$ of $(Y,\Delta_Y)$ that is minimal for the property $x\in f(Z)$, an lc center $W\subset Y$ such that $x\in f(W)$, then there exists an lc center $Z_W\subset W$ such that $Z$ and $Z_W$ are weakly $\bP^1$-linked.

By <ref> this claim implies the result, taking $x$ to be the generic point of $T$.

First we prove our claim holds after passing to an elementary étale neighborhood of $(x\in X)$. Then we will show that the weak $\bP^1$-link descends along the étale base-change.

By <cit.>, the fibers of

$$\Supp\Delta_Y^{=1}=\Nklt(Y,\Delta_Y)\longrightarrow X$$

are geometrically connected. After passing to an elementary étale neighborhood of $(x\in X)$, we may assume that $\Delta_Y^{=1}=\sum_i \Delta_i$, that each $\Delta_i$ has a connected fiber above $x$ and that every lc center of $(Y,\Delta_Y)$ intersects $f^{-1}(x)$. Relabelling the $\Delta_i$ is necessary, we assume that $Z\subset \Delta_1,W\subset \Delta_r$ and $\Delta_i\cap \Delta_{i+1}\cap f^{-1}(x)\neq \emptyset$ for all $i=1,\dots,r-1$.

Now we prove that $Z\subset \Delta_1$ is $\bP^1$-linked (after the étale base-change) to an lc center contained in $\Delta_1\cap \Delta_2$. For ease of notation, we let $S:=\Delta_1$, $D:=\Diff^*_{S}\Delta_Y$, $E:=\Delta_1\cap \Delta_2$. Then $(S,D)$ is dlt, admits a projective morphism $g=f|_S\colon S\to X$ with connected fibers such that $K_S+D\sim_{\bQ,g}0$. Notice that $ E\subset \Supp D^{=1}$ and that $Z\subset \Supp D^{=1}\cap g^{-1}(x)$. We distinguish a few cases according to the dimension of $g(S)$.

* Suppose that $\dim g(S)=2$. Then $R^1g_*\sO_S(-D^{=1})=0$ by <ref>, so the natural map $g_*\sO_S\to g_*\sO_{D^{=1}}$ is surjective. If $D^{=1}\cap g^{-1}(x)$ was disconnected, then we would get a section of $g_*\sO_S$ that vanishes on one connected component of $g^{-1}(x)\cap D^{=1}$, and is identically $1$ on another. But $g^{-1}(x)$ is connected and sections of $\sO_{g^{-1}(x)_\text{red}}$ are supported on entire components, so we get a contradiction. Thus $D^{=1}\cap g^{-1}(x)$ is connected.

The scheme $D^{=1}\cap g^{-1}(x)$ is either $0$-dimensional or $1$-dimensional. If it is $0$-dimensional then $D^{=1}\cap g^{-1}(x)=Z$ and we are done.

Assume that $D^{=1}\cap g^{-1}(x)$ is $1$-dimensional. Then $Z$ is an intersection point of two components of $D^{=1}$ above $g^{-1}(x)$. If $Z$ is the only such point, then we are done. If not, then by adjunction we easily see that each component of $D^{=1}\cap g^{-1}(x)$ must support two such points. In particular these points are $k(x)$-points, and the components of $D^{=1}\cap g^{-1}(x)$ are isomorphic to $\bP^1_{k(x)}$.

* Suppose that $\dim g(S)=1$. First we consider the case where $x$ is the generic point of $C:=g(S)$. Then $D^{=1}\cap g^{-1}(x)$ is the set of generic points of those components of $D^{=1}$ that dominate $C$. If it is connected, then $D^{=1}\cap g^{-1}(x)=Z$ and we are done. If $D^{=1}\cap g^{-1}(x)$ is not connected, then since $D^{=1}$ dominates $C$ we may use <ref> to obtain that, after possibly a further elementary étale base-change, $D^{=1}\cap g^{-1}(x)$ has two connected components that are $\bP^1$-linked to each other.

Now assume that $x\in C$ is a closed point. If $D^{=1}\cap g^{-1}(x)$ is connected, we can apply the same analysis as in the previous point.
If $D^{=1}\cap g^{-1}(x)$ is disconnected and $D^{=1}$ dominates $C$, then we can use <ref>.

Finally, if $D^{=1}$ does not dominate $C$, we show that $D^{=1}\cap g^{-1}(x)$ is connected, hence all cases have been covered. Indeed, if $D^{=1}$ does not dominate $C$ then it is supported on fibers, and $K_S+D^{<1}\sim_{\bQ,g}-D^{=1}$ is pseudo-effective on the generic one. By [54] we can run a $(K_S+D^{<1})$-MMP over $C$, which terminates with a birational model $S'$ of $S$, say

(S,D^{<1},D^{=1})\arrow[dr, "g" below left]\arrow[rr, "\varphi", dotted] && (S',\Delta':=\varphi_*D^{<1},\Gamma':=\varphi_*D^{=1})\arrow[dl, "g'"] \\
& C &
\end{tikzcd}$$

such that $K_{S'}+\Delta'\sim_{\bQ,g'}-\Gamma'$ is $g'$-nef. (The map $\varphi$ is actually a morphism.) Notice that:

* By <ref> the map $\varphi\colon (S,D)\dashrightarrow (S',\Delta'+\Gamma')$ is crepant;

* Since $K_{S'}+\Delta'+\Gamma'\sim_{\bQ,g'}0$ and $\Gamma'$ is vertical over $C$, we see that $(K_{S'}+\Delta')\cdot (g')^*M=0$ for any $\bQ$-Cartier divisor $M$ on $C$;

* $\Gamma'\neq 0$. For if $\varphi$ would contract $D^{=1}$, then looking at in intermediate step of the MMP we may assume that $D^{=1}$ is irreducible. But then $(K_S+D^{<1})\cdot D^{=1}=-(D^{=1})^2$ is non-negative by <ref>, which is a contradiction.

Now we claim that $\Gamma'\cap (g')^{-1}(x)$ is connected for any $x\in C'$. By <ref> we have $(\Gamma')^2\leq 0$, with equality if and only if $\Gamma'$ is a weighted sum of fibers. In any case we can write

$$-\Gamma'\sim_{\bQ} (g')^*N+E$$

where $N$ is $\bQ$-Cartier on $C$ and $E$ is an effective $\bQ$-divisor supported on the fibers. Hence we have

$$0\geq (\Gamma')^2=(K_{S'}+\Delta')\cdot (-\Gamma')=(K_{S'}+\Delta')\cdot ((g')^*N+E)\geq 0,$$

thus $(\Gamma')^2=0$. Since $g'\colon S'\to C$ has connected fibers, we deduce that every $\Gamma'\cap (g')^{-1}(x)$ is connected.

Since $(S,D^{<1})$ is klt and $\varphi$ is a $(K_S+D^{<1})$-MMP, the pair $(S',\Delta')$ is also klt. In particular $\Nklt(S',\Delta'+\Gamma')=\Gamma'$. On the other hand, since $(S,D)$ is dlt we have $\Nklt(S,D)=D^{=1}$. Therefore, since $\varphi\colon (S,D)\dashrightarrow (S',\Delta'+\Gamma')$ is a crepant $C$-map, by <ref> we have a bijection between the connected components of $D^{=1}\cap g^{-1}(x)$ and the connected components of $\Gamma'\cap (g')^{-1}(x)$. In particular $D^{=1}\cap g^{-1}(x)$ is connected.

* Finally, suppose that $\dim g(S)=0$. If $D^{=1}\cap g^{-1}(x)$ is connected, we apply the same analysis as in the first point. If $D^{=1}\cap g^{-1}(x)$ is disconnected, we can use <ref>.

It remains to show that the weak $\bP^1$-link descends the étale base-change. More precisely, let $Z_i\subset Y$ be the lc centers that are minimal for the property $x\in f(Z_i)$. We have proved that after an elementary étale base-change $(x'\in X')\to (x\in X)$, the $Z_i':=Z_i\times_XX'$ are weakly $\bP^1$-linked. In particular they have the same image in $X'$, and thus the $Z_i$ have the same image in $X$, say $V$. Let $v\in X$ be the generic point of $V$. Notice that the lc centers of $(Y,\Delta_Y)$ that are minimal above $v$ are exactly the $Z_i$'s. Thus we can refine the earlier statement: there is a elementary étale neighborhood $(\tilde{v}\in \tilde{X})\to (v\in X)$ such that the $\tilde{Z}_i:=Z_i\times_X\tilde{X}$ are $\bP^1$-linked. Since $k(\tilde{v})=k(v)$, the morphisms $\tilde{Z}_i\to Z_i$ are generical isomorphisms. Thus the weak $\bP^1$-links between the $\tilde{Z}_i$'s descend to weak $\bP^1$-links between the $Z_i$'s.

