Proof. Recall that a variety X over k has the property $(S_n)$ if and only if $X_{\overline {k}}$ also has, by faithfully flat descent. As Y is $(S_2)$ , by Serre’s criterion [Sta, Tag 031S], Y is geometrically normal if and only if it is geometrically $(R_1)$ . Suppose by contradiction that there exists a codimension 1 point $\eta \in Y$ such that the localisation $\mathcal {O}_{Y, \eta }$ is not geometrically regular. As Y is normal, $\pi $ is an isomorphism over codimension 1 points of Y, and thus, X is not geometrically $(R_1)$ , reaching the contradiction.

Proof. Rationality of klt surface singularities follows from [Reference KollárKol13, Proposition 2.28], and $\mathbb {Q}$ -factoriality of rational singularities is proven in [Reference LipmanLip69, Proposition 17.1]. The last statement is obvious by considering a common resolution of X and Y.

Proof. If X is canonical, then it is rational by Proposition 2.6. Suppose now that X is rational and let $f \colon Y \to X$ be a resolution of singularities. As X is Gorenstein and X has rational singularities, we have that $f_*\omega _Y = \omega _X$ by [Reference KollárKol13, Proposition 2.77], which in turn implies that X has canonical singularities by [Reference KollárKol13, Claim 2.3.1].

Proof. We can suppose k is separably closed and Y is the spectrum of a local henselian ring $(R, \mathfrak {m})$ by the existence of resolution of singularities [Reference LipmanLip78]. Let be the punctured spectrum. Since Y is rational, the group $ {\mathrm {Pic}}(U)$ is finite by [Reference LipmanLip69, Proposition 17.1]. Therefore, also $X:=Y_{\overline {k}}$ is $\mathbb {Q}$ -factorial by [Reference TanakaTan18a, Lemma 2.5], and thus, $ {\mathrm {Pic}}(U_{\overline {k}})$ is a torsion group. Let $f \colon W \to X$ be the minimal resolution with exceptional divisor $E=\sum _{i=1}^n E_i$ . As defined in [Reference LipmanLip69], $ {\mathrm {Pic}}^0(W)$ is the group of line bundles L on W such that $L \cdot E_i=0$ for every i and there is an exact sequence of groups $0 \to {\mathrm {Pic}}^0(W) \to {\mathrm {Pic}}(W) \to \bigoplus \mathbb {Z}[E_i] \to 0$ . By [Reference LipmanLip69, Proