Proof. Let $f \colon W \to X$ be a thrifty log resolution of $(X,\Delta )$ [Reference KollárKol13, Lemma 2.79]. As $X$ is a surface, $f$ being thrifty means that it is an isomorphism at the nodes of $\lfloor \Delta \rfloor$ . This we can achieve by running our resolution algorithm by excluding the nodes of $\lfloor \Delta \rfloor$ .

Then run a $(K_W + f_*^{-1}\Delta )$ -MMP over $X$ ending with $\pi \colon Y \to X$ such that $K_Y+\pi _*^{-1}\Delta$ is $\pi$ -nef. As this is also a $K_W$ -MMP, we deduce that $Y$ is regular by [Reference KollárKol13, Theorem 2.29]. As $f$ does not extract log canonical places, (ii) is immediate. To finally verify (iii), we argue as in the proof of [Reference KollárKol13, Theorem 2.25.a].

Proof. Without loss of generality we can suppose $k=H^0(C, \mathcal{O}_C)$ . Suppose by contradiction that $\deg _k \omega _{C/k} \lt 0$ . By [Reference KollárKol13, Lemma 10.6], $C$ embeds as a conic in $\mathbb{P}^2_k$ . Taking the base change to $\overline{k}$ , we still get an embedding $C_{\overline{k}} \rightarrow \mathbb{P}^2_{\overline{k}}$ . As $k$ is separably closed, $C_{\overline{k}}$ is still an irreducible conic, and by the classification of conics over an algebraically closed field, either $C_{\overline{k}}$ is regular or it is a double line. Note that the case of a double line cannot appear as ${\mathrm{char}}(k) \neq 2$ by [Reference Bernasconi and TanakaBT22, Lemma 2.17]. Finally, if $C_{\overline{k}}$ is regular, we deduce $C$ is regular by descent for faithfully flat morphisms [Sta, Tag 033E], getting a contradiction.

Proof. The proof is case by case.

$\fbox{$There\ exists\ a\ singular\ exceptional\ curve\ E \subset{\mathrm{Ex}}(\pi )\textit{:}$}$ By Lemma 2.18, $\deg _k \omega _{C/k} \geq 0$ in this case. By [Reference KollárKol13, 3.30.1], then $\Delta =0$ , and $E$ is the unique exceptional divisor. Let $-1 \leq a \in{\mathbb{Q}}$ be the discrepancy of $E$ . By adjunction we have

(1) \begin{equation} 0 \leq \deg K_E \leq (K_X +E) \cdot E = (K_X - aE) \cdot E + (1+a) E^2 = (1+ a ) E^2 \leq 0. \end{equation}

In particular, we have equality everywhere. Taking into account that $E^2\lt 0$ , this means that $a=-1$ . We obtain that $(Y,E)$ is log canonical. Taking into account that $Y$ is regular, we deduce that $(Y,E)$ is étale-snc. To see $K_Y + E \equiv _X 0$ , we simply note that $(K_Y +E ) \cdot E =0$ by Equation (1).

$\fbox{$All\ irreducible\ component\ E_{i}\ of\ {\mathrm{Ex}}(\pi )\ are\ regular\textit{:}$}$ Note that $\pi _*^{-1} \lfloor{\Delta \rfloor }$ is regular and that ${\mathrm{Ex}}(\pi ) + \pi _*^{-1} \lfloor{\Delta \rfloor }$ is snc at intersection points by construction of the minimal resolution, in this case. We are left to prove that ${\mathrm{Ex}}(\pi )$ is snc. We can suppose that all irreducible components are conics by [Reference KollárKol13, 3.30.2], and we fix $r_i:=\dim _k H^0(E_i, \mathcal{O}_{E_i})$ .

$\fbox{$All\ the\ E_{i}\ are\ regular,\ and\ (E_i \cdot E_j) \gt \max \left \{r_i, r_j\right \}\ for\ some\ i \neq j\textit{:}$}$ In this case $\Delta =0$ and there are two exceptional curves $E_1$ and $E_2$ , by [Reference KollárKol13, 3.30.3]. Consider the following computation:

\begin{align*} &0 \leq E_1 \cdot E_2 + \deg K_{E_1} \leq (K_X +E_1 + E_2) \cdot E_1\\ &= (K_X - a_1E_1 - a_2E_1) \cdot E_1 + (1+a_1) E_1^2 + (1+a_2) E_1 \cdot E_2 = (1+a_1) E_1^2 + (1+a_2) E_1 \cdot E_2 \end{align*}

This implies that

(2) \begin{equation} -(1+a_1) E_1^2 \leq (1+ a_2) E_1 \cdot E_2. \end{equation}

By applying the same argument to $E_2$ instead of to $E_1$ , we obtain

(3) \begin{equation} -(1+a_2) E_2^2 \leq (1+ a_1) E_1 \cdot E_2. \end{equation}

Multiplying (2) and (3) together, we obtain the following, where we are also using that both sides of the two inequalities are non-negative:

(4) \begin{equation} (1+a_2) (1+a_1) (E_1^2)( E_2^2) \leq (1+ a_1) (1+a_2) (E_1 \cdot E_2)^2. \end{equation}

In other words, either one of the $a_i$ is equal to $-1$ , or the determinant of the intersection matrix is non-positive. The latter contradicts the negative definiteness of the intersection matrix; hence, we obtain that one of the $a_i$ is $-1$ . By symmetry we can assume that $a_1=-1$ . However, then (3) says that $(1+a_2) E_2^2 \geq 0$ . As $E_2^2\lt 0$ , this implies also that $a_2=-1$ .

In particular, $(Y, E_1 + E_2)$ are log canonical, and hence, by adjunction, so are $(E_1, E_1 \cap E_2)$ and $(E_2, E_1 \cap E_2)$ . This means that the coefficients of $E_1 \cap E_2$ are $1$ on both $E_1$ and $E_2$ . As $k$ is separably closed, all finite non-trivial extensions of $k$ have degree divisible by $p$ . As $E_1 \cdot E_2 =2$ and $p \neq 2$ , in fact $E_1 \cap E_2$ contains only points with residue field equal to $k$ . As above we have seen that the coefficients of these points cannot be more than $1$ , so we obtain that $E_1 \cap E_2$ has two distinct points with coefficient $1$ and, hence, that the intersection of $E_1$ and $E_2$ is transversal. In particular, $(Y, E_1 + E_2)$ is snc (and the singularity is a cusp with the exceptional divisor of the minimal resolution being a cycle of two conics).

$\fbox{$All\ the\ E_{i}\ are\ regular,\ and\ (E_i \cdot E_j) =\max \left \{r_i, r_j\right \}\ for\ all\ i\ and\ j\textit{:}$}$ fix two components $E_i$ and $E_j$ . We may assume by symmetry that $r_i \geq r_j$ . In particular, the intersection scheme $E_i \cap E_j$ is a length one Artinian scheme over $H^0(E_i, \mathcal{O}_{E_i})$ . This implies that $E_i \cap E_j$ is reduced. Hence, the intersection of $E_i$ and $E_j$ is transversal, which concludes our proof.

Proof. Let $C$ be a curve on $Z$ , mapping to a closed point of ${\mathrm{Spec}}(R)$ . By projection formula for the Mumford pull-back, we conclude that $f_*D \cdot C =D \cdot f^*C \geq 0$ .

Proof. First, we may assume that $R$ is strictly Henselian. Second, let $\pi \colon Y \to X$ be as in Definition 2.14. By Proposition 2.20, there are two cases. In case i, $\Delta =0$ $\big (Y,{\mathrm{Ex}}(\pi )\big )$ is étale-snc and ${\mathrm{Ex}}(\pi )$ is a single exceptional divisor; necessarily, then, anti- $f$ -nef. In this case, $f:=\pi$ satisfies the assertion of the theorem.