The same argument shows that <ref> also holds for an arbitrary quasi-projective lc surface pair $(S,\Delta)$ over a field and a crepant dlt blow-up $(S',\Delta')\to (S,\Delta)$. This can also be shown by considering the minimal resolution of $(S,\Delta)$.

Let $f\colon (Y,\Delta_Y)\to (X,\Delta)$ be as in <ref>, and $S\subset Y$ a lc center of $(Y,\Delta_Y)$. Denote by

$$S\overset{f_S}{\longrightarrow} X_S\overset{\pi}{\longrightarrow} X$$

the Stein factorization of $f|_S$. If $Z\subset X$ is an lc center of $(X,\Delta)$, then every irreducible component of $\pi^{-1}Z\subset X_S$ is the image of an lc center of $(S,\Delta_S:=\Diff^*_S\Delta_Y)$.

Choose a minimal lc center $V\subset Y$ above $Z$. By <ref>, there is a minimal lc center $V'\subseteq Y$ that dominates $Z$ and that is contained in $S$. By adjunction for dlt pairs, $(V',\Diff^*_{V'}\Delta_Y)$ is an lc center of $(S,\Delta_S)$. Then $f_S(V')\subset X_S$ is one of the irreducible components of $\pi^{-1}Z$.

Now let $\eta\in X$ be the generic point of $Z$. After passing to an elementary étale neighborhood of $(\eta\in X)$, we can assume that $S$ is the union of irreducible components $S_j$ where each $f^{-1}(\eta)\cap S_j$ is connected. The previous argument show that the irreducible components of $\pi^{-1}Z$ are images of lc centers of the $S_j$, and these lc centers descend to $S$ by <cit.>.

Let $(X,\Delta)$ be a quasi-projective slc threefold pair over a perfect field of characteristic $>5$. Then any intersection of lc centers is a union of lc centers.

The normalization $(\bar{X},\bar{D}+\bar{\Delta})\to (X,\Delta)$ is crepant and $(\bar{X},\bar{D}+\bar{\Delta})$ is lc. Thus we may assume that $(X,\Delta)$ is lc. Let $Z,Z'\subset X$ be lc centers, and pick a point $x\in Z\cap Z'$. If $f\colon (Y,\Delta_Y)\to (X,\Delta)$ is a crepant dlt blow-up, then by <ref> we can find a minimal lc center $W$ of $(Y,\Delta_Y)$ above $x$ whose image $f(W)$ is contained in $Z\cap Z'$. Since $f(W)$ is an lc center of $(X,\Delta)$, the result is proved.

Let $(X,\Delta)$ be a quasi-projective slc threefold pair over a perfect field of characteristic $>5$. Then the minimal lc centers of $(X,\Delta)$ are normal up to universal homeomorphism.

Let $Z\subset X$ be a minimal lc center of $(X,\Delta)$, and $z\in Z$ be any point. Choose an étale neighborhood $(z'\in Z')\to (z\in Z)$. After shrinking it if necessary, we may assume that it is a standard étale neighborhood <cit.> and thus there is an étale neighborhood $(z'\in X')\to (z\in X)$ such that $Z'=Z\times_XX'$. Hence $Z'$ is an lc center of $(X',\Delta')$ and it is connected. If it was reducible, then the intersection of its components would be a union of lc centers by <ref>, and their images in $X$ would also be lc centers <cit.>. This contradicts the minimality of $Z$. Thus $(z'\in Z')$ is irreducible.

By <cit.> we obtain that $Z$ is geometrically unibranch. So the normalization morphism $Z^n\to Z$ is universally bijective <cit.>. Since it is surjective and universally closed, we obtain that that $Z^n\to Z$ is a universal homeomorphism.

§.§.§ Springs and sources for the reduced boundary

We are now able to define sources of lc centers, analogously to <cit.>. The theory of sources should be thought of as higher codimension version of adjunction for divisors. However we only define sources for lc centers that are contained in the reduced boundary: see <ref> below.

Let $f\colon (Y,\Delta_Y)\to (X,\Delta)$ be as in <ref>. Let $Z\subset X$ be a lc center contained in $\Delta^{=1}$ with normalization $n\colon Z^n\to Z$.

Let $(S,\Delta_S:=\Diff^*_{S}\Delta_Y)\subset Y$ be a minimal lc center above $Z$, with Stein factorization $f^n_S\colon S\to Z_S\to Z^n$.

* Uniqueness of sources. The crepant birational class of $(S,\Delta_S)$ over $Z$ does not depend on the choice of $S$. We call it the source of $Z$, and denote it by $\src(Z, Y,\Delta_Y)$.

* Uniqueness of springs. The isomorphism class of $Z_S$ over $Z$ does not depend on the choice of $S$. We call it the spring of $Z$, and denote it by $\spr(Z,Y,\Delta_Y)$.

* Crepant log structure. $(S,\Delta_S)$ is dlt, $K_S+\Delta_S\sim_{\bQ,Z}0$ and $(S,\Delta_S)$ is klt on the general fiber above $Z$.

* Galois property. The field extension $k(Z)\subset k(Z_S)$ is Galois and the morphism $\Bir^c_Z(S,\Delta_S)\to\Gal(Z_S/Z)$ is surjective.

* Poincaré residue map. For $m>0$ divisible enough, there are well-defined isomorphisms

$$\omega_Y^{[m]}(m\Delta_Y)|_S\cong \omega_S^{[m]}(m\Delta_S)$$

$$n^*(f_*(\omega_Y^{[m]}(m\Delta_Y)))\cong ((f^n_S)_*\omega_S^{[m]}(m\Delta_S))^{\Bir^c_Z(S,\Delta_S)}.$$

* Birational invariance. Let $(Y',\Delta_{Y'})$ be another crepant dlt blow-up of $(X,\Delta)$. Then

\begin{eqnarray*}
\src(Z,Y,\Delta_Y)&\overset{\emph{cbir}}{\cong}& \src(Z,Y',\Delta_{Y'}), \\
\spr(Z,Y,\Delta_Y)&\cong & \spr(Z,Y',\Delta_{Y'})
\end{eqnarray*}

over $Z$. In particular, we may write $\src(Z,X,\Delta)$ for the source, and $\spr(Z,X,\Delta)$ for the spring of $Z\subset X$.

For clarity, we divide the proof into four steps and several claims.

Step 1: Sources, springs and invariance. If $S'\subset Y$ is another minimal lc center above $Z$, then by <ref> and <ref> there is a crepant birational map $(S,\Delta_S)\dashrightarrow (S',\Delta_{S'})$ over $Z$ given by a composition of weak direct $\bP^1$-links. Since $Z_S=\Spec_{Z^n}f_*\sO_S$, we obtain a $Z$-isomorphism $Z_S\cong Z_{S'}$. This shows the uniqueness of $(S,\Delta_S)$ up to crepant birational maps over $Z$, and the uniqueness of $Z_S$ up to isomorphisms over $Z$.

The fact that $(S,\Delta_S)$ is dlt follows from <ref>, and that fact that it is klt on the generic fiber above $Z$ holds by <ref> and the fact that $S$ is minimal.

To obtain birational invariance, we apply <cit.> to a common log resolution of $(Y,\Delta_Y)$ and $(Y',\Delta_{Y'})$. (The proof of <cit.> uses <cit.>, which we replace by <ref>).

The following observation will be crucial for the rest of the proof:

If $\dim Z=2$ then $\dim \src(Z,Y,\Delta_Y)=2$. If $\dim Z\leq 1$ then $\dim \src(Z,Y,\Delta_Y)\leq 1$.

Indeed, if $Z$ is a divisor, then $S$ can be chosen to be the strict transform $Z_Y$ of $Z$. If $Z$ is a curve or a single point, contained in a component $\Gamma$ of $\Delta^{=1}$, then by <ref> we can choose $S$ to be contained in $\Gamma_Y$. Since $\Gamma_Y$ is $2$-dimensional and not $f$-exceptional, $S$ is at most $1$-dimensional. Since the dimension of $S$ depends only on $Z$, our claim is proved.

