Proof. Let $\sigma \colon \widetilde S \to S$ be the minimal resolution and $(\widetilde {\mathcal {S}}, \mathcal E)$ be an R-lifting of $(\widetilde S, E_\sigma )$ . Shrinking the field of definition of the generic fiber of $(\widetilde {\mathcal {S}}, \mathcal E)$ if necessary, we obtain the base change $(\widetilde {S}_{\mathbb {C}}, E_{\mathbb {C}})$ of the generic fiber to $\operatorname {Spec} \mathbb {C}$ . Since $E_{\mathbb {C}}$ have the same intersection matrix as $E_\sigma $ , we have a contraction $\sigma _{\mathbb {C}} \colon \widetilde {S}_{\mathbb {C}} \to S_{\mathbb {C}}$ of $E_{\mathbb {C}}$ by [Reference Bădescu2, Theorem 3.9]. Since $H^1(\widetilde {S}_{\mathbb {C}}, \mathcal {O}_{\widetilde {S}_{\mathbb {C}}}) \cong H^1(\widetilde {S}, \mathcal {O}_{\widetilde {S}}) \cong 0$ and $H^0(\widetilde {S}_{\mathbb {C}}, 2K_{\widetilde {S}_{\mathbb {C}}}) \cong H^0(\widetilde {S}, 2K_{\widetilde {S}}) \cong 0$ , the Castelnuovo’s criterion shows that $\widetilde {S}_{\mathbb {C}}$ is a rational surface. Since $K_{\widetilde {S}_{\mathbb {C}}}^2=K_{\widetilde {S}}^2$ , the Picard rank of $\widetilde {S}_{\mathbb {C}}$ coincides with that of $\widetilde {S}$ , and hence the Picard rank of $S_{\mathbb {C}}$ is one. On the other hand, take $\mathcal H$ as a Cartier divisor on $\widetilde {\mathcal {S}}$ relatively ample over R and write $H_k$ (resp. $H_{\mathbb {C}}$ ) as the restriction of $\mathcal H$ to $\widetilde S$ (resp. $\widetilde {S}_{\mathbb {C}}$ ). Then

$$ \begin{align*} 0>(K_S \cdot \sigma_* H_k)=(\sigma^* K_{\widetilde S} \cdot H_k)=(\sigma^*_{\mathbb{C}} K_{\widetilde{S}_{\mathbb{C}}} \cdot H_{\mathbb{C}})=(K_{S_{\mathbb{C}}} \cdot (\sigma_{\mathbb{C}})_* H_{\mathbb{C}}), \end{align*} $$

which implies that $-K_{S_{\mathbb {C}}}$ is ample. Hence $S_{\mathbb {C}}$ is the desired surface.

Proof. Suppose that $H^1(S, \mathcal {O}_S (-A)) \neq 0$ . Then $H^0(S, (\Omega _S^{[1]} \otimes \mathcal {O}_S(-p^e A))^{**}) \neq 0$ for some $e \in \mathbb {Z}_{>0}$ by [Reference Kawakami8, Lemma 2.5], where $\Omega _S^{[1]}$ is the reflexive hull of $\Omega _S^1$ . In other words, we have an injection $s \colon \mathcal {O}_S(p^e A) \hookrightarrow \Omega _S^{[1]}$ . Let $C \in |-rK_S|$ be a smooth member. Then C is connected and the exact sequence $0 \rightarrow \mathcal {O}_S \rightarrow \mathcal {O}_S(-rK_X) \rightarrow \mathcal {O}_C(-rK_X|_C) \to 0$ gives that $\dim |-rK_S| \geq \frac {r+1}{2} (-rK_S)^2 \geq 1$ . In particular, $|-rK_S|$ has no fixed component and, by replacing k as an infinite field if necessary, we may assume that C avoids the associated locus of $\mathrm {Coker(s)}$ . Then $s|_C \colon \mathcal {O}_C(p^e A) \rightarrow \Omega _S^{[1]}|_C$ is still injective and $\Omega _S^{[1]}|_C=\Omega _S^1|_C$ . If the composition $t \colon \mathcal {O}_C(p^e A) \to \omega _C$ of $s|_C$ and the canonical map $\Omega _S^1|_C \to \omega _C$ is zero, then the conormal bundle sequence with respect to $C \subset S$ gives an injection $\mathcal {O}_C(p^e A) \hookrightarrow \mathcal {O}_C(-C)$ , a contradiction since $(p^e A \cdot C)>0> -C^2$ . Hence t is nonzero, which implies that $\deg \mathcal {O}_C(p^e A) \leq \deg \omega _C$ . Hence $p^e(-rK_S \cdot A) \leq (r-1) rK_S^2$ , and the assertion holds.

Proof. By [Reference Lacini14, Theorem 7.2], such surfaces lifts to characteristic zero over a smooth base. [Reference Arvidsson, Bernasconi and Lacini1, Proposition 2.5] now shows that they do not satisfy (NL). Hence the assertion follows from [Reference Kawakami and Nagaoka11, Proposition 3.4] and Lemma 3.4.

Proof. (1): By [Reference Lee, Park and Schenzel18, Theorem 3.1], we can choose the coordinates $[x:y:z]$ of $\mathbb {P}^2_k$ such that $C = \{x^3=y^2z\}$ and $M_u=\{y=0\}$ . Since the automorphism $[x:y:z] \longmapsto [\alpha x:y:\alpha ^3 z]$ of $\mathbb {P}^2_k$ with $\alpha \in k^*$ fixes C, we may assume that $t=[0:1:0]$ or $[1:1:1]$ . Since Q does not pass through $u = [0:0:1]$ , we can write $Q = \{ax^2+by^2+z^2+dxy+eyz+fzx=0\}$ for some $a, b,d,e,f \in k$ . Since

either $0$ or $1$ is the root of multiplicity at least five of $x^6+fx^4+ex^3+ax^2+dx+b=0$ . If the former holds, then $a=b=d=e=f=0$ and hence $Q=\{z^2=0\}$ , a contradiction. Hence the latter holds. Since the coefficient of $x^5$ is zero, we obtain $x^6+fx^4+ex^3+ax^2+dx+b=(x-1)^5(x+5)$ . Comparing coefficients, we obtain the assertion.

(2): As in the proof of the assertion (1), if $C = \{x^3=y^2z\}$ , then $Q=\{-45x^2-5y^2+z^2+24xy+40yz-15zx=0\}=\{(y+z)^2=0\}$ , a contradiction. Hence we can choose the coordinates $[x:y:z]$ of $\mathbb {P}^2_k$ such that $C = \{x^3=y^2z+x^2y\}$ and $M_u=\{y=0\}$ by [Reference Lee, Park and Schenzel18, Theorem 3.1]. Since the automorphism $[x:y:z] \longmapsto [x+\alpha y:y:\alpha x+(\alpha ^3-\alpha ^2)y+z]$ of $\mathbb {P}^2_k$ with $\alpha \in k$ fixes C, we may assume that $t=[0:1:0]$ . Since Q does not pass through $u=[0:0:1]$ , we can write $Q = \{ax^2+by^2+z^2+dxy+eyz+fzx=0\}$ for some $a, b,d,e,f \in k$ . Since

$0$ is the root of multiplicity five of $x^6+x^5+(1+f)x^4+(e-f)x^3+(a-e)x^2+dx+b=0$ . Comparing coefficients, we obtain the assertion.

(3): In $p \neq 3$ (resp. $p=3$ ), we have $L_{tu}=\{x=y\}$ (resp. $\{x=0\}$ ) and $L_{tu} \cap Q=\{[1:1:1], [1:1:-26]\}$ (resp. $\{[0:1:0], [0:1:1]\}$ ). Hence the assertion holds.

(4): In $p \neq 3$ (resp. $p=3$ ), we have $C \cap Q=\{[1:1:1], [-5:1:-125]\}$ (resp. $\{[0:1:0], [-1:1:1]\}$ ). Hence the assertion holds.

(5): In $p \neq 3$ (resp. $p=3$ ), $Q \cap M_u$ is isomorphic to (resp. ), whose discriminant equals $405=5 \cdot 3^4$ (resp. $5$ ). Hence the assertion holds.

(6): Suppose that $L_{su}$ is the tangent line of B at s. Assume that $p = 3$ in addition. Then $s=[-1:1:1]$ and $L_{su}=\{x=-y\}$ . However, $Q \cap L_{su} \cong \{x=-y, (x-z)(x+z)=0\}$ consists of two points, a contradiction.

Hence $p \neq 3$ , $s=[-5:1:-125]$ , and $L_{su}=\{x=-5y\}$ . Since

consists of one point, it holds that $125+10 = 135=5 \cdot 3^3$ equals zero in k. Hence the assertion holds.

Proof. Let be the irreducible decomposition and . Take non-negative rational vectors $\mathbf {a} = (a_i)_{i=1}^n, \mathbf {b} = (b_i)_{i=1}^n, \mathbf {d} = (d_i)_{i=1}^n, \mathbf {e} = (e_i)_{i=1}^n$ and a rational vector $\mathbf {f} = (f_i)_{i=1}^n$ so that $a_i= (\widetilde C \cdot E_i)$ , $\sigma ^* C = \widetilde C + \sum _{i=1}^n d_i E_i$ , $\sigma ^* K_S =K_{\widetilde S} +\sum _{i=1}^n e_i E_i$ , $\mathbf {b}=\mathbf {d}+\mathbf {e}$ , and $\mathbf {f}=\mathbf {1} - \mathbf {b}$ , where $\mathbf {1}=(1)_{i=1}^n$ . We will use the symbol $\langle $ -,- $\rangle $ to denote the inner product of vectors. Write $M =((E_i \cdot E_j))_{i=1, j=1}^{n, n}$ for the intersection matrix of $E_\sigma $ . By assumption, we have

$$ \begin{align*} 0 &\geq (K_S +C \cdot C) = (K_{\widetilde S}+ \widetilde C \cdot \sigma^* C) = (K_{\widetilde S}+\widetilde C \cdot \widetilde C) + \sum_{i=1}^n d_i (K_{\widetilde S} \cdot E_i) + \langle \mathbf{d}, \mathbf{a} \rangle \\ &\geq (K_{\widetilde S}+\widetilde C \cdot \widetilde C) = 2 (p_a(\widetilde C)-1). \end{align*} $$

Suppose that $p_a(\widetilde C) \geq 1$ . Then we have $p_a(\widetilde C)=1$ . Moreover $d_i=0$ or $a_i=0$ for each $1 \leq i \leq n$ , and hence $C \subset S^\circ $ . In particular, the last assertion holds.

In what follows, we may assume that $p_a( \widetilde C)=0$ and $C \not \subset S^\circ $ . Taking a resolution of $S \setminus C$ if necessary, we may also assume that $\operatorname {Sing} S \subset C$ . Then

$$ \begin{align*} (K_S +C \cdot C) &= (\sigma^*(K_S + C) \cdot \widetilde C) = (K_{\widetilde S}+\widetilde C \cdot \widetilde C) + \langle \mathbf{b}, \mathbf{a} \rangle \\ &= -2+ \langle \mathbf{b}, \mathbf{a} \rangle = -2+\sum_{i=1}^n a_i - \langle \mathbf{f}, \mathbf{a}\rangle \end{align*} $$

and hence

$$ \begin{align*} 0 \geq (K_S+C \cdot C) \iff \langle \mathbf{a}, \mathbf{b} \rangle \leq 2 \iff \langle \mathbf{a}, \mathbf{f}\rangle \geq \sum_{i=1}^n a_i -2. \end{align*} $$

Set $\operatorname {Sing} S = \{x_1, \ldots , x_m\}$ and take the decomposition $\mathbf {a}=\sum _{j=1}^m \mathbf {a}_j$ and $\mathbf {b} = \sum _{j=1}^m \mathbf {b}_j$ with respect to $\{x_1, \ldots , x_m\}$ so that $\langle \mathbf {a}, \mathbf {b} \rangle = \sum _{j=1}^m \langle \mathbf {a}_j, \mathbf {b}_j \rangle $ . If $(K_S+C \cdot C) \leq 0$ , then $\langle \mathbf {a}_j, \mathbf {b}_j \rangle \leq 2$ for all $1 \leq j \leq m$ . Since the condition (2) is local, we may assume that $\operatorname {Sing} S$ consists of one point contained in C, and

(5.1) $$ \begin{align} \langle \mathbf{a}, \mathbf{f} \rangle \geq \sum_{i=1}^n a_i -2. \end{align} $$

On the other hand, $M^{-1}$ has only negative entries by [Reference Kollár13, 2.19.3 Lemma]. If $\mathbf {a}$ becomes smaller, then so does $\langle \mathbf {a}, \mathbf {b} \rangle $ since $\mathbf {d}=-M^{-1} \mathbf {a}$ and $\mathbf {b}=\mathbf {d}+\mathbf {e}$ . In particular, if (5.1) holds for some $\mathbf {a}$ , then it also holds for smaller $\mathbf {a}$ . Similarly, if the equality of (5.1) holds for some $\mathbf {a}$ , then (5.1) does not hold for larger $\mathbf {a}$ .

Now let us show the first assertion by induction on $\mathbf {a}$ . Set $\Gamma $ as the weighted dual graph of $E_\sigma $ with the vertex $v_i$ corresponding to $E_i$ . Without further reference, we will use [Reference Kollár13, (3.1.10)] to calculate the log discrepancies $\mathbf {f}$ with respect to $(S, C)$ .

Case 1: Suppose that $\Gamma $ is a chain.

Case 1a: Suppose further that $a_i \neq 0$ if and only if $i=1$ . The subgraph $\Gamma - v_1= \Gamma _1+\Gamma _2$ is the disjoint union of two chains (each $\Gamma _i$ could be empty). Then

$$ \begin{align*} f_1 =\frac{\Delta(\Gamma_1)\Delta(\Gamma_2)}{\Delta(\Gamma)}(\frac1{\Delta(\Gamma_1)}+\frac1{\Delta(\Gamma_2)}-a_1). \end{align*} $$

If $a_1=1$ , then $\sigma $ is a log resolution of $(S, C)$ . Suppose further that $a_1 \geq 2$ . Then $f_1 \geq 0$ since $a_1 f_1 = \langle \mathbf {a}, \mathbf {f} \rangle \geq \sum _{i=1}^n a_i -2 \geq 0$ . Hence $\Delta (\Gamma _1)=\Delta (\Gamma _2)=1$ and $a_1=2$ . Hence $E_\sigma $ is irreducible. Moreover, either $\sigma $ is a log resolution of $(S, C)$ or $E_\sigma $ satisfies (a) or (b) of Definition 5.1. Note that when $a_1=2$ , we have $\langle \mathbf {a}, \mathbf {f} \rangle = \sum _{i=1}^n a_i -2$ and hence (5.1) does not hold for larger $\mathbf {a}$ .

Case 1b: Suppose further that $a_i \neq 0$ if and only if $i \leq 2$ . The subgraph $\Gamma - v_1-v_2= \Gamma _1+\Gamma _2+\Gamma _3$ is the disjoint union of three chains (each $\Gamma _i$ could be empty) such that and are connected. Since (5.1) also holds for $\mathbf {a}-(a_1, 0^{n-1})$ and $\mathbf {a}-(0, a_2, 0^{n-2})$ , the consequence of Case 1a shows that $a_1 =a_2=1$ . Hence (5.1) for $\mathbf {a}$ is equivalent to $f_1+f_2 \geq 0$ . Hence we may assume that $f_1 \geq 0$ . On the other hand, we have

$$ \begin{align*} f_1 = \frac{\Delta (\Gamma_1) \Delta(\Gamma_{2,3})}{\Delta(\Gamma)}\left( \frac{1-\Delta(\Gamma_1)}{\Delta(\Gamma_1)}+\frac{1-\Delta(\Gamma_3)}{\Delta(\Gamma_{2,3})}\right) \geq 0. \end{align*} $$

Hence $\Delta (\Gamma _1)=\Delta (\Gamma _3)=1$ . In particular, $\Gamma _1=\Gamma _3=\emptyset $ . Hence either $\sigma $ is a log resolution of $(S, C)$ or $E_\sigma $ satisfies (c) of Definition 5.1. Since it holds that $\langle \mathbf {a}, \mathbf {f} \rangle = \sum _{i=1}^n a_i -2$ , (5.1) does not hold for larger $\mathbf {a}$ .

Case 1c: Suppose further that $a_{i} \neq 0$ if $i \leq 3$ . Then (5.1) also holds for $(a_1, a_2, 0^{n-2})$ , a contradiction with the consequence of Case 1b.

Case 2: Suppose that $\Gamma $ is not a chain. Let $v_n$ be the center of $\Gamma $ and set $\Gamma _1, \Gamma _2$ , and $\Gamma _3$ be the branches of $\Gamma $ .

Case 2a: Suppose further that $a_i \neq 0$ if and only if $i=n$ . Then

$$ \begin{align*} f_n = \frac{\Delta(\Gamma_1) \Delta(\Gamma_2) \Delta(\Gamma_3)}{\Delta(\Gamma)} \left( \frac1{\Delta(\Gamma_1)} + \frac1{\Delta(\Gamma_2)} + \frac1{\Delta(\Gamma_3)} - (1+a_n)\right). \end{align*} $$

If $a_n \geq 2$ , then $f_n \geq 0$ since $a_n f_n = \langle \mathbf {a}, \mathbf {f} \rangle \geq \sum _{i=1}^n a_i -2 \geq 0$ . Hence (5.1) shows that

$$ \begin{align*} \frac32 \geq \frac1{\Delta(\Gamma_1)} + \frac1{\Delta(\Gamma_2)} + \frac1{\Delta(\Gamma_3)} \geq 1+a_n \geq 3, \end{align*} $$

a contradiction. Hence $a_1=1$ and $\sigma $ is a log resolution of $(S, C)$ .

Case 2b: Suppose further that $v_1 \in \Gamma _1$ and $a_i \neq 0$ if and only if $i=1$ . The subgraph $\Gamma _1 - v_1=\Gamma _{1,1}+\Gamma _{1,2}$ is the disjoint union of two chains (each $\Gamma _{1,i}$ could be empty) such that $\Gamma _{1,2}$ is adjacent to $v_n$ . Let . Then

$$ \begin{align*} f_1 = \frac{\Delta(\Gamma_{1,1}) \Delta(\Gamma')}{\Delta(\Gamma)} \left( \frac{1-(\Delta(\Gamma_3)-1)(\Delta(\Gamma_2)-1)}{\Delta(\Gamma')} + \frac{1}{\Delta(\Gamma_{1,1})} - a_1 \right). \end{align*} $$

If $a_1 \geq 2$ , then $f_1 \geq 0$ since $a_1 f_1 = \langle \mathbf {a}, \mathbf {f} \rangle \geq \sum _{i=1}^n a_i -2 \geq 0$ . Hence (5.1) shows that

$$ \begin{align*} 1 \geq \frac{1-(\Delta(\Gamma_3)-1)(\Delta(\Gamma_2)-1)}{\Delta(\Gamma')} + \frac{1}{\Delta(\Gamma_{1,1})} \geq a_1 \geq 2, \end{align*} $$

a contradiction. Hence $a_1 = 1$ and $\sigma $ is a log resolution of $(S, C)$ .

Case 2c: Suppose further that $v_1 \in \Gamma _1$ , $v_2 \in \Gamma _2$ and $a_i \neq 0$ if and only if $i \leq 2$ . Then for $i=1, 2$ , $\Gamma _i - v_i=\Gamma _{i,1}+\Gamma _{i,2}$ is the disjoint union of two chains (each $\Gamma _{i,j}$ could be empty) such that $\Gamma _{i,2}$ is adjacent to $v_n$ . Set . Since (5.1) holds for $\mathbf {a}-(a_1, 0^{n-1})$ and $\mathbf {a}-(0, a_2, 0^{n-2})$ , the consequence of Case 2b show that $a_1 =a_2=1$ . Hence (5.1) for $\mathbf {a}$ is equivalent to $f_1+f_2 \geq 0$ . Hence we may assume that $f_1 \geq 0$ . However, we have

$$ \begin{align*} f_1 &= \frac{\Delta(\Gamma') \Delta(\Gamma_{1,1})}{\Delta(\Gamma)} \left( \frac{1 - (\Delta(\Gamma_2) -1) (\Delta(\Gamma_3) - 1)}{\Delta(\Gamma')} + \frac{1-\Delta(\Gamma_{1,1})}{\Delta(\Gamma_{1,1})} - \frac{\Delta(\Gamma_{2,1}) \Delta(\Gamma_3)}{\Delta(\Gamma')}\right)\\ & < 0, \end{align*} $$

a contradiction.

Case 2d: Suppose further that $v_1, v_2 \in \Gamma _1$ and $a_i \neq 0$ if and only if $i \leq 2$ . Then $\Gamma _1 -v_1 - v_2 = \Gamma _{1,1}+\Gamma _{1,2} + \Gamma _{1,3}$ is the disjoint union of three chains (each $\Gamma _{1,i}$ could be empty). We may assume that $\Gamma _{1,3}$ is adjacent to both $v_2$ and $v_n$ , and $\Gamma _{1,2}$ is adjacent to both $v_1$ and $v_2$ . Set . , and . Since (5.1) holds for $\mathbf {a}-(a_1, 0^{n-1})$ and $\mathbf {a}-(0, a_2, 0^{n-2})$ , the consequence of Case 2b show that $a_1 =a_2=1$ . Hence (5.1) for $\mathbf {a}$ is equivalent to $f_1+f_2 \geq 0$ . However, we have

$$ \begin{align*} f_1 &= \frac{\Delta (\Gamma_{1,1}) \Delta (\Gamma_a)}{\Delta (\Gamma)} \left( \frac{1 - (\Delta (\Gamma_2) - 1) (\Delta (\Gamma _3) -1)}{\Delta (\Gamma_a)} + \frac{1- \Delta(\Gamma_{1,1})}{\Delta (\Gamma_{1,1})} - \frac{\Delta (\Gamma_b)}{\Delta (\Gamma_a)} \right) <0,\\ f_2 &= \frac{\Delta (\Gamma_{b}) \Delta (\Gamma_c)}{\Delta (\Gamma)} \left( \frac{1 - (\Delta (\Gamma_2) - 1) (\Delta (\Gamma _3) -1)}{\Delta (\Gamma_b)} + \frac{1- \Delta(\Gamma_c)}{\Delta (\Gamma_{c})} - \frac{\Delta (\Gamma_{1,1})}{\Delta (\Gamma_c)} \right) <0, \end{align*} $$

a contradiction.

Case 2e: Suppose further that $v_1 \in \Gamma _1$ and $a_i \neq 0$ if and only if $i = 1$ or n. The subgraph $\Gamma _1 - v_1=\Gamma _{1,1}+\Gamma _{1,2}$ is the disjoint union of two chains (each $\Gamma _{1,i}$ could be empty) such that $\Gamma _{1,2}$ is adjacent to $v_n$ . Set . Since (5.1) holds for $\mathbf {a}-(a_1, 0^{n-1})$ and $\mathbf {a}-(0^{n-1}, a_n)$ , the consequences of Cases 2a and 2b show that $a_1 =a_n=1$ . Hence (5.1) for $\mathbf {a}$ is equivalent to $f_1+f_n \geq 0$ . However, we have

$$ \begin{align*} f_1&=\frac{\Delta (\Gamma') \Delta (\Gamma_{1,1})}{\Delta (\Gamma)} \left( \frac{1 - (\Delta (\Gamma_2) - 1) (\Delta (\Gamma _3) -1)}{\Delta (\Gamma')} + \frac{1- \Delta(\Gamma_{1,1})}{\Delta (\Gamma_{1,1})} - \frac{\Delta (\Gamma_2) \Delta(\Gamma_3) }{\Delta (\Gamma')} \right)<0,\\ f_n&=\frac{\Delta(\Gamma_1)\Delta(\Gamma_2)\Delta(\Gamma_3)}{\Delta(\Gamma)} \left( \frac{1}{\Delta(\Gamma_1)} + \frac{1}{\Delta(\Gamma_2)} + \frac{1}{\Delta(\Gamma_3)} -2 - \frac{\Delta(\Gamma_{1,1})}{\Delta(\Gamma_1)}\right) <0, \end{align*} $$

a contradiction.

Case 2f: Suppose further that there is an integer $l \geq 3$ such that $a_{1}, a_{2}, a_{l} \neq 0$ . Then (5.1) holds for $(a_1, a_2, 0^{n-2})$ , a contradiction with the consequences of Cases 2c and 2d.

Combining these results, we obtain the assertion.

Proof. (1): We follow the notation of Lemma 2.6. By definition, there is an effective $\mathbb {Q}$ -divisor $\alpha $ such that $K_S+\alpha $ is anti-nef and $K_T+E+f^{-1}_*(\alpha ) = f^*(K_S+\alpha )$ . We may assume that E has maximal coefficient for $(S, \emptyset )$ among exceptional tigers in $\widetilde S$ .

Let us show that we can replace E so that $K_T+E$ is plt. For this, we may assume that $x = f(E)$ is a non-chain singularity and E is distinct from the central curve, say $E_0$ . Then there is a unique non-plt point of $K_T+E$ , say y. Let $\Gamma $ be the dual graph of the exceptional divisor in $\widetilde S$ over y and v (resp. $v_0$ ) the vertex of $\Gamma $ corresponding to E (resp. $E_0$ ). Then the subgraph $\Gamma - v_0 =\Gamma _1+\Gamma _2+\Gamma _3$ is the disjoint union of three chains ( $\Gamma _1$ could be empty) such that $\Gamma _1$ is adjacent to v and $v_0$ . By [Reference Kollár13, (3,1,10)], the log discrepancy of $E_0$ with respect to $K_T+E$ is

$$ \begin{align*} \frac{\Delta(\Gamma_1)\Delta(\Gamma_2)\Delta(\Gamma_3)}{\Delta(\Gamma)} \left( \frac{1}{\Delta(\Gamma_2)}+ \frac{1}{\Delta(\Gamma_3)} -1 \right) \leq 0. \end{align*} $$

Hence $e(E_0; S, \alpha ) = e(E_0; T, E+f^{-1}_*(\alpha )) \geq 1$ , and $E_0$ is an exceptional tiger. By [Reference Keel and McKernan10, 8.3.9 Lemma], we have $e(E_0; S, \emptyset ) \geq e(E; S, \emptyset )$ . Thus we can replace E as $E_0$ .

Suppose that T is a net. Let F be a general $\pi $ -fiber. Then

$$ \begin{align*} 0 \geq (f^*(K_S+\alpha) \cdot F) = (K_T+E+f^{-1}_*(\alpha) \cdot F) \geq -2+(E \cdot F). \end{align*} $$

Hence we obtain the description (a).

Suppose that $\pi \colon T \to S_1$ is birational. Let $q_1 = \pi (\Sigma ) \in S_1$ and $e = e(E; S, \emptyset )$ . Since $E^2 <0$ in T and $0<C^2$ in $S_1$ , E intersects with $\Sigma $ . Hence $q_1 \in C$ . By Lemma 2.6 (4), $K_T+aE$ is $\pi $ -trivial for some $a>e$ . Since $K_T+E \sim -f^{-1}_*(\alpha ) +f^*(K_S+\alpha )$ is numerically equivalent to an anti-effective $\mathbb {Q}$ -Weil divisor, so is $f_*(K_T+E)=K_S+C$ . Hence $K_S+C$ is anti-nef.

If $a \geq 1$ , then $K_{S_1}+C$ is plt since so is $K_T+E$ , and we obtain the description (c). Hence we may also assume that $a<1$ . By the choice of E, $K_{S_1}+C$ is plt away from $q_1$ . Lemma 5.2 now shows that either $C \subset S^\circ $ , $K_{S_1}+C$ is almost log canonical of one of types (a)–(c) in Definition 5.1 at $q_1$ , or the minimal resolution of $S_1$ is a log resolution of $(S_1, C)$ .

Suppose further that the first case occurs. If C is smooth, then we obtain the description (c). If C is singular, then C has a singularity of genus one at $q_1$ since $2(p_a(C)-1) \leq (K_{S_1}+C \cdot C) \leq 0$ . Hence we obtain the description (b).

Suppose further that the second case occurs. Take $E_1$ as an exceptional divisor in $\widetilde S_1$ over $q_1$ . Set $f(t) = e(E_1; S_1, tC)$ . Then $f(0) \geq 0$ . Moreover, $f(1) =1$ because the singularity $q_1 \in (S_1, C)$ has the same numerical property as a log canonical pair with analytically reducible boundary. Since f is an affine function, we obtain $f(a) \geq a$ . Now consider . Then $g(0) \geq 0$ . We have just seen that $g(a)=f(a) \geq a$ . Hence $g(e) \geq e$ . On the other hand, define . Then

$$ \begin{align*} f^*(K_S+\alpha) = K_T+E+f^{-1}_*(\alpha) = \pi^*(K_{S_1}+C + \pi_*f^{-1}_*(\alpha)) + d\Sigma. \end{align*} $$

Since $e(E_1; S, \alpha )=e(E_1; T, E+f^{-1}_*(\alpha )) \geq e(E_1; S_1, C)=1$ , the divisor $E_1$ is an exceptional tiger of S in $\widetilde S$ . Then $g(e) = e$ by the choice of E. Hence $g(t) \geq t$ for all $t \geq 0$ . However, we have $g(1) < 1$ since $(T, E)$ is plt, a contradiction.