In case ii, the pair $\big (Y, \pi _*^{-1}\lfloor \Delta \rfloor +{\mathrm{Ex}}(\pi )\big )$ is snc. By [Reference TanakaTan18, Theorem 1.1.(QF)], we can run a $\big (K_Y+\pi _*^{-1}\Delta +{\mathrm{Ex}}(\pi ) \big ) \equiv _X \big (\sum _i (1+a(E_i, X, \Delta ) E_i\big )$ -MMP over $X$ , denoted by $\rho \colon Y \to Z$ , ending with a minimal model $f \colon Z \to X$ . By a standard application of the negativity lemma [Reference Bhatt, Ma, Patakfalvi, Schwede, Tucker, Waldron and WitaszekBMP⁺23]*Lemma 2.16, the birational contraction $\rho$ contracts exactly the $\pi$ -exceptional divisors with discrepancy $a(E,X, \Delta )\gt -1$ , and thus $Z$ is a $\mathbb{Q}$ -factorial surface with $(K_Z + f_*^{-1}\Delta +{\mathrm{Ex}}(f)) = f^*(K_X+\Delta )$ . As a $\big (K_Y+\pi _*^{-1}\Delta +{\mathrm{Ex}}(\pi )\big )$ -MMP over $X$ is a $\big (K_Y+\pi _*^{-1}\lfloor \Delta \rfloor +{\mathrm{Ex}}(\pi )\big )$ -MMP,Footnote ¹ the pair $\big (Z, f_*^{-1}\lfloor \Delta \rfloor +{\mathrm{Ex}}(f)\big )$ remains dlt. By the definition of the log minimal resolution, $-\sum _i a(E_i, X, \Delta ) E_i$ is nef over $X$ , and therefore, so is $-{\mathrm{Ex}}(f)=\rho _*(-\sum _i (1+a(E_i, X, \Delta )) E_i)$ by Lemma 2.21.

Proof. This is [Reference Filipazzi and WaldronFW20, Corollary 1.7] (see also [Reference PosvaPos21b, Corollary 5.2.16] for a proof in the case $X$ is defined over $\mathbb{F}_p$ ).

Proof. Suppose by contradiction that $(X,\Delta )$ is not plt around $D$ . By definition, there exists a log canonical centre $S$ such that $Z:=S \cap D$ is not-empty and that ${\mathrm{codim}}_X Z \geq{\mathrm{codim}}_X S \gt 1$ . As $Z$ is a union of log canonical centres by Proposition 2.26, this contradicts the minimality of $D$ .

Proof. Let $E$ be a log canonical place over $(X, \Delta )$ . As $(Y,\Delta _Y)$ is dlt, the generic point of $Z={\mathrm{cent}}_Y(E)$ is a stratum of $(Y, \Delta _Y^{=1})$ , and $E$ is already a log canonical place of $(Y, \Delta _Y^{=1})$ . This in particular implies that $Z \not \subseteq{\mathrm{Supp}} \Delta _Y^{\lt 1}$ . However, as both $X$ and $Y$ are $\mathbb{Q}$ -factorial, the exceptional locus of $\pi$ is purely divisorial. So, putting the last two sentences and the assumption on the discrepancies of the $\pi$ -exceptional divisors together, we obtain that $Z \not \subseteq{\mathrm{Ex}}(\pi )$ . However, that means that $\pi$ is an isomorphism around the generic point of $ \pi (Z)$ , and therefore $\pi (Z)$ is also a stratum of $(X, \Delta ^{=1})$ .

Proof. For i, it is sufficient to localise at codimension 1 points of $E_i$ and apply [Reference KollárKol13, Theorem 2.31]. Then ii follows immediately from i and iii is proven in [Reference Hacon and WitaszekHW23, Lemma 2.1].

Proof. If $E=0$ , then $\lfloor \Delta \rfloor =0$ , and hence $(X, \Delta )$ is klt. This is covered by point 1. Hence, we may assume that $E \neq 0$ . As we work in the local case, this means that $x \in{\mathrm{Supp}} E$ . By [Reference KollárKol13, Proposition 2.15] $(X,\Delta )$ has log canonical singularities. If there is an irreducible component of $E$ that is singular, then $E$ is an irreducible nodal curve and $\Delta =0$ by Theorem 2.16. As $(X, \Delta )$ is étale-dlt, in this case $X$ is regular, so we are in case ii. Thus we may also assume that every irreducible component of $E$ is regular. We may also assume that $x \not \in{\mathrm{etsnc}}(X, \Delta )$ . Note that ${\mathrm{Spec}} R \setminus \{x \} \subseteq{\mathrm{snc}}(X, \Delta )$ , and that all discrepancies over the point $x$ are greater than $-1$ by the étale snc assumption. Hence, $(X, \Delta )$ is actually dlt at $x$ .

For the assertion about $\mathbb{Q}$ -factoriality, we conclude in case i by combining [Reference KollárKol13, Proposition 2.28] and [Reference KollárKol13, Proposition 10.9]; case ii is then immediate.

Proof. Let $x \in (X, \Delta )$ be a closed point. We divide the proof into two cases. If $(X,\Delta )$ is dlt near $x$ , we take the resolution of singularities at $x$ constructed in Lemma 2.16. If $(X,\Delta )$ is not dlt near $x$ , we do not perform any blow-up as $X$ is already regular around $x$ by Lemma 2.30.

Proof. Recall that $X$ is $\mathbb{Q}$ -factorial by Lemma 2.30. By the assumption that no irreducible component of $ E$ and no point of the singular locus of $E$ is contained in the support of $D$ , the divisor $D|_E$ is a well-defined Mumford $\mathbb{Q}$ -divisor on $E$ .

Let $\pi \colon Y \to X$ be the resolution of the pair $ (X, E)$ given by Lemma 2.31, and write $K_Y+E_Y+\Gamma _Y=\pi ^{*}(K_X+E)$ , where $\lfloor E_Y + \Gamma _Y \rfloor = E_Y=\pi _*^{-1}(E)$ . As $\pi$ extracts no divisor of discrepancy $-1$ , $\pi$ is an isomorphism around the singular points of $E$ . Hence, $ \pi |_{E_Y} : E_Y \to E$ is an isomorphism. For similar reasons, $\lceil \pi ^* D\rceil |_{E_Y}$ does make sense, i.e. the support of $\lceil \pi ^* D\rceil$ intersects $E_Y$ only along its regular locus. Let $D_E$ be the divisor on $E$ corresponding to the divisor $\lceil \pi ^* D\rceil |_{E_Y}$ on $E_Y$ via the isomorphism $\pi |_{E_Y}$ .

As $Y$ is regular, we have the following exact sequence on $Y$ :

(5) \begin{equation} 0 \to \omega _Y(\lceil \pi ^*D \rceil ) \to \omega _Y(E_Y+ \lceil \pi ^{*}D \rceil ) \to \omega _{E_Y}\left (\lceil \pi ^{*}D \rceil |_{E_Y}\right ) \to 0. \end{equation}

Note the following properties:

1. (i) since $\pi$ does not extract any divisor of discrepancy $-1$ , we have $K_Y+\lceil \pi ^{*}D \rceil \geq \lfloor K_Y+\Gamma _Y+\pi ^*D\rfloor \geq \lfloor \pi ^{*}(K_X+D)\rfloor,$ so $\pi _*(\omega _Y(\lceil \pi ^*D \rceil ))= \omega _{X}(D)$ . Similarly, $\pi _*(\omega _Y(E_Y+ \lceil \pi ^{*}D \rceil ))=\omega _X(E+D)$ ;

2. (ii) By the above choice of $D_E$ , we have $\pi _* \omega _{E_Y}\left (\lceil \pi ^{*}D \rceil |_{E_Y}\right ) \cong \left ( \pi |_{E_Y} \right )_* \omega _{E_Y}\left (\lceil \pi ^{*}D \rceil |_{E_Y}\right ) \cong \omega _E (D_E)$ .

3. (iii) by GR vanishing for surfaces [Reference KollárKol13, Theorem 10.4], $R^1\pi _*\omega _Y(\lceil \pi ^*D \rceil )=0$ .

Thus, pushing forward 5 via $\pi$ , we obtain the short exact sequence

\begin{equation*}0 \to \omega _X(D) \to \omega _X(E+D) \to \omega _E(D_E) \to 0.\end{equation*}

We are left to check only that $D_E \sim _{{\mathbb{Q}}} D|_E+\Gamma _E$ for some $0 \leq \Gamma _E \leq{\mathrm{Diff}}_E(0)$ . Note that via the isomorphism $\pi |_{E_Y}$ , $\Gamma _E$ identifies with $(\lceil \pi ^{*}D \rceil -\pi ^*D)|_{E_Y} \geq 0$ . Let $x$ be a point of $X$ , and let $i_x$ be the determinant of the dual graph of the minimal resolution of $X$ at $x$ . By possibly restricting to a neighbourhood of $x$ , we have that $i_x D$ is Cartier by [Reference KollárKol13, Prop 10.9.(3)]. Additionally, by [Reference KollárKol13, Corollary 3.45], the following equality holds:

\begin{equation*} {\mathrm {Diff}}_E(0) =\begin {cases} \left (1-\frac {1}{i_x}\right )x, & \mbox {if } (X,E) \mbox { is plt near } x \\ x, & \mbox {if } (X,E) \mbox { is not plt near } x, \end {cases}\end{equation*}

Since $\lfloor \Gamma _E \rfloor =0$ and $i_x \Gamma _E$ is integral, we finally conclude that $\Gamma _E \leq{\mathrm{Diff}}_E(0) \leq{\mathrm{Diff}}_E(\Delta ^{\prime}).$

Proof. By Lemma 2.30, at the codimension $2$ singular points of $E$ , the $ \mathbb{Z}$ -divisor $D$ is Cartier. Hence, up to replacing $D$ by another divisor in its linear equivalence class, we may assume that $D$ does not contain any component of $E$ and that it also does not contain any singular point of $E$ that has codimension $2$ in $X$ . By localising at codimension 2 points of $X$ and applying Lemma 2.32, there exists a canonically defined Mumford $\mathbb{Z}$ -divisor $D_E$ on $E$ . As the irreducible components of $E$ are $(R_1)$ , by taking the preimage of $D$ in $E^{\nu }$ we obtain a globally defined Mumford $\mathbb{Z}$ -divisor $D_{E^\nu }$ on the $(S_2)$ surface $E^\nu$ .

Consider the natural exact sequence

\begin{equation*}0 \to \omega _X(D) \to \omega _X(E+D) \to \mathcal {Q} \to 0,\end{equation*}

where $\mathcal{Q}$ is a sheaf supported on $E$ . Note that $\mathcal{Q}$ is a torsion-free $\mathcal{O}_E$ -module of rank 1 by [Reference KollárKol13, Corollary 2.61], and therefore, the $(S_2)$ -hull $\mathcal{Q} \to \mathcal{Q}^{(**)}$ is an injection. So we obtain the exact sequence

\begin{equation*}0 \to \omega _X(D) \to \omega _X(E+D) \to \mathcal {Q}^{**}.\end{equation*}

We now claim that $\mathcal{Q}^{(**)}$ is isomorphic to $\nu _*(\omega _{E^{\nu }}(D_{E^{\nu }}))$ . By construction of the residue map, there is a natural homomorphism $\psi \colon \mathcal{Q} \to \nu _*(\omega _{E^{\nu }}(D_{E^{\nu }}))$ . As $\nu _*(\omega _{E^{\nu }}(D_{E^{\nu }}))$ is $S_2$ , then there is a natural map $\mathcal{Q}^{(**)} \to \nu _*(\omega _{E^{\nu }}(D_{E^{\nu }}))$ . As both $\mathcal{O}_X$ -modules are $(S_2)$ , it is sufficient to show equality at codimension 1 points of $E$ , which has been proved in Lemma 2.32. The linear equivalence $D_{E^{\nu }} \sim _{{\mathbb{Q}}} D|_{E^{\nu }} + \Gamma _E$ and $0 \leq \Gamma _{E^{\nu }} \leq{\mathrm{Diff}}_{E^{\nu }}(\Delta ^{\prime})$ is a codimension 2 statement and it is a consequence of Lemma 2.32.

For the last claim, if $\omega _X(D)$ is $(S_3)$ , then $\mathcal{Q}$ is $(S_2)$ by [Reference KollárKol13, Lemma 2.60], thus concluding that $r$ is surjective.

Proof. The proof is similar to [Reference PosvaPos21a, Lemma 3.6.5]. We fix $t \in \mathfrak{m}$ to be a uniformiser.

$\fbox{$Suppose\ t -\tau (t) \notin \mathfrak{m}^2.$}$ In this case, we set $\pi :=t - \tau (t)$ . Note that $\tau (\pi )=\tau (t)-\tau ^2(t)=i$ $\tau (t)-t=-\pi$ , and we conclude.