Step 2: Poincaré residue. To obtain the Poincaré residue map, let $m>0$ be even and sufficiently divisible such that $\omega_Y^{[m]}(m\Delta_Y)\sim f^*L$ for some line bundle $L$ on $X$. By <ref> we have isomorphisms

$$\sR^m_{Y,S}\colon f^*L|_S\cong \omega_Y^{[m]}(m\Delta_Y)\overset{\sim}{\longrightarrow} \omega_S^{[m]}(m\Delta_S).$$

However the choice of $S$ is in general not unique. The following claim shows how the maps $\sR^m_{Y,S}$ relate for different choices of $S$.

Let $(S',\Delta_{S'})$ be another minimal lc center above $Z$. Then there is a crepant birational map $\phi\colon (S,\Delta_S)\dashrightarrow (S',\Delta_{S'})$ such the diagram

\begin{equation}\label{diagram:Poincare_residue_diagram}
\begin{tikzcd}
\omega_Y^{[m]}(m\Delta_Y)
\arrow[r, "\sim"] \arrow[d, "\sR_{Y,S}^m"]
& f^*L
\arrow[r, "\sim"]
& \omega_Y^{[m]}(m\Delta_Y)
\arrow[d, "\sR_{Y,S'}^m"]
\\
\omega_S^{[m]}(m\Delta_S)
\arrow[rr, "\phi^*"]
&& \omega_{S'}^{[m]}(m\Delta_{S'})
\end{tikzcd}
\end{equation}

is commutative.

Indeed, we may assume that $S$ and $S'$ are directly weakly $\bP^1$-linked. Then there is a lc center $T\subset Y$ containing both $S$ and $S'$, which is birational to the total space of a weak standard $\bP^1$-link $T'\to W$ whose sections map birationally to $S$ and $S'$. Moreover, the map $T\dashrightarrow W$ factors the morphism $T\to f(T)$. The induced projections $S\dashrightarrow W \dashleftarrow S'$ are birational, and induce a birational map $\phi\colon S\dashrightarrow S'$, which we claim is the map we are looking for.

Since $\phi$ is obtained from a $\bP^1$-link, by <ref> it is crepant.

To prove the commutativity of the diagram, since $\sR^m_{T,S}\circ\sR^m_{Y,T}=\sR^m_{Y,S}$, we may assume that $Y=T$. In this case, note that $S,S'\subset \Delta_Y^{=1}$. Moreover we are dealing with torsion-free sheaves, so it is enough to check commutativity generically. Thus we may assume that we have a standard weak $\bP^1$-link $X\to W$ factorizing $f$, with sections $S$ and $S'$. Localizing at the generic point of $W$, we may furthermore assume that $W$ is the spectrum of a field $L$ and that $X=\bP^1_L$ <cit.>. In this case $\Delta_Y=S+S'$ and we may choose coordinates $x,y$ on $X$ such that $S=[0;1]$ and $S'=[1;0]$. Then a generator of $H^0(\bP^1_L,\omega_{\bP^1_L}(S+S'))$ is $dx/x$, and

$$\sR^1_{Y,S}(dx/x)=1,\quad \sR^1_{Y,S'}(dx/x)=-1$$

while $\phi^*$ is the identity map on $L$. Thus <ref> indeed commutes for $m$ even.

If we think about $(S,\Delta_S)$ as a crepant birational class, then <ref> shows that we can define a Poincaré residue map $\sR^m_{Y,S}\colon \omega_Y^{[m]}(m\Delta_Y)\to \omega_S^{[m]}(m\Delta_S)$, up to the action of the group $\Bir^c_Z(S,\Delta_S)$ on the target. We can remedy to this ambiguity using the following claim.

The group $\Bir^c_Z(S,\Delta_S)$ acts on $\omega_S^{[m]}(m\Delta_S)$ as a finite group of roots of unity in $k(Z)$.

We are dealing with torsion-free sheaves and with a group action that commutes with the projection to $Z$. So to understand the action, we can localize over the generic point of $Z$. Then we obtain a proper $k(Z)$-pair $(S_{k(Z)},\Delta_{S_{k(Z)}})$ such that $\omega_{S_{k(Z)}}^{[m]}(m\Delta_{S_{k(Z)}})$ is trivial, and we must show that the action of $\Bir^c_{k(Z)}(S_{k(Z)},\Delta_{S_{k(Z)}})$ on the $1$-dimensional $k(Z)$-vector space $H^0(S_{k(Z)},\omega_{S_{k(Z)}}^{[m]}(m\Delta_{S_{k(Z)}}))$is finite. If $\dim S=\dim Z$ then $S_{k(Z)}$ is the spectrum of a finite field extension of $k(Z)$, and so $\Aut_{k(Z)}(S_{k(Z)})$ is finite. By <ref>, the only case left is when $Z=\{x\}$ is a closed point of $X$ and $(S,\Delta_S)$ is a proper $1$-dimensional klt pair  over $k(x)$ such that $\omega_S^{[m]}(m\Delta_S)$ is trivial. Since $k(x)$ is a perfect field, it follows from <ref> that $\Bir^c_{k(x)}(S,\Delta_S)$ acts finitely on the $1$-dimensional vector space $H^0(S,\omega_S^{[m]}(m\Delta_S))$.

So in every case $\Bir^c_Z(S,\Delta_S)$ acts finitely on the generic stalk of $\omega_S^{[m]}(m\Delta_S)$, hence through the multiplication with some $r^\text{th}$-root of unity in $k(Z)$.

Replacing $m$ by $mr$, the action becomes trivial and the ambiguity about the Poincaré residue map disappears.

Step 3: the Galois property. We wish to prove that the field extension $k(Z_S)/k(Z)$ is Galois.

The case where $Z$ is a divisor is the easiest, we treat it first. By <cit.> there is a unique lc center $S$ above $Z$ (namely, its strict transform) and the morphism $S\to Z$ has connected fibers in a neighborhood of the generic point of $Z$. Thus $S\to Z$ is birational, $\Bir^c_Z(S,\Delta_S)$ is trivial and $k(Z_S)=k(Z)$.

From now on, we assume that $\dim Z\leq 1$. To prove that the finite morphism $Z_S\to Z$ induces a Galois extension on the function fields, we may localize at the generic point of $Z$. Then the situation is the following: $(x\in X,\Delta)$ is a local lc pair of dimension $2$ or $3$, $(Y,\Delta_Y)\to (X,\Delta)$ is a crepant dlt blow-up and $(S,\Delta_S)\subset Y$ is an lc center of dimension $\leq 1$ such that the morphism $S\to X$ factorizes through the closed point, and $S$ is minimal for this property.

In this set-up, the fields of definition of the geometric connected components of $S$ are the same, and it is a Galois extension of $k(x)$.

Indeed, let $K^s$ be a separable closure of $k(x)$. Then $(Y,\Delta_Y)\times_{k(x)}K^s$ is a dlt $\bQ$-factorial pair, and every component of $S_{K^s}$ is an lc center that is minimal. Let $W$ be one of them, with field of definition $F$. Then every lc center containing $W$ is also defined over $F$ <cit.>, and so any lc center that is weakly $\bP^1$-linked to $W$ is also defined over $F$. By <ref> (see <ref>) we obtain that all the components of $S_{K^s}$ are defined over $F$. If $\sigma$ is an element of the Galois group of the Galois closure of $k(x)\subset F$, then $W^\sigma$ is a minimal lc center defined over the conjugate field $F^\sigma$. Therefore $F^\sigma=F$, so we see that $k(x)\subset F$ is Galois.

$H^0(S,\sO_S)$ is a separable field extension of $k(x)$.

If $(x\in X)$ has dimension $3$, this holds because $k(x)$ is perfect (since in this case we localized at a closed point). If $(x\in X)$ has dimension $2$, then $(x\in X,\Delta)$ is a local lc surface singularity with the property that $\Delta^{=1}\neq 0$. (By <cit.>, these $(x\in X)$ are rational surface singularities.) The possible dual graphs of the minimal resolutions $(T,\Gamma)$ of such pairs are classified, see for example <cit.>. Inspecting them, we see that if $C\subset T$ is an exceptional proper curve above $x$, then $\dim_{k(x)}H^0(C,\sO_C)\leq 4$, and if $C'$ is another exceptional proper curve, then $C\cdot C'=\length_{k(x)}C\cap C'\leq 4$. Hence if $E\subset T$ is a lc place of $(x\in X,\Delta)$ then $H^0(E,\sO_E)$ is a separable extension of $k(x)$ provided that $\Char k(x)\geq 5$.