Suppose further that the last case occurs. If $q_1 \in S_1^\circ $ , then $(S_1, C)$ is log smooth at $q_1$ by assumption and hence we have the description (c). Hence we may assume that $q_1 \not \in S_1^\circ $ . Let c be the log canonical threshold of $(S_1, C)$ at $q_1$ . Then $1 \geq c \geq a$ since $K_T+aE = \pi ^*(K_{S_1}+aC)$ . Assume that there is an exceptional divisor $E_1$ in $\widetilde S_1$ over $q_1$ such that $e(E_1; S_1, cC) =1$ . Set . Then $f(0) \geq 0$ and $f(c)=1 \geq c$ by assumption. Hence $f(a) \geq a$ . We now apply the same argument as in the second case to obtain a contradiction. Since $q_1 \not \in S_1^\circ $ , we conclude that $c=1$ and $(S, C)$ is plt at $q_1$ by [Reference Keel and McKernan10, 8.3.3 Lemma]. Hence we have the description (c).

Combining these results, we obtain the assertion.

(2): Let $C \subset S$ be a tiger. Then $K_S+C$ is anti-nef by definition. First suppose that $C \not \subset S^\circ $ . Then $(S, C)$ is log smooth around $S^\circ $ by Lemma 5.2. If there is an exceptional divisor E in $\widetilde S$ such that $e(E; S,C) \geq 1$ , then E is an exceptional tiger, a contradiction with the assumption. Now [Reference Keel and McKernan10, 8.3.3 Lemma] shows that $(S, C)$ is plt. Next suppose that $C \subset S^\circ $ . If $p_a(C)=0$ , then $K_S+C$ is plt. Hence we may assume that $p_a(C) \geq 1$ . Then $p_a(C) = 1$ , $C \sim -K_S$ , and S is a Du Val del Pezzo surface. If $K_S^2\geq 8$ , then $S \cong \mathbb {P}^2_k$ or a quadric cone in $\mathbb {P}^3_k$ and we can replace C as a conic so that $(S, C)$ is plt. Otherwise, we can replace C as a $(-1)$ -curve so that $C \not \subset S^\circ $ and hence $(S, C)$ is plt. Combining these results, we obtain the assertion.

Proof. The assertion follows from Theorem 5.3 and [Reference Lacini14, §5.1]. We note that [ibid., Example 7.6] belongs to LDP (b) by construction.

Proof. The assertion (1) $\Rightarrow $ (2) is obvious and (3) $\Rightarrow $ (4) follows from Lemma 3.4. Since each Dynkin type violates the Bogomolov bound [Reference Keel and McKernan10, 9.2 Corollary], the assertion (2) $\Rightarrow $ (3) holds. We can check that the arguments as in [Reference Lacini14, §4] hold in characteristic both zero and five except for [ibid., Proposition 4.4 (2) and (3)], that is, the existence of Example 4.2 in characteristic five. Hence the assertion (4) $\Rightarrow $ (1) holds.

Proof. The arguments as in [Reference Lacini14, §5.1] hold in characteristic both zero and five except for [ibid, Proposition 5.1 (2)], which shows the existence of [ibid, Example 7.6]. Since this example also belongs to LDP (b), we may assume that S belongs to LDP (a), (b), or (c) by Theorem 5.8 and the non-log liftability of S.

Suppose that S belongs to LDP (a). Let $\sigma \colon \widetilde S \to S$ be the minimal resolution. Then there is a $\mathbb {P}^1_k$ -fibration $\widetilde \pi \colon \widetilde S \to \mathbb {P}^1_k$ which factors through $\pi \colon T \to \mathbb {P}^1_k$ as in Definition 5.4. By [Reference Kawakami9, Remark 4.2 and Lemma 4.11], we obtain $H^2(\widetilde S, T_{\widetilde S}(-\operatorname {log} E_\sigma ))=0$ . Since $\widetilde S$ is rational, we have $H^2(\widetilde S, \mathcal {O}_{\widetilde S})=0$ . Thus $(\widetilde S, E_\sigma )$ lifts to $W(k)$ by [Reference Kawakami and Nagaoka11, Theorem 2.3], a contradiction.

Hence S belongs to LDP (b) or (c). By assumption, any log resolution of the pair $(S, \emptyset )$ does not lift to $W(k)$ . Then neither does any log resolution of the pair $(S_1, C)$ as in Definitions 5.5 and 5.6.

Suppose S belongs to LDP (c). Then, by [Reference Lacini14, §5.1], $(S_1, C)$ must be the surface pairs as in [ibid., Example 7.6]. By construction, the hunt step for $(S_1, C)$ gives the Du Val del Pezzo surface $S_2$ with Dynkin type $2[2^4]$ and its cuspidal anti-canonical divisor $C'$ contained in $S_2^\circ $ . Hence S also belongs to LDP (b).

Suppose S belongs to LDP (b). Repeated blow-ups at the intersection of C and a $(-1)$ -curve gives an extremal rational elliptic fibration Z with singular fiber F of type $II$ or $I_1$ . By [Reference Lang15, Theorem 4.1], the non-log liftability of $(S, \emptyset )$ shows that the singular fibers of Z are of type $II$ , $I_5$ , $I_5$ . Hence $\operatorname {Dyn} (S_1) =2 [2^4]$ and C is cuspidal, and the assertion holds.

Proof. (1): Take $\pi \colon T \to S_1$ and $E \subset T$ as in Definition 5.5. Note that the contraction of E always gives a log del Pezzo surface of rank one by [Reference Keel and McKernan10, 23.3 Lemma]. Since each $[2^4]$ -singularity is disjoint from C, we can write $\operatorname {Dyn} (S) = 2[2^4] + (\dagger )$ for some symbol $(\dagger )$ . When $\pi $ is one of configuration I, II, and III of [Reference Keel and McKernan10, §11.2], we obtain $(\dagger ) = [3], [2,4]$ , and $[2]+[3]+[5]$ , respectively.

Now we may assume that $\pi $ is none of them. In what follows, we use the notation of Definition 5.6. Then S also belongs to LDP (c) with $(S_1, C)$ the same as [Reference Lacini14, Example 7.6]. We also follow the notation of [Reference Keel and McKernan10, §11.4] to indicate a series of blow-ups constructing $\pi \colon T \to S_1$ . Note that interior blow-ups cannot appear as a component $\pi $ , since otherwise S contains a non-klt non-cyclic singularity.

Suppose that $q_1 \in C \cap S_1^\circ $ . Then blow-ups at farthest points cannot appear as a component $\pi $ , since otherwise S contains a non-klt non-cyclic singularity. Hence $\pi $ is indicated by $(n)$ or $(n^{a+1})$ and we have $(\dagger ) = [2; [2], [3], [5]]$ or $[2^a]+[2+a; [2], [3], [5]]$ for some $a \in \mathbb {Z}_{\geq 1}$ .

Suppose that $q_1$ is the $[2]$ -singularity. Then blow-ups at farthest points can appear only as the last component, since otherwise S contains a non-klt non-cyclic singularity. Hence $\pi $ is indicated by $(n^{a+1})$ or $(n^{a+1};f)$ for some $a \in \mathbb {Z}_{\geq 0}$ . When $a=0$ (resp. $a>0$ ), we have $(\dagger ) = [3]+[3, 2, 5]$ or $[4]+[2; [2], [3], [5]]$ (resp. $[2^a, 3]+[3, 2+a, 5]$ or $[3, 2^{a-1}, 3]+[2+a; [2], [3], [5]]$ ).