$\fbox{$Suppose\ t -\tau (t) \in \mathfrak{m}^2.$}$ Then there exists $f \in \mathfrak{m}$ such that $\tau (t)=(1+f)t$ . Moreover, it is easy to see that $\tau (t^k)-t^k \in \mathfrak{m}^{k+1}$ . We distinguish two cases.

$\fbox{$1.\ \tau\ acts\ non-trivially\ on\ k.$}$ Let $\alpha \in k \setminus k^{\tau }$ . Let $\widetilde{\alpha }$ be a lifting of $\alpha$ to $R$ . Note that $\widetilde{\alpha }-\tau (\widetilde{\alpha })(1+f)$ is invertible. Indeed, as $R$ is local, it is sufficient to note that $\widetilde{\alpha }-\tau (\widetilde{\alpha })(1+f) \equiv \alpha -\tau (\alpha ) \neq 0 \mod \mathfrak{m}$ by choice of $\alpha$ . Consider $s:= \widetilde{\alpha } t$ . Note that

\begin{equation*}s-\tau (s) \equiv \big (\widetilde {\alpha } - \tau (\widetilde {\alpha })\big ) t \neq 0 \mod \mathfrak {m}^2, \end{equation*}

as $\alpha -\tau (\alpha ) \neq 0$ . Thus $s-\tau (s) \notin \mathfrak{m}^2$ , and thus, we can conclude by the previous step.

$\fbox{$2.\ \tau\ acts\ trivially\ on\ k.$}$ We verify this contradicts the non-triviality of $\tau$ . We construct a recursive sequence $t_k$ such that $t_k -\tau (t_k) \in \mathfrak{m}^{k+1}$ . Fix $t_0=0$ and $t_1:=t$ . Suppose $t_k$ is defined. We have $t_k -\tau (t_k)= a t^{k+1} \mod \mathfrak{m}^{k+2}$ for some $a \in R$ . As $2$ is invertible in $R$ , we can define

\begin{equation*}t_{k+1}:=t_k - \frac {a}{2} t^{k+1}.\end{equation*}

Note that $\tau (\frac{a}{2}) \cong \frac{a}{2} \mod \mathfrak{m}$ by hypothesis. Therefore,

\begin{equation*} \tau (t_{k+1})-t_{k+1} \equiv -a t^{k+1}+\frac {a}{2} t^{k+1} - \tau \left (\frac {a}{2} \right ) t^{k+1} \mod \mathfrak {m}^{k+2} \equiv 0, \end{equation*}

as $\tau (a/2) \equiv a/2 \mod \mathfrak{m}$ . Now the sequence $(t_k)$ is a Cauchy sequence and thus converges to $s \in \mathfrak{m}$ . Note that by construction, $\tau (s)=s$ , reaching a contradiction (as $\tau$ is non-trivial).

Proof. The relation $R(\tau )$ is finite; thus the quotient $p \colon S \to U:=S/R(\tau )$ exists, and it is demi-normal by [Reference PosvaPos21a, Lemma 2.3.13]. Moreover, by [Reference KollárKol13, 9.13], the diagram

is a push-out square. We may assume that $U={\mathrm{Spec}}(R)$ is the spectrum of a local ring with maximal ideal $\mathfrak{m}_R$ . Therefore, $S$ is an affine regular scheme ${\mathrm{Spec}}(A)$ , and there exists a Cartier divisor $f \in A$ (resp. $h \in A$ ) such that $D=(f=0)$ (resp. $H=(h=0)$ ).

If $D \to D/R(\tau )$ is a $(\mathbb{Z}/2\mathbb{Z})$ -quotient we have two cases:

1. (i) $A$ has exactly two maximal ideals. In this case, up to an étale base change, we may assume $A=A_1 \oplus A_2$ , where $A_1$ and $A_2$ are local rings. Let $f_i \in A_i$ (resp. $h_i \in A_i$ ) be the local equations of $D|_{{\mathrm{Spec}}(A_i)}$ (resp. $H|_{{\mathrm{Spec}}(A_i)}$ ) for $i=1,2$ . Note that the transversality hypothesis on $H$ and $D$ implies that $(f_i, h_i)= \mathfrak{m}_{A_i}$ . Then the push-out property implies that $\mathfrak{m}_U=R \cap (\mathfrak{m}_{A_1} \oplus \mathfrak{m}_{A_2})$ . Let $\tau \colon A_1/f_1 \to A_2/f_2$ be the involution; therefore, $x:=(f_1,0), y=(0,f_2)$ and $z:=(h_1,h_2)$ generate $\mathfrak{m}_U$ . If $\tau$ is trivial, then we have the relations $x=y$ and $H/R(\tau )=(z=0)$ , and we are in case i of Definition 2.38. If $\tau$ is not-trivial, then we have the relation $xy=0$ , and therefore, ${\mathrm{Spec}}(R)$ is nodal and $H/R(\tau )=(z=0)$ , thus ending in case ii of Definition 2.38.

2. (ii) $A$ is a local ring such that $\tau (\mathfrak{m}_A)=\mathfrak{m}_A$ , and let $\tau \colon A/(f) \to A/(f)$ be an involution. The involution $\tau$ extends to the completion of $A$ and the completion of $R$ is the preimage of the $\tau$ -invariant elements of $A/(f)$ . As $\tau$ fixes $\mathfrak{m}_A$ , the residue field of $R$ is isomorphic to $k^{\tau }$ . The completion $\hat{A}/(f)$ is a complete DVR, and thus there exists a uniformiser $\pi \in \hat{A}/(f)$ such that $\tau (\pi )=-\pi$ by Lemma 2.39. Let $\widetilde{\pi }$ be a lifting of $\pi$ to $\hat{A}$ such that $H=(\widetilde{\pi }=0)$ (note that $h/\widetilde{\pi }$ is invertible). We distinguish two cases:

   1. (a) Suppose $k=k^{\tau }$ . Then $R \subset A$ is the subalgebra generated by $f, \widetilde{\pi }$ and $\widetilde{\pi }f$ . Moreover, $H/R(\tau )$ is given by the equations $ (\widetilde{\pi }=\widetilde{\pi }f=0),$ and thus we are in case iii of Definition 2.38.

   2. (b) Suppose $k^{\tau } \subsetneq k$ . Then there exists $\alpha \in k$ such that $k=k^{\tau }(\alpha )$ and $\tau (\alpha )=-\alpha .$ Let $A^{\prime}$ to be the preimage of $k^{\tau }$ under the projection $A \to k$ . In this case, consider the subalgebra of $A^{\prime}$ generated by $x:=\alpha \widetilde{\pi }, y:=f, z:= \alpha f$ . Therefore, we have the relation $ \alpha ^2y^2=z^2$ , showing that $R$ is a nodal singularities and that $H/R(\tau )$ is described by $(x=0)$ , showing we end up in case ii of Definition 2.38.

Proof. Let $\nu \colon (S^\nu, D_{S^\nu }) \to S$ be the normalisation morphism where $D_{S^\nu }$ is the conductor subscheme of $S^\nu$ . Let $f\colon X \to (S^\nu, D_{S^\nu }+\nu ^*H)$ be a log resolution of $(S^\nu, D_{S^\nu }+\nu ^*H)$ and let $D_X:=f_*^{-1}D_{S^\nu }$ . The involution $\tau$ lifts to an involution of $D_X$ , and we can apply Lemma 2.40 to construct a projective birational contraction $q \colon Y \to S$ fitting in the commutative diagram

such that $q \colon Y \to (S,H)$ is a semi-regular resolution of $(S,H)$ .

If $(S,H)$ is semi-dlt, then $(S^\nu, D_{S^n}+\nu ^*H)$ is a dlt pair. In this case, we can take $f \colon X \to (S^\nu, D_{S^n}+\nu ^*H)$ to be a thrifty log resolution of $(S^n, D_{S^n})$ , and end-product $T \to (S,H)$ is clearly a thrifty semi-regular resolution.

Proof. Consider $q \colon Y \to (S,H)$ be the semi-regular resolution constructed in Proposition 2.41. The only problem is around the pinch points of $Y$ , which are isolated by dimension reasons. Therefore, we can localise to a neighbourhood of $y \in Y$ , where $y$ is a pinch point, and we let $D_y$ be the local component of the non-normal locus $D_T$ . By blowing-up $D_y$ we obtain our desired semi-log resolution as explained in Remark 2.37.