$H^0(S,\sO_S)$ is a Galois extension of $k(x)$.

By <ref> and <ref>, every connected component of $S_{K^s}$ is already defined over the Galois closure $K$ of $k(x)\subset H^0(S,\sO_S)$. On the other hand,

the field of definition of these components is a Galois extension of $k(x)$ by <ref>. So we deduce that $K$ is the field of definition of every connected component of $S_{K^s}$.

It remains to show that $H^0(S,\sO_S)=K$. Consider the
Cartesian diagram

S\arrow[d] & S_K=\bigcup_i S_K^{(i)}\arrow[l]\arrow[d] \\
k(x) & K\arrow[l]
\end{tikzcd}$$

where the $S_K^{(i)}$ are the irreducible components of $S_K$. As before, $(Y,\Delta_Y)\times_{k(x)} K$ is a dlt pair above $K$, and each $S_K^{(i)}$ is an lc center. Thus the intersection of the $S_K^{(i)}$ are also lc centers. By minimality of $S$, we obtain that the $S_K^{(i)}$ are disjoint. So the $S_K^{(i)}$ are also the connected components of $S_K$. Since $K$ is Galois over $k(x)$, the Galois group $G=\Gal(K/k(x))$ acts on $S_K$. Assume that some $S_K^{(j)}$ is stable under the action of a non-trivial subgroup of $G$. Then by <ref>, $S_K^{(j)}$ would be defined over a proper sub-extension $K'$ of $k(x)\subset K$. Then every $S_K^{(i)}$ would be defined over $K'$, which is a contradiction with the previous paragraph. Thus $G$ acts freely on the set of $S_K^{(i)}$'s. The action is also transitive, since $S_K\to S$ is the geometric quotient by $G$.

By flat base-change, we have

and the $G$-action is given by the action on $K$. On the other hand, since $S_K$ is the disjoint union of the $S_K^{(i)}$ and $G$ permutes them freely, we also have

$$H^0(S_K,\sO_{S_K})=\prod_{\sigma\in G} K$$

where $G$ acts by permuting the factors. Taking $G$-invariants, we obtain

$$H^0(S,\sO_S)=H^0(S,\sO_S)\otimes_{k(x)}K^G=\left(H^0(S,\sO_S)\otimes_{k(x)}K\right)^G=\left(\prod_{\sigma\in G}K\right)^G=K$$

as desired.

Step 4: Galois group and crepant birational maps.
It remains to show that every element of the Galois group $\Gal(Z_S/Z)$ is induced by a crepant birational self-map of $(S,\Delta_S)$.

There is nothing to prove if $\dim Z=2$, as noticed at the beginning of the previous step.

From now on assume that $\dim Z\leq 1$. We have proved in the previous step that $K:=k(Z_S)$ is Galois over $k(Z)$. Let $W$ be a component of $S\times_{k(Z)}K$ and pick $\sigma\in \Gal(K/k(Z))$. The proof of <ref> shows that $W$ is defined over $K$, so it has a conjugate $W^\sigma\subset S\times_{k(Z)}K$. Fix a weak $\bP^1$-link between $W$ and $W^\sigma$ inside $Y\times_X K$: then the $\Gal(K/k(Z))$-orbit of this link descends to an element of $\Bir^c_{k(Z)}((S,\Delta_S)\times_{Z}k(Z))$, and in turn this crepant birational map induces $\sigma$ on $H^0(S\times_{k(Z)}K,\sO)=K$.

This proves that Galois automorphisms of $k(Z)\subset k(Z_S)$ are induced by birational self-maps of $S$ which are generically $(K_S+\Delta_S)$-crepant over $Z$. We need to show that these maps are crepant, not only generically crepant. If $\dim Z=0$ there is nothing to show, so assume $\dim Z=1$. Then we make the following observation: $Z$ is contained in a component $D$ of $\Delta^{=1}$ and by adjunction any preimage of $Z$ in $D^n$ is a codimension one lc center of $(D^n,\Diff_{D^n}(\Delta-D))$. By the classification of lc surface singularities <cit.>, we see that $D$ is regular at the generic point of $Z$. Hence the strict transform of $Z$ must appear in $\lfloor \Diff_{D^n}(\Delta-D)\rfloor$, and by adjunction we get a natural boundary $\Gamma$ on $Z^n$. Moreover the equality $\dim S=\dim Z$ implies $S=Z_S$, so $f_S^n\colon S\to Z^n$ is a finite morphism. By the definition of $(S,\Delta_S)$ we must have

\begin{equation}\label{eqn:source_boundary_in_finite_case}
\end{equation}

On the other hand, the fact that $S=Z_S$ shows that the birational self-maps of $S$ we found are just the Galois automorphisms of $Z_S$ over $Z^n$. Since $\Delta_S$ can be defined by <ref>, we see that Galois automorphisms are crepant.

The proof is complete.

Let $(X,D+\Delta)$ be a quasi-projective lc threefold pair above a perfect field of characteristic $>5$, where $D$ is a reduced divisor with normalization $n\colon D^n\to D$. Let $Z\subset D$ be an lc center of $(X,D+\Delta)$, and $Z_D\subset D^n$ be an irreducible variety such that $n(Z_D)=Z$. Then:

* $Z_D$ is an lc center of $(D^n,\Diff_{D^n}\Delta)$,

* there is a commutative diagram

\src(Z_D, D^n,\Diff_{D^n}^*\Delta) \arrow[rr, dashed, "\emph{cbir}"] \arrow[d] && \src(Z,X,D+\Delta)\arrow[d] \\
D^n \arrow[rr, "n"] && D
\end{tikzcd}$$

* There is an isomorphism $\spr(Z_D,D^n,\Diff_{D^n}^*\Delta)\cong \spr(Z,X,D+\Delta)$.

By enlarging $\Delta$, we may assume that $D$ is irreducible. Choose a $\bQ$-factorial dlt crepant blow-up $(Y,\Delta_Y)$ of $(X,D+\Delta)$ and let $D_Y$ be the strict transform of $D$. The morphism $D_Y\to D$ is birational, so its Stein factorization is precisely

$$D_Y\overset{\sim}{\longrightarrow} D^n\to D,$$

which implies that $(D_Y,\Diff_{D_Y}^*\Delta_Y)=(D^n,\Diff_{D^n}\Delta)$. By <ref> there is a lc center $W\subset D_Y$ of $(Y,\Delta_Y)$ that dominates (actually is equal to) $Z_D\subset D^n$. By <ref> $W$ is also an lc center of $(D_Y,\Diff_{D_Y}^*\Delta_Y)$, which proves the first item. Moreover $W$ is a representative for both $\src(Z_D, D^n,\Diff_{D^n}\Delta)$ and $\src(Z,Y,\Delta_Y)$, and the second item follows by uniqueness of the source up to crepant birational map. The third item is a consequence of the second item.

The same method gives sources, springs, crepant log structure, adjunction and birational invariance of arbitrary lc centers of $(X,\Delta)$. However:

* The Galois property is problematic: if $Z$ is $1$-dimensional and not contained in any component of $\Delta^{=1}$, then $\sO_{X,Z}$ might by an elliptic or a cusp singularity, and the degrees of the exceptional curves might be arbitrarily large <cit.>. For example, we can take a product of such singularities with $\bP^1$. So inseparable extensions might appear.

* The Poincaré residue map can be defined, but it is not clear how to get rid of the $\Bir^c_Z(S,\Delta_S)$-ambiguity. Indeed, in constrast to <ref>, one more case can show up, namely $\dim S=2$ and $\dim Z=1$, and we were not able to show finiteness of representations with sufficient generality in this case.

§.§ Gluing theorems for threefolds in characteristic $>5$

We now prove the gluing theorems for lc threefolds. Our proofs follow closely those of Kollár <cit.>.

Let $(X,\Delta)$ and $(X',\Delta')$ be quasi-projective lc threefold pairs over a perfect field of characteristic $>5$, and $\phi\colon (X,\Delta)\cong (X',\Delta')$ a log isomorphism. Let $Z\subset X$ be an lc center of $(X,\Delta)$. Then:

* $Z':=\phi(Z)$ is an lc center of $(X',\Delta')$, and

* we have a commutative square

\src(Z,X,\Delta) \arrow[d]\arrow[r, dotted, "\emph{cbir}"] & \src(Z',X',\Delta')\arrow[d] \\
Z\arrow[r, "\phi"] & Z'
\end{tikzcd}$$

Since $\phi$ is a log isomorphism, $Z'$ is an lc center of $(X',\Delta')$. The second point follows from the birational invariance property proved in <ref>. The source is defined up to crepant birational map over the lc center, so there is no ambiguity about the commutativity of the diagram.

First we show that the geometric quotient exists.

Let $(X,D+\Delta)$ be a projective lc threefold pair over a perfect field $k$ of characteristic $>5$. Let $\tau\colon (D^n, \Diff_{D^n}\Delta)\cong (D^n, \Diff_{D^n}\Delta)$ be a generically fixed point free involution. Assume that $K_X+D+\Delta$ is ample. Then the induced equivalence relation $R(\tau)\rightrightarrows X$ is finite, the geometric quotient $X/R(\tau)$ exists, and $X/R$ is proper over $k$.

By <cit.>, the geometric quotient exists as soon as $R:=R(\tau)$ is finite. Notice that, by construction, a point of $\Supp(D)$ and a point of $X\setminus \Supp(D)$ cannot be in the same equivalence class. Moreover, $R$ restricts to the identity relation away from $\Supp(D)$. Thus we only need to prove that $R$ is finite on $\Supp(D)$.

For every lc center $Z$ of $(X,D+\Delta)$ contained in $D$, let $Z^0:= Z\setminus \text{(lower dimensional lc centers)}$ and let $\spr^0(Z,X,D+\Delta)$ be the preimage of $Z^0$ through the finite morphism $\spr(Z,X,D+\Delta)\to Z$. Set

$$p\colon \spr^0(X,D+\Delta,\subseteq D):=\bigsqcup_{Z\subseteq D}\spr^0(Z,X,D+\Delta)\longrightarrow X,$$

it is a quasi-finite morphism mapping surjectively onto $D$.

The equivalence relation $R\rightrightarrows D$ pullbacks to an equivalence relation $(p\times p)^*R\rightrightarrows\spr^0(X,D+\Delta,\subseteq D)$ that commutes with the projections to $D$, and since $p$ is surjective onto $D$ it is sufficient to show that $(p\times p)^*R$ is finite. We describe how the generators of $R$ pullbacks to $\spr^0(X,D+\Delta,\subseteq D)$. To make book-keeping easier, we let $\{Z_j\}_j$ be the set of lc centers of $(X,D+\Delta)$ contained in $D$.

By the Galois property of <ref>, over the normal locus of $Z_j^0$ the morphism $\spr^0(Z_j,X,D+\Delta)\to Z_j^0$ is the quotient by the Galois group $G_j:=\Gal(\spr(Z_j,X,D+\Delta)/Z_j)$. Thus the preimage of the diagonal $Z_j^0\times Z_j^0$ under $p$ is a union of the graphs of the $G_j$-action, together with other components that do not dominate $Z_j^0$ (their images are contained in the locus where $Z_j^0$ is not normal).

Next we understand the pullback of $\tau$.

Let $Z_{jh}\subset D^n$ be a subvariety dominating $Z_j\subset X$. Let $Z_l:=n(\tau(Z_{jh}))$. Then $Z_{jh}$ is a lc center of $(D^n,\Diff_{D^n}\Delta)$. Moreover

* $\tau$ induces a crepant birational map

$$\tilde{\tau}_{jhl}\colon\src(Z_j,X,D+\Delta)\overset{\emph{cbir}}{\sim} \src(Z_l,X,D+\Delta)$$

determined up to the left and right action of $\Bir^c_{Z_j}\src(Z_j,X,D+\Delta)$ and $\Bir^c_{Z_l}\src(Z_l,X,D+\Delta)$;

* $\tilde{\tau}_{jhl}$ induces an isomorphism

$$\tau_{jhl}\colon \spr^0(Z_j,X,D+\Delta)\cong \spr^0(Z_l,X,D+\Delta)$$

determined up to left and right multiplication by $G_j$ and $G_l$.

By <ref>, $Z_{jh}$ is an lc center of $(D^n,\Diff_{D^n}\Delta)$ and

\begin{equation}\label{eqn:adjunction_for_sources}
\src(Z_{jh},D^n,\Diff_{D^n}\Delta)\overset{\text{cbir}}{\sim}\src(Z_j,X,D+\Delta).
\end{equation}

By <ref> the automorphism $\tau$ of $(D^n,\Diff_{D^n}\Delta)$ induces a crepant birational map between $\src(Z_{jh},D^n,\Diff_{D^n}\Delta)$ and $\src(\tau(Z_{jh}),D^n,\Diff_{D^n}\Delta)$. Then by <ref> we obtain crepant birational maps

$$\tilde{\tau}_{jkl}\colon \src(Z_j,X,D+\Delta)\overset{\text{cbir}}{\sim} \src(Z_l,X,D+\Delta).$$

Since $\tilde{\tau}_{jhl}$ preserves the non-klt locus and since $\spr^0(Z_j,X,D+\Delta)$ is precisely the image of the klt locus of $\src(Z_j,X,D+\Delta)$, the $\tilde{\tau}_{jhl}$ descend to an isomorphism

$$\tau_{jhl}\colon \spr^0(Z_j,X,D+\Delta)\cong \spr^0(Z_l,X,D+\Delta)$$

as claimed. However $\tau_{jhl}$ is not uniquely defined, since $\tilde{\tau}_{jhl}$ is determined up to left and right multiplication by $\Bir^c_{Z_j}\src(Z_j,X,D+\Delta)$ and $\Bir^c_{Z_l}\src(Z_l,X,D+\Delta)$. By <ref> we obtain that $\tau_{jhl}$ is determined up to left and right multiplication by $G_j$ and $G_l$.

Since $n(n^{-1}Z_j)=\bigcup_i D_i\cap Z_j$ and each $D_i\cap Z_j$ is a union of lc centers by <ref>, we see that each component of $n^{-1}Z_j$ dominates a lc center of $(X,D+\Delta)$. So, thanks to <ref>, we have found all the generators of $(p\times p)^*R$.

To show that $(p\times p)^*R$ is finite, by <cit.> it is sufficient to show that it is finite over the generic point of every $Z_j^0$. Therefore we may assume that $(p\times p)^*R$ is the groupoid generated by the $G_j$ and the $\tau_{jhl}$, and the stabilizer of $\spr^0(Z_j,X.D+\Delta)$ is generated by the sets $\{\tau^{-1}_{jh'l}G_l\tau_{jhl}\}_{h,h',l}$.

The Galois property of <ref> shows that $G_j$ is a subgroup of

$$\Aut^s_k\spr(Z_j,X,D+\Delta):= \im\left[ \Bir^c_{k}\src(Z_j,X,D+\Delta)\to \Aut_k\spr(Z_j,X,D+\Delta)\right].$$

By <ref>, the $\tau^{-1}_{jh'l}G_l\tau_{jhl}$ are also subgroups of $\Aut^s_k\spr(Z_j,X,D+\Delta)$. To complete the proof we will show that these groups of $k$-automorphisms are finite. This is where the ampleness assumption on $K_X+D+\Delta$ will be used.

* If $\dim Z_j=2$ then $\src(Z_j,X,D+\Delta)\to Z_j$ is birational and therefore $\Aut^s_k(Z_j,X,D+\Delta)$ is finite by <ref>.

* If $\dim Z_j=1$ then $\dim \src(Z_j,X,D+\Delta)=1$ by <ref>. Hence $\src(Z_j,X,D+\Delta)$ is a $1$-dimensional pair of general type, since its log canonical divisor is the pullback of the ample divisor $K_X+D+\Delta$ through the composition of finite morphisms

$$\src(Z_j,X,D+\Delta)\to Z_j\hookrightarrow X.$$

Thus $\Bir^c\src(Z_j,X,D+\Delta)$ is finite by <ref>.

* If $\dim Z_j=0=\dim \src(Z_j,X,D+\Delta)$ then finiteness is clear.

* The only case left, according to <ref>, is $\dim Z_j=0$ and $\dim \src(Z_j,X,D+\Delta)=1$. Then $\src(Z_j,X,D+\Delta)$ is a Calabi-Yau curve and finiteness follows from <ref>.

Thus $X/R$ exists and is a $k$-scheme. Since $X\to X/R$ is finite, $X/R$ is proper over $k$ <cit.>. This completes the proof.

With the notations of the proof of <ref>, let us observe that the stabiliser of $\spr(Z_j^0,X,D+\Delta)$ is not contained in the smaller group

\begin{equation*}\label{eqn:source_automorphisms_over_the_base}
\Aut_{Z_j}\spr(Z_j,X,D+\Delta):=\im\left[ \Bir^c_{Z_j}\src(Z_j,X,D+\Delta)\to\Aut_{Z_j}\spr(Z_j,X,D+\Delta)\right]
\end{equation*}

even though $G_j$ belongs to it and each $\tau_{jhl}$ commutes with the projections to $Z_j$ and $Z_l$. Indeed the stabiliser is generated by the groups $\tau_{jh'l}^{-1}G_l\tau_{jhl}$ where it may happen that $h\neq h'$. In this case the corresponding automorphisms of $\spr(Z_j,X,D+\Delta)$ may not commute with the projection to $Z_j$. This happens if $Z_j$ is dominated by several lc centers of $(D^n,\Diff_{D^n}\Delta)$ whose images under $\tau$ dominate the same lc center $Z_l\subset X$.

Notice that $\Aut_{Z_j}\spr(Z_j,X,D+\Delta)$ is a finite group since $\spr(Z_j,X,D+\Delta)\to Z_j$ is a finite morphism. On the other hand, it is not clear that $\Aut^s_k\spr(Z_j,X,D+\Delta)$ should be finite, and this is where the ampleness assumption comes into the picture.

Now we show that the log canonical divisor descends to the geometric quotient.

Let $(X,D+\Delta)$ be a quasi-projective lc threefold pair over a perfect field $k$ of characteristic $>5$. Let $\tau\colon (D^n, \Diff_{D^n}\Delta)\cong (D^n, \Diff_{D^n}\Delta)$ be an involution. Assume that $R(\tau)\rightrightarrows X$ is finite, let $q\colon X\to Y:=X/R(\tau)$ be the geometric quotient, and let $\Delta_Y:=q_*\Delta$. Then $K_Y+\Delta_Y$ is $\bQ$-Cartier.

By <ref> the $\bQ$-divisor $K_Y+\Delta_Y$ is $\bQ$-Cartier in codimension $2$. Hence we may localize over a closed point of $Y$, and assume that $Y$ is local with closed point $y$, and that $K_Y+\Delta_Y$ is $\bQ$-Cartier on $Y^0:=Y\setminus\{y\}$. Since $q\colon X\to Y$ is an isomorphism above the away from $q(D)$, we may assume that $y$ belongs to the nodal locus. Since an $\sO_Y$-module is locally free if and only if it is locally free after an étale base-change, we may base-change along the strict henselization of $Y$, and assume that $k(y)$ is separably closed. Since $k$ is perfect, we may therefore assume that $k(y)$ is algebraically closed.

We are going to descend the total space of a multiple of $K_X+D+\Delta$ to $Y$, and use the theory of Seifert bundles (see <cit.>) to conclude that it defines a line bundle on $Y$.

Choose an even integer $m>0$ such that $m\Delta$ is a $\bZ$-divisor, and both $\omega_X^{[m]}(mD+m\Delta)$ and $\omega_{Y^0}^{[rm]}(rm\Delta_Y|_{Y^0})$ are invertible sheaves. We consider the $\bA^1$-bundle over $X$ given by

$$X_L:=\Spec_X\sum_{r\geq 0}\omega_X^{[rm]}(rmD+rm\Delta)\overset{p}{\longrightarrow} X.$$

Set $D_L:=p^{-1}D$ and $\Delta_L:=p^{-1}\Delta$. Clearly $(X_L,D_L+\Delta_L)$ is lc. Since $X_L\to X$ is an $\bA^1$-bundle, we see that the normalization $D^n_L\to D_L$ is equal to $D^n\times_DD_L$. By adjunction and functoriality of the relative spectrum, this gives the alternative description

$$D^n_L=\Spec_{D^n}\sum_{r\geq 0} \omega_{D^n}^{[rm]}(rm\Diff_{D^n}\Delta).$$

The fiber product description shows that $\Diff_{D^n_L}\Delta_L=p^*\Diff_{D^n}\Delta$ and that the lc centers of $(D^n_L,\Diff_{D^n_L}\Delta_L)$ are the preimages of the lc centers of $(D^n,\Diff_{D^n}\Delta)$. As $\Diff_{D^n}\Delta$ is $\tau$-invariant, the relative spectrum description shows that $\tau$ lifts to an involution $\tau_L$ of the pair $(D^n_L,\Diff_{D^n_L}\Delta_L)$.

Now we wish to show that the induced equivalence relation $R_L:=R(\tau_L)\rightrightarrows X_L$ is finite, so that we can form the quotient $X_L/R_L$.

Denote by $p'\colon Y^0_L\to Y^0$ the total space of the invertible sheaf $\omega_{Y^0}^{[rm]}(rm\Delta_Y|_{Y^0})$, $X^0:=q^{-1}Y^0$ and  $X^0_L:=(p\circ q)^{-1}Y^0$. Then we have a natural finite morphism of $\bA^1$-bundles $q'\colon X^0_L\to Y^0_L$ making the diagram

X^0_L\arrow[r, "p"]\arrow[d, "q'"] & X^0\arrow[d, "q"] \\
Y^0_L\arrow[r, "q'"]& Y^0
\end{tikzcd}$$

commutative. Since $X^0_L$ is the total space of the line bundle $\omega_{X^0}^{[m]}(mD|_{X^0}+m\Delta|_{X^0})$ which descends to $Y^{0}$, we have that $Y^0_L=X^0_L/R_L^0$ where $R^0_L$ is the restriction of $R_L$ to $X^0_L$ <cit.>. Therefore $R^0_L$ is finite, and we only need to prove the finiteness of $R_L$ over the complement of $X^0_L$.

Let $x_1,\dots,x_s\in X$ be the preimages of $y$. Since $y$ belongs to the nodal locus of $Y$, we have $x_1,\dots,x_s\in D$. If none of the $x_i$ are lc centers of $(X,D+\Delta)$, then every lc center of $(X_L,D_L+\Delta_L)$ intersects $X^0_L$ and therefore $R_L$ is finite by <cit.>.

Assume that one of the $x_i$ is an lc center. Then every $x_i$ is an lc center: for the $x_i$ form an equivalence class of $\tau$, by adjunction one of them corresponds to an lc center of $(D^n,\Diff_{D^n}\Delta)$ and $\tau$ permutes these lc centers.

Since $k(y)$ is algebraically closed, we have $k(x_i)=k(y)$ for each $i$. The fiber of $p\colon X_L\to X$ above $x_i$ is the spectrum of the symmetric algebra of the $1$-dimensional $k(y)$-vector space

$$V_i:=\omega_X^{[m]}(mD+m\Delta)\otimes_{\sO_X} k(x_i)$$

and $X_L\setminus X^0_L=\bigcup_iV_i$. If $x'_i\in D^n$ is a preimage of $x_i$, then the data of $\tau(x_i')=x_j'$ defines an isomorphism $\tau_{ijl}\colon V_i\to V_j$ (the index $l$ accounts for the fact that $x_i$ might have several preimages in $D^n$). The collection of isomorphisms $\{\tau_{ijl}\colon V_i\to V_j\}$ generates a groupoid, and $R_L$ is finite if and only if each group $\Stab(V_i)\subset \Aut(V_i)$ of all possible compositions $\tau_{ij_1l_1}\circ\dots\circ\tau_{j_nil_n}\colon V_i\to V_i$, is finite.