Suppose that $q_1$ is the $[3]$ -singularity. By a similar argument, $\pi $ is indicated by $(n^{a+1})$ , $(n^{a+1};f^b)$ or $(n^{a+1};f;n)$ for some $a \in \mathbb {Z}_{\geq 0}$ and $b \in \{1, 2\}$ . When $a=0$ (resp. $a>0$ ), we have $(\dagger )=[4]+[2, 2, 5]$ , $[4+b]+[2; [2], [2^b], [5]]$ , or $[2, 5]+[2;[2], [3], [5]]$ (resp. $[2^a, 4]+[2, 2+a, 5]$ , $[2+b, 2^{a-1}, 4]+[2+a; [2], [2^b], [5]]$ , or $[2, 3, 2^{a-1}, 4]+[2+a; [2], [3], [5]]$ ).

Suppose that $q_1$ is the $[5]$ -singularity. By a similar argument, $\pi $ is indicated by $(n^{a+1})$ , $(n^{a+1};f^b)$ , $(n^{a+1};f;n^c)$ , $(n^{a+1};f;n;f)$ , or $(n^{a+1};f^2;n)$ for some $a \in \mathbb {Z}_{\geq 0}$ , $b \in \{1, 2, 3, 4\}$ , and $c \in \{1,2,3\}$ . When $a=0$ (resp. $a>0$ ), we have $(\dagger )=[6]+[2, 2, 3]$ , $[6+b]+[2; [2], [3], [2^b]]$ , $[2^c, 7]+[2; [2], [3], [2+c]]$ , $[3, 7]+[2; [2], [3], [3, 2]]$ , or $[2, 8]+[2; [2], [3], [2, 3]]$ (resp. $[2^a, 6]+[2, 2+a, 3]$ , $[2+b, 2^{a-1}, 6]+[2+a;[2], [3], [2^b]$ , $[2^c, 3, 2^{a-1}, 6]+[2+a; [2], [3], [2+c]]$ , $[3, 3, 2^{a-1}, 6]+[2+a; [2], [3], [3, 2]]$ , or $[2, 4, 2^{a-1}, 6]+[2+a; [2], [3], [2, 3]]$ ).

Combining these results, we obtain the first assertion.

(2): When $(\dagger ) \neq [2^n]+[2+n; [2], [3], [5]]$ for any $n \geq 0$ , we see at once that $\operatorname {Dyn}(S)$ determines $\pi $ uniquely. Now suppose that $(\dagger ) = [2^n]+[2+n; [2], [3], [5]]$ for some $n \geq 0$ . Then S also belongs to LDP (c), and we can replace $(S_1, C)$ as [Reference Lacini14, Example 7.6]. Then there is a point $q_1 \in C \cap S_1^\circ $ such that $\pi $ is induced by the blow-up of $S_1$ at $q_1 (n+1)$ times along C. By construction, the induced morphism $T \to S$ is the extraction of the central curve of the $[2+n; [2], [3], [5]]$ -singularity. Hence the automorphism group of S equals that of T, and the group of automorphisms of $S_1$ fixing $q_1$ . On the other hand, since every automorphism of $S_1$ fixes the $[2]$ , $[3]$ , and $[5]$ -singularities and C, they fix each point of $C \cap S_1^\circ $ . Therefore the isomorphism classes of del Pezzo surfaces with Dynkin type $[2^n]+[2+n; [2], [3], [5]]$ are parametrized by $C \cap S_1^\circ \cong \mathbb {P}^1_k \setminus \{0,1, \infty \}$ . Combining these results, we obtain the second assertion.

Proof. If $(\dagger ) \neq [2; [2], [3], [5]]$ in addition, then $\operatorname {Dyn} (S)$ violates the Bogomolov bound [Reference Keel and McKernan10, 9.2 Corollary], and the assertion holds. Hence we may assume that $(\dagger ) = [2; [2], [3], [5]]$ .

Suppose by contradiction that there is a log del Pezzo surface $S_{\mathbb {C}}$ of rank one over $\mathbb {C}$ with $\operatorname {Dyn} (S_{\mathbb {C}})=2[2^4]+[2; [2], [3], [5]]$ . Run the hunt step for $(S_{\mathbb {C}}, \emptyset )$ , that is, take $f_0 \colon T_1 \to S_{\mathbb {C}}$ as the extraction of the central curve $E_1$ of $[2; [2], [3], [5]]$ and $\pi _1 \colon T_1 \to S_{1}$ be the $K_{T_1}$ -negative contraction. From now on, we follow the notation of [Reference Keel and McKernan10, 8.2.10]. An easy computation shows that $e_0=28/29<a_1$ .

Assume that $T_1$ is a net. Let F be a general $\pi _1$ -fiber. Since $0=(K_{T_1}+a_1 E_1 \cdot F)> -2+28/29(E_1 \cdot F)$ , we have $(E_1 \cdot F)=1$ or $2$ . If the former holds, then the $\pi _1$ -fiber intersecting with $E_1$ at the $[2]$ -singularity must contain another $[2]$ -singularity by [Reference Keel and McKernan10, 3.4 Lemma], a contradiction. Hence the latter holds. Since any $\pi _1$ -fibers cannot contain two of $[2]$ , $[3]$ and $[5]$ -singularities, there are three distinct $\pi _1$ -fibers $F_2$ , $F_3$ , and $F_5$ such that each $F_i$ passes through the $[i]$ -singularity. Since these fibers are multiple, $F_i \cap E_1$ supports on the $[i]$ -singularity for $i=2,3,5$ . It implies, however, that the double covering $\pi _1|_{E_1}$ ramifies at three points, a contradiction.

Hence $\pi _1$ must be birational. Next let us show that $(S_1, A_1)$ is plt. If $a_1 \geq 1$ , then $(S_1, A_1)$ is plt since so is $(T, E_1)$ . Hence, to this end, we may assume that $a_1<1$ . Then $(S_1, a_1 A_1)$ is flush by Lemma 2.8. If $q_1 \in S_1$ is singular, then $(S_1, A_1)$ is plt by Lemma 2.9. Suppose that $q_1 \in S_1$ is smooth. Then [Reference Keel and McKernan10, 8.3.7 Lemma (4)] shows that $A_1$ is normal crossing at $q_1$ since $a_1 \geq 4/5$ . Hence $A_1$ is either smooth or has a simple node at $q_1$ . Assume that the latter case holds. By [Reference Keel and McKernan10, 11.1.1 Lemma] and $a_1> 1/2$ , $E_1 \cap \Sigma _1$ consists of either one smooth point and the $[2]$ -singularity, or two singular points. In the former case, we obtain $a_1=2/3 < 28/29$ , a contradiction. In the latter case, $E_1 \cap \Sigma _1$ consists of the $[2]$ and $[3]$ -singularities since $S_1$ is smooth at $q_1$ . However, $(K_T+a_1 E_1 \cdot \Sigma _1)=0$ gives $a_1=4/5 < 28/29$ , a contradiction.

Hence $A_1$ is smooth at $q_1$ and $(S_1, A_1)$ is plt. If $\Sigma _1$ contains no $[2^4]$ -singularities, then $(S_1, A_1)$ violates the Bogomolov bound [Reference Keel and McKernan10, 9.2 Corollary]. From this fact, $\Sigma _1$ contains one $[2^4]$ -singularity. By the contractibility of $\Sigma _1$ , this point is the unique singularity of T which $\Sigma _1$ passes through. Hence $\operatorname {Dyn} (S_1)=[2^4]+[2]+[3]+[5]$ and $A_1$ passes through $[2]$ , $[3]$ , and $[5]$ -singularities. In particular, $K_{S_1}+A_1$ is anti-ample. However, this contradicts [Reference Keel and McKernan10, 23.5 Proposition] since $[2^4]$ is a cyclic singularity. Hence the assertion holds.

Proof. We follow the notation of [Reference Keel and McKernan10, 8.2.10]. Let $f_0 \colon T_1 \to S$ and $\pi _1 \colon T_1 \to S_1$ be the first hunt step. Since $T_1$ is a Du Val surface containing at least $2[2^4]$ singularities, [Reference Keel and McKernan10, 3.4 Lemma] shows that $T_1$ is not a net. Hence $\pi _1$ is birational.

By the contractibility of the $\pi _1$ -exceptional divisor $\Sigma _1$ , it contains at most one singularity and $K_{T_1}+\Sigma _1$ is plt. Then $\operatorname {Dyn} (S_1)$ is one of $[2]+2[2^4]$ , $2[2^4]$ , $[2]+[2^4]$ or $[2^4]$ . Comparing [Reference Lacini14, Theorem B.6], we have $\operatorname {Dyn} (S_1) \neq [2]+2[2^4]$ or $[2]+[2^4]$ . In particular, $q_1 \in A_1 \subset S_1^\circ $ . If $a_1 \geq 1$ , then $K_{S_1}+A_1$ is anti-ample and hence $A_1$ is a smooth rational curve in $S_1^\circ $ , a contradiction with [Reference Keel and McKernan10, 13.7 Lemma]. Hence $a_1 < 1$ and $(S_1, a_1 A_1)$ is flush by Lemma 2.8. Since $1/3 \leq e_0<a_1$ , [Reference Keel and McKernan10, 8.3.7 Lemma (1)] shows that the multiplicity of $A_1$ at $q_1$ is at most three. By [Reference Keel and McKernan10, 13.7 Lemma], the multiplicity cannot be one. Since $T_1$ is Du Val and $E_1 \cap \operatorname {Sing} T_1$ consists of at most one $[2]$ -singularity, [Reference Keel and McKernan10, §11.1–§11.3] shows that the multiplicity is two and $T_1$ has type I or II.

Assume that $\operatorname {Dyn} (S_1) =[2^4]$ . Then $\operatorname {Dyn} (S) =2[2^4]+[3]$ , $T_1$ has type I and $\Sigma _1$ contains one $[2^4]$ -singularity. In particular, $A_1$ has a double point of genus five at $q_1$ . On the other hand, write $A_1 \sim -nK_{S_1}$ for some $n \in \mathbb {Q}_{>0}$ . Then the genus formula yields $5 = p_a(A_1) = \frac 12 (A_1+K_{S_1} \cdot A_1)+1=\frac 52 n(n-1)+1$ . However, this has no rational solution, a contradiction.

Hence $\operatorname {Dyn} (S_1) =2[2^4]$ . If $\operatorname {Dyn} (S) =2[2^4]+[3]$ , then $T_1$ has type I and $\Sigma _1 \subset T_1^\circ $ . If $\operatorname {Dyn} (S) =2[2^4]+[2, 4]$ , then $T_1$ has type II and $\Sigma _1 \cap \operatorname {Sing} T_1$ contains the $[2]$ -singularity. In each case, $A_1$ has a double point of genus one at $q_1$ . Hence $A_1$ is a singular member of $|-K_{S_1}|$ , and the assertion holds.

Proof. Let $\sigma \colon \widetilde S \to S$ be the minimal resolution. Suppose by contradiction that S is log liftable over $W(k)$ . Let $E_\sigma = \sum _{i=1}^n E_i$ be the irreducible decomposition and take as a $W(k)$ -lifting of $(\widetilde S, E_\sigma )$ . Let K be a subfield of the fractional field of $W(k)$ such that the generic fiber of $(\widetilde {\mathcal {S}}, \mathcal {E})$ is defined over K and $K \subset \mathbb {C}$ . For a field extension $K \subset L$ , write and for each i. Let $\sigma _{\mathbb {C}} \colon \widetilde S_{\mathbb {C}} \to S_{\mathbb {C}}$ be the contraction of $E_{\mathbb {C}}$ . Take $D \subset \widetilde S$ and $D_{\mathbb {C}} \subset \widetilde {S}_{\mathbb {C}}$ as in Remark 6.7. Since $(D_{\mathbb {C}} \cdot -K_{\widetilde {S}_{\mathbb {C}}}) = (D \cdot -K_{\widetilde S})$ and $(D_{\mathbb {C}} \cdot E_{i, \mathbb {C}}) = (D \cdot E_i)$ for each $1 \leq i \leq n$ , $D_{\mathbb {C}}$ descends to $\operatorname {Spec} K$ and the closure $\mathcal {D}$ of $D_K$ in $\widetilde {\mathcal {S}}$ is a lift of D.

On the other hand, D spans an extremal lay of the Mori cone $\overline {\mathrm {NE}}(\widetilde S)$ . Since every line bundle on $\widetilde S$ is liftable to $\widetilde {\mathcal {S}}$ by [Reference Fantechi, Göttsche, Illusie, Kleiman, Nitsure and Vistoli7, Corollary 8.5.6(a)], D also spans an extremal lay R of $\overline {\mathrm {NE}}(\widetilde {\mathcal {S}}/\operatorname {Spec} W(k))$ . By [Reference Takamatsu and Yoshikawa19, Proposition 5.6], there is a contraction $\Phi \colon \widetilde {\mathcal {S}} \to \widetilde {\mathcal {S}}_1$ of R. Since the special fiber of $\widetilde {\mathcal {S}}_1$ is the contraction of D, which is smooth, $\widetilde {\mathcal {S}}_1$ is also smooth over $\operatorname {Spec} W(k)$ . Since $\Phi $ contracts the special fiber of $\mathcal {D}$ to a point, $\Phi (\mathcal {D})$ is a section over $\operatorname {Spec} W(k)$ .

Now suppose that $\operatorname {Dyn} (S)=2[2^4]+[2, 4]$ . Then there are exactly two irreducible components of $\mathcal {E}$ , say $\mathcal {E}_1$ and $\mathcal {E}_2$ which intersects with $\mathcal {D}$ as we have seen in the proof of Lemma 6.6. We may assume that $\mathcal {E}_2$ is the lift of the $(-4)$ -curve. Then is a simple normal crossing divisor over $\operatorname {Spec} W(k)$ and $(\widetilde {\mathcal {S}}_1, \mathcal {F})$ is a log lift of a log del Pezzo surface with Dynkin type $2[2^4]+[3]$ .

Hence we may assume that $\operatorname {Dyn} (S)=2[2^4]+[3]$ to prove the assertion. Then there is a unique irreducible component of $\mathcal {E}$ , say $\mathcal {E}_1$ which intersects with $\mathcal {D}$ as we have seen in the proof of Lemma 6.6. Then is a log lift of the Du Val del Pezzo surface $S_1$ with $\operatorname {Dyn} (S_1) = 2[2^4]$ . Lemma 6.6 also shows that defines a singular anti-canonical member in both the special fiber and the generic fiber. As the generic fiber, we get a weak del Pezzo surface $S_{1,K}$ , the union of all $(-2)$ -curves $F_K = \mathcal {F} \otimes _{W(k)} K$ , and one of two nodal anti-canonical members $A_{1,K} = \mathcal {A}_1 \otimes _{W(k)} K$ . In particular, the other nodal anti-canonical member of $S_{1, K}$ , say B, is also defined over K. Let $Z_K$ be the blow-up of $S_{1, K}$ at the basepoint of $|-K_{S_{1, K}}|$ , which is a K-rational point. Then $|-K_{Z_K}|$ defines an elliptic fibration $g \colon Z_K \to \mathbb {P}^1_K$ . Let G be the strict transform $F_K \cup A_{1, K} \cup B$ in $Z_K$ . Then $g(G) \subset \mathbb {P}^1_K$ consists of four K-rational points and $g^{-1}(g(G))$ is a disjoint union of four singular g-fibers. We fix coordinates $s, t$ of $\mathbb {P}^1_K$ such that $g(G)=\{st(t-s)(t-\alpha s)=0\}$ for some $\alpha \in k \setminus \{0,1\}$ .