Proof. Using the relative Kawamata–Viehweg vanishing theorem for surfaces [Reference TanakaTan18, Theorem 3.3], we can repeat the same steps of the proof of [Reference KollárKol13, Corollary 10.34].

Proof. The same proof of [Reference KollárKol13, Proposition 10.36] applies as thrifty semi-log resolutions by Theorem 2.43 exist for excellent surfaces and the necessary vanishing theorems hold by Proposition 2.44.

Proof. Let $g \colon Y \to S$ be a proper birational morphism such that the pair $(Y, \Delta _{D,Y})$ and the Cartier divisor $D_Y$ on $Y$ satisfy the conditions of Lemma 2.45, and denote by $h \colon Y \to T$ the natural composition. By Proposition 2.44, we conclude that $R^1h_*\mathcal{O}_Y(D_Y)=0$ . As $g_*{\mathcal{O}}_Y(D_Y)={\mathcal{O}}_S(D)$ and $R^ig_*{\mathcal{O}}_Y(D_Y)=0$ for $i\gt 0$ by Proposition 2.44 we deduce $R^1f_*{\mathcal{O}}_S(D)=0$ by the Leray spectral sequence.

Proof. We argue by contradiction. Let $E$ be an exceptional divisor such that $a(E, S, \Delta )=-1$ and ${\mathrm{cent}}_S(E) \subset{\mathrm{nsnc}}(S, \Delta )$ . The hypothesis (a) guarantees that $(S,\Delta )$ is snc at codimension 1 points. Therefore ${\mathrm{cent}}_S(E)$ is a closed point, contradicting (b).

Proof. Let $p \colon \widetilde{S} \to S$ be the double cover of Proposition 2.11. As $2$ is invertible, $\mathcal{O}_S(D)$ is a direct summand of $p_*\mathcal{O}_{\widetilde{S}}(p^*D)$ . We can thus assume that the irreducible components of $S$ are regular in codimension 1. In this case, $(S, \Delta )$ is semi-dlt by Lemma 2.47 and we conclude by Proposition 2.46.

Proof. We prove the result by induction on the number $n$ of irreducible components of $S$ . If $n=1$ , we conclude by Proposition 2.44.

We prove the induction step. Let $E$ be an irreducible component of $S$ such that $A|_E$ is big over $C$ . Let $T$ be the union of the irreducible components of $S$ except $E$ . Denote $B:=E \cap T$ , and we consider the short exact sequence:

\begin{equation*}0 \to \mathcal {O}_{E}(D|_{E}-B) \to \mathcal {O}_S(D) \to \mathcal {O}_{T}(D|_{T}) \to 0. \end{equation*}

Taking the long exact sequence in cohomology, it is sufficient to show that $R^1g_*\mathcal{O}_{E}(D-B)=R^1g_*\mathcal{O}_{T}(D|_{T})=0$ to conclude that $R^1g_*\mathcal{O}_S(D)=0$ .

Since $K_S|_{E}=K_{E}+B$ , we have $ D|_{E}-B \sim _{g,{\mathbb{Q}}} K_{E} +\Delta ^{\prime}|_{E}+A|_{E}$ ; so by Proposition 2.44, we conclude that $R^1g_*\mathcal{O}_{E}(D-B)=0$ .

Since $K_S|_{T}=K_{T}+B$ , we have that $ D_{T} \sim _{{\mathbb{Q}}, g} K_{T}+\Delta ^{\prime}|_{T}+B+A|_{T},$ and that $B$ is not trivial on some irreducible component of every connected component of $T$ , as $g$ has connected fibres. By hypothesis, $B$ must be a non-empty horizontal divisor; thus it is nef over $C$ , and for every connected component of $T$ , there exists an irreducible component $F$ such that $B|_F$ is $g|_F$ -big. Therefore, we apply the induction hypothesis to deduce $R^1g_*\mathcal{O}_{T}(D|_{T})=0$ .

Proof. As in the proof of Proposition 2.46, it sufficient to combine Lemma 2.45 and Proposition 2.49 with the Leray spectral sequence.

Proof. We can repeat the same proof of Corollary 2.48 using Proposition 2.50.