To show this property, consider the sources $(S_i,\Delta_i):=\src(x_i,X,D+\Delta)$. These are proper Calabi-Yau varieties over $k(y)$. If we pullback $\omega_X^{[m]}(mD+m\Delta)$ to a crepant dlt blow-up, restrict it to (a model of) $S_i$ and take global sections, we obtain $V_i$. Thus the Poincaré residue maps constructed in <ref> give canonical isomorphisms

$$V_i\cong H^0(S_i,\omega_{S_i}^{[m]}(m\Delta_i)).$$

Moreover <ref> shows that each isomorphism $\tau_{ijl}\colon V_i\to V_j$ is induced by a crepant birational map $\phi_{ijl}\colon (S_j,\Delta_j)\dashrightarrow (S_i,\Delta_i)$. Hence we conclude that

\begin{equation}\label{eqn:sources_and_descent_of_canonical_bundle}
\Stab(V_i)\subseteq \im\left[ \Bir^c(S_i,\Delta_i)\to \Aut_{k(y)}H^0(S_i,\omega_{S_i}^{[m]}(m\Delta_i))\right].
\end{equation}

Now observe that the $S_i$ are at most $1$-dimensional by <ref>, and therefore by <ref> the right-hand side in <ref> is finite.

It follows that $R_L$ is finite, and thus the quotient $X_L/R_L$ exists. By <cit.> the complement of the zero section is a Seifert bundle over $Y$, and by <cit.> this implies that its define a line bundle on $Y$. By construction this line bundle is equal to $\omega_{Y^0}^{[m]}(m\Delta_Y|_{Y^0})$ over $Y^0$. Hence $\omega_Y^{[m]}(m\Delta_Y)$ is invertible, as was to be shown.

Let $k$ be a perfect field of characteristic $> 5$. Then normalization gives a one-to-one correspondence

\begin{equation*}
(\Char k > 5) \quad
\begin{pmatrix}
\text{Proper slc threefold pairs}\\
(X,\Delta) \text{ such that}\\
K_X+\Delta\text{ is ample}
\end{pmatrix}
\overset{1:1}{\longrightarrow}
\begin{pmatrix}
\text{Proper lc threefold pairs } (\bar{X},\bar{D}+\bar{\Delta})  \\
\text{plus a generically fixed point free}\\
\text{involution }\tau\text{ of }(\bar{D}^n,\Diff_{\bar{D}^n}\bar{\Delta}) \\
\text{such that }K_{\bar{X}}+\bar{D}+\bar{\Delta} \text{ is ample.}
\end{pmatrix}
\end{equation*}

Given $(\bar{X},\bar{D}+\bar{\Delta},\tau)$ as in the right-hand side, by <ref> the equivalence relation $R(\tau)$ is finite and we can form the geometric quotient $q\colon \bar{X}\to X:=\bar{X}/R(\tau)$. Set $\Delta:=q_*\bar{\Delta}$. By <ref> the scheme $X$ is demi-normal, and by <ref> the $\bQ$-divisor $K_X+\Delta$ is $\bQ$-Cartier. Therefore $(X,\Delta)$ is slc. This gives a map $\{(\bar{X},\bar{D}+\bar{\Delta},\tau)\}\to \{(X,\Delta)\}$. It is an inverse to the normalization map by <ref> and <cit.>.

§ APPLICATION TO THE MODULI THEORY OF STABLE SURFACES

In this section we apply the theory of gluing to the moduli functor of stable varieties. Our discussion will be conditional, since some technical results are not known yet in positive characteristic.

Let $k$ be an algebraically closed field. We define families of stable log varieties and the moduli functor of stable varieties following [47].

A projective connected pure-dimensional $k$-scheme $X$ together with a $\bQ$-divisor is a stable log pair if $(X,D)$ is slc and $K_X+D$ is and ample. If $D=0$, we simply say that $X$ is a stable variety.

Let $T$ be a $k$-scheme. A family of pairs over $T$ is a flat morphism of $k$-schemes $X\to T$ together with a $\bQ$-divisor $D$ on $X$ such that: for every $t\in T$, none of the irreducible components of $X_t$ is contained in $\Supp D$, and none of the irreducible components of $X_t\cap \Supp D$ is contained in $\Sing X_t$. This allows us to define a restricted divisor $D_t$ on $X_t$.

Let $T$ be a $k$-scheme. A family of stable log pairs over $T$ is a family of pairs $f\colon (X,D)\to T$ such that $K_{X/T}+D$ is $\bQ$-Cartier and the geometric fiber $(X_{\bar{t}}, D_{\bar{t}})$ is a stable log pair for every $t\in T$.

The moduli functor $\overline{\sM}_{n,v,k}$, where $v\in \bQ_+$, is defined on $\text{Sch}_k$ by the values

\overline{\sM}_{n,v,k}(T)=\left\{
\begin{tabu}{c | l l}
\begin{tikzcd}
X \arrow[dd, "f" left] \\  \\T
\end{tikzcd}
& &
\begin{aligned}
1) \; &f \text{ is a flat morphism of }k\text{-schemes},\\
2) \; &\left(\omega_{X/T}^{[m]}\right)_S\cong \omega_{X_S/S}^{[m]} \text{ for every }S\to T\text{ and }m\in \bN,\\
3) \; & \text{for every }t\in T, X_{\bar{t}} \text{ is a stable variety of dimension } n\\ & \text{ with } \Vol(K_{X_{\bar{t}}})=v.
\end{aligned}
\end{tabu}
\right\}

and for $T'\to T$, the corresponding map $\overline{\sM}_{n,v}(T)\to \overline{\sM}_{n,v}(T')$ is given by pullbacks.

There are subtle differences between families of stable varieties and families parametrized by the functor $\overline{\sM}_{n,v,k}$:

* At least when $n=2$, if $(X\to T)\in \overline{\sM}_{2,v,k}(T)$ and $T$ is normal, then $X\to T$ is a family of stable surfaces, see <cit.>.

* If $(X\to T)\in \overline{\sM}_{n,v,k}(T)$, then $X\to T$ need not be a family of stable varieties, see <cit.>.

We have only defined the moduli functor in the boundary-free case. To define a moduli functor of stable log pairs we need a good notion of family of divisors above arbitrary bases. A good notion, at least in characteristic $0$, is developped in [40]. However, to avoid technical difficulties, we will restrict ourselves to the boundary-free case in what follows. This way, the only pairs we will have to deal with are the one arising from the normalization of demi-normal varieties.

From now on we consider the case of stable surfaces (that is, $n=2$) over an algebraically closed field $k$ of characteristic $p>5$. Then it is known that $\overline{\sM}_{2,v,k}$ is a separated Artin stack of finite type over $k$ with finite diagonal <cit.>. We discuss the valuative criterion of properness using the methods of <cit.>.

We are interested in the following situation. Let $T$ be an affine one-dimensional regular scheme of finite type over $k$, $t\in T$ a closed point and $T^0:=T\setminus\{t\}$. Suppose we are given a family $(f^0\colon X^0\to T^0)\in \overline{\sM}_{2,v}(T^0)$. Then we are looking for a finite morphism $\pi\colon T'\to T$ and a family $(f'\colon X'\to T')\in \overline{\sM}_{2,v}(T')$ such that the pullback family $f^0\times_T \pi\colon X^0\times_T T'\to T'$ is isomorphic to $X'\to T'$ over $\pi^{-1}T^0$.

The method of <cit.> can be outlined as follows: establish a dictionnary between the fiberwise properties and the global properties of a family of stable varieties, and apply methods of the MMP on $X^0\to T^0$ to produce the desired completed family. In general, the scheme $X^0$ is only demi-normal, and the MMP works best for normal varieties: here the gluing theory developped in [38] is useful to go back and forth between slc varieties and their normalizations.

One must solve several problems to carry out this program in positive characteristic:

* The dictionnary between fiberwise and global properties works well in one direction. Suppose $f\colon (X,D)\to T$ is a family of stable log surfaces over a one-dimensional normal base. Since $T$ and every fiber $X_t$ are $S_2$, we see that $X$ is $S_2$. Points of codimension one of $X$ that do not dominate $T$ are regular, and those which dominate $T$ are $G_1$ since the generic fiber is demi-normal. Thus $X$ is demi-normal. Now inversion of adjunction implies that $(X,D+X_t)$ is slc for every $t\in T$.

Notice that if $(f\colon X\to T)\in\overline{\sM}_{2,v,k}(T)$ and $(\bar{X},\bar{D})\to X$ is the normalization, then it follows from <ref> and <ref> that the morphism $(\bar{X},\bar{D})\to T$ is a family of stable log surfaces (with $\bar{D}$ dominating $T$ if $X$ is not already normal).