On the other hand, $g(G)$ is projectively equivalent to $\{st(t^2+11st-s^2)=0\}$ by Example 2.10. Hence there is an automorphism $\iota $ of $\mathbb {P}^1_K$ which sends $g(G)$ to $\{st(t^2+11st-s^2)=0\}$ . When $\iota $ sends $[1:0]$ , $[0:1]$ , $[1:1]$ , and $[1:\alpha ]$ to $[a_0:a_1]$ , $[b_0,b_1]$ , $[c_0:c_1]$ , and $[d_0:d_1]$ respectively, then an easy computation shows that

$$ \begin{align*} \alpha=\frac{b_0c_1-b_1c_0}{a_1c_0-a_0c_1} \cdot \frac{d_0a_1-a_0d_1}{b_0d_1-b_1d_0}. \end{align*} $$

In particular, $\alpha $ is a solution of one of $x^2-123x+1=0$ , $121x^2-121x-1=0$ , or $x^2+121x-121=0$ . Since we can transform those equations as

$$ \begin{align*} \left( \frac{1}{55} (2x-123)\right)^2=5, \left(\frac{11}{5} (2x-1) \right)^2=5, \text{ and } \left( \frac{1}{55} (2x+121) \right)^2=5 \end{align*} $$

respectively, there is a solution of the equation $y^2=5$ in K, a contradiction to the Eisenstein’s criterion. Hence the assertion holds.

Proof of Theorem 1.2

If S has no tiger, then the assertion holds from Lemma 6.1. Hence we may assume that S has a tiger. By Lemmas 6.2 and 6.3, to show the assertions (1) and (2), it suffices to show that S is not log liftable over $W(k)$ if $\operatorname {Dyn}(S)=2[2^4]+(\dagger )$ , where $(\dagger )$ is listed in Table 1. When $(\dagger ) \neq [3]$ or $[2,4]$ (resp. $(\dagger ) = [3]$ or $[2,4]$ ), we get the assertions by Lemmas 3.4 and 6.4. (resp. Proposition 6.8). The assertion (3) follows from Lemma 6.4 and Example 6.5.

Proof. (1): By Lemma 6.1, S is not log liftable over $W(k)$ . If S has a tiger, then Lemmas 6.2 and 6.3 shows that S contains $2[2^4]$ -singularities, a contradiction with Example 4.2.

(2): Let U be the blow-up of $\mathbb {P}^2_k$ as in Example 4.2 with $(n_t, n_u)=(5,2)$ . Then there is the induced birational morphism $\widetilde S \to U$ from the minimal resolution $\widetilde S$ of S and we can check that U is the minimal resolution of a log del Pezzo surface with Dynkin type $2[2^4]+[3]$ . Hence the assertion follows from Lemma 6.6.

Proof of Theorem 1.3

When S has a tiger (resp. no tiger), then Lemma 6.2 (resp. Lemma 6.1 and Corollary 6.9) gives the assertion.

Proof. We follow the notation of [Reference Keel and McKernan10, 8.2.10]. Since $-rK_S$ is Cartier, it suffices to find a smooth member of $|-rK_S|$ . When $\operatorname {Dyn} (S) =2 [2^4] +[3]$ , we have $f_0^*(-3K_S) \sim -3K_{T_1} -E_1 \sim 3(\pi _1^*(-K_{S_1})-\Sigma _1)-(\pi _1^*(A_1)-2\Sigma _1) \sim \pi _1^*(-2K_{S_1})-\Sigma _1$ . On the other hand, when $\operatorname {Dyn} (S) =2 [2^4] +[2, 4]$ , we have $f_0^*(-7K_S) \sim -7K_{T_1} -4E_1 \sim 7(\pi _1^*(-K_{S_1})-2\Sigma _1)-4(\pi _1^*(A_1)-3\Sigma _1) \sim \pi _1^*(-3K_{S_1})-2\Sigma _1$ . Hence, to show the assertion, it suffices to find a smooth member $D_i \in |-iK_{S_1}|$ such that $D_i \cap A_1$ supports on the cusp of $A_1$ for $i=2$ and $3$ .

By Lemma 6.6 and Example 2.10, $S_1$ is defined by $y^2-(x^3+2t^4x+4s^5t+2t^6)=0$ in $\mathbb {P}(1,1,2,3)$ , where s, t, x, and y are coordinates of weight $1$ , $1$ , $2$ , and $3$ , respectively. Moreover, $A_1$ is defined by $t=0$ and its cusp is $[1:0:0:0]$ . Then we can choose the desired $D_2$ (resp. $D_3$ ) as the curve defined by $x-t(s+2t)=0$ (resp. $y-t(x+t^2+st+3s^2)=0$ ), and the assertion holds.

Proof. Let A be an ample $\mathbb {Z}$ -Weil divisor on S. Suppose that $\operatorname {Dyn} (S) =2 [2^4] +[3]$ . Then S is of index $3$ and $K_S^2=1/3$ . Since $p=5> 3 \cdot (3-1) \cdot K_S^2=2$ , Lemmas 3.5 and 6.10 yield $H^1(S, \mathcal {O}_S(-A))=0$ .

Now suppose that $\operatorname {Dyn} (S) = 2[2^4] +[2, 4]$ . Then S is of index $7$ and $K_S^2=1/7$ . From now on, we follow the notation of Example 2.10 and Lemma 6.6. Then $U_2$ is the minimal resolution of $S_1$ and $C_2$ is the strict transform of $A_1$ in $\mathbb {P}^2_k$ . For abbreviation, we continue to write $C_2$ for its strict transform in $U_2$

Now take $\widetilde \pi _1 \colon \widetilde S \to U_2$ as the blow-up of $U_2$ at the cusp of $C_2$ two times along $C_2$ . Then Lemma 6.6 shows that $\widetilde S$ is the minimal resolution of S. There are two $\widetilde \pi _1$ -exceptional curves $G_1$ and $G_2$ such that $G_1^2=-2$ and $G_2^2=-1$ . To simplify notation, we continue to write $L_{ab}, \ldots , L_{cd}, E_{a}, \ldots , E_d, F_a, \ldots , F_d, C_2$ for the strict transform of these curves in $\widetilde S$ . Note that the minimal resolution $\sigma \colon \widetilde S \to S$ is the contraction of $L_{ab}$ , $L_{bc}$ , $L_{cd}$ , $L_{ad}$ , $E_a$ , $E_b$ , $E_c$ , $E_d$ , $G_1$ , and $C_2$ . We denote $L_{ac, V}, L_{bd, V}, F_{a, V}, \ldots F_{d, V}$ and $G_{2, V}$ the strict transform of $L_{ac}, L_{bd}, F_a, \ldots , F_d$ and $G_2$ in a birational model V of S respectively. By construction, the class divisor group of S is generated by $L_{ac, S}, L_{bd, S}, F_{a, S}, \ldots F_{d, S}$ and $G_{2, S}$ . Easy calculations gives that