Problems appear with the converse implication. Assume that $(X,D+X_t)$ is slc. Then the deformation theory of nodes show that $X_t$ is $G_1$ <cit.>, and adjunction would then imply that $(X_t,D_t)$ is slc, provided that $X_t$ is $S_2$. That $X_t$ is $S_2$ in characteristic $0$ follows from a non-trivial result of Alexeev (see <cit.>). This is not known at the moment in positive characteristic, so let us formulate the condition

(S2)         If $(X,D)\to T$ is a flat family of geometrically reduced surface pairs over a one-dimensonal normal base such that $(X,D+X_t)$ is slc for every $t\in T$, then every $X_t$ is $S_2$.

* To produce the completed family, one first extends $X^0\to T^0$ to a flat family $X_1\to T$. In general the central fiber is not even reduced: so one looks for a base-change along a finite $T'\to T$ such that the fibers of $X'_1:=X_1\times_TT'\to T'$ are reduced. If we find one such base-change, we have to make sure that $(X_1',(X_1')_{t'})$ is still slc for every $t'\in T'$. In positive characteristic, this is a problem if $T'\to T$ is wildly ramified or inseparable, see <cit.>. So we formulate the following condition of semi-stable reduction:

(SSR)         Let $X\to T$ is a flat morphism where $X$ is a regular threefold and $T$ a one-dimensional curve. Let $E$ be a reduced effective divisor on $X$ such that $(X,E+\red X_t)$ is snc for every closed $t\in T$. Then there exists a finite morphism $T'\to T$ such that: if $Y$ is the normalization of $X\times_T T'$ and $E_Y$ is the pullback divisor, then every closed fiber $Y_{t'}$ is reduced and every $(Y, E_Y+Y_{t'})$ is lc.

Modulo these two conditions, we can prove the valuative criterion of properness for $\overline{\sM}_{2,v,k}$, following the method of <cit.>.

Let $(X,D+\Delta)\to T$ be a family of stable log surfaces over a normal one-dimensional base, where $D$ is a reduced divisor with normalization $n\colon D^n\to D$. Then every componenent of $D^n$ dominates $T$ and $(D^n,\Diff_{D^n}\Delta)\to T$ is a (disjoint union of) families of stable log curves.

By definition of families of pairs, every irreducible component of $D$ dominates $T$. In particular every component of $D^n$ dominates $T$ and $D^n\to T$ is flat. So the fibers $D^n_t:=(D^n)_t$ are of pure dimension one. By inversion of adjunction the pair $(X,D+\Delta+X_t)$ is slc. Passing to the normalization (recall that $X$ is normal at the generic points of $D$), we may assume it is lc. Since $X_t$ is Cartier we have

By adjunction, this implies that $(D^n,\Diff_{D^n}\Delta+D^n_t)$ is lc. Since $D^n$ is a surface, classification of surface lc singularities show that $D^n_t$ is Gorenstein. Moreover, as $D^n$ is $S_2$ (it is normal), the $D^n_t$ are $S_1$. Thus $D^n_t$ is a demi-normal curve, and by adjunction we deduce that $(D^n,\Diff_{D^n}\Delta)\to T$ is a family of log curves. As

we obtain that $K_{D^n/T}+\Diff_{D^n}\Delta$ is ample over $T$.

We work over an algebraically closed field $k$ of characteristic $>5$. Assume that the conditions (S2) and (SSR) hold. Then $\overline{\sM}_{2,v,k}$ is proper.

We consider anew an affine one-dimensional regular $k$-scheme of finite type $T$, a closed point $t\in T$ and a family $(f_0\colon X^0\to T^0:=T\setminus\{t\})\in \overline{\sM}_{2,v,k}(T^0)$. Since separatedness holds, we have to show that there exists a finite morphism $\pi\colon T'\to T$ such that the pullback family $f^0\times_T\pi\colon X^0\times_TT'\to \pi^{-1}T^0$ extends to a family in $\overline{\sM}_{2,v,k}(T')$. By <ref>, we may think of $f^0$ as a family of stable surfaces.

Step 1: Normal case. First assume that $X^0$ is actually normal. We can extend $f^0$ to a flat morphism $f_1\colon X_1\to T$. Let $g_1\colon Y_1\to X_1$ be a log resolution with $E_1:=\Exc(g_1)$ such that $(Y_1,E_1+\red (Y_1)_t)$ is snc for every $t\in T$. Such a resolution exists for threefolds: the proof is the same as in <cit.>, using <ref> at the appropriate places.

By (SSR), there is a finite morphism $\pi\colon T'\to T$ such that: if $Y_2$ is the normalization of $Y_1\times_TT'$ with induced morphism $f_2\colon Y_2\to T'$ and $E_2$ is the pullback divisor of $E_1$, then every fiber $(Y_2)_{t'}$ is reduced and $(Y_2,E_2+(Y_2)_{t'})$ is lc for every closed $t'\in T'$.

By [57], the family $(Y_2,E_2)\to T'$ admits a relative canonical model $(X_{\can}, E_{\can})\to T'$. Notice that the pullback family $X^0\times_TT'\to \pi^{-1}T^0$ is a relative canonical model of $(Y_2,E_2)\to T'$ over a dense open susbset of $T'$. By uniqueness of canonical models, these two families are generically isomorphic; by separatedness, the isomorphism extends to the whole $\pi^{-1}T^0$ (see <cit.> for the technical statement), and this implies that $E_{\can}=0$. By <ref> the pairs $(X_{\can}, (X_{\can})_{t'})$ is lc for every closed $t'\in T'$. The condition (S2) then ensures that $(X_{\can}\to T')\in\overline{\sM}_{2,v,k}(T')$.

Step 2: Demi-normal case. Now we consider the case where $X^0$ is only demi-normal. Let $(\bar{X}^0,\bar{D}^0)\to T^0$ be the normalization, with induced involution $\tau^0$. Then we can apply the same argument as before (all the results we used are available for log pairs). We obtain a finite morphism $\pi\colon T'\to T$ and a family of surface pairs $\bar{f}_{\can}\colon (\bar{X}_{\can},\bar{D}_{\can})\to T'$ which extends the pullback of $(\bar{X}^0,\bar{D}^0)\to T^0$ and such that $(\bar{X}_{\can},\bar{D}_{\can}+(\bar{X}_{\can})_{t'})$ is lc for every closed $t'\in T'$.

By <ref>, the morphism $(\bar{D}^n_{\can},\Diff_{\bar{D}^n_{\can}}(0))\to T'$ is a family of stable log curves. Hence by <cit.>, the pullback of the involution $\tau^0$ extends to an involution $\tau_{\can}$ on $(\bar{D}^n_{\can},\Diff_{\bar{D}^n_{\can}}(0))$.

By <cit.>, none of the lc centers of $(\bar{X}_{\can},\bar{D}_{\can})$ are disjoint from $\bar{X}^0\times_T T'$. Moreover, since $R(\tau^0)\rightrightarrows \bar{X}^0$ is a finite equivalence relation and $T'\to T$ is finite as well, then $R(\tau_{\can})\rightrightarrows \bar{X}_{\can}$ is finite on $\bar{f}_{\can}(\pi^{-1}T^0)$. Thus <cit.> implies that $R(\tau_{\can})$ is finite. Thus there exists a geometric quotient $\bar{X}_{\can}\to X_{\can}$. Since $\tau^0$ commutes with the projection to $T$, the involution $\tau_{\can}$ commutes with the projection to $T'$, and therefore $\bar{f}_{\can}$ factorises through a morphism $f_{\can}\colon X_{\can}\to T'$.

By <ref> the scheme$X_{\can}$ is demi-normal, and by <ref> the divisor $K_{X_{\can}}$ is $\bQ$-Cartier. Since $(X_{\can},(X_{\can})_{t'})$ pullbacks to $(\bar{X}_{\can}, \bar{D}_{\can}+(\bar{X}_{\can})_{t'})$, we see that $(X_{\can},(X_{\can})_{t'})$ is slc for every closed $t'\in T'$. Since $\bar{X}^0\times_T T'\to \bar{X}^0$ is flat and the formation of geometric quotient commutes with flat base-change <cit.>, we see that the pullback of $X^0$ along $\pi^{-1}T^0\to T^0$ is isomorphic to $f_{\can}^{-1}(\pi^{-1}T^0)$. The condition (S2) ensures that $(f_{\can}\colon X_{\can}\to T')\in \overline{\sM}_{n,v,k}(T')$. This finishes the proof.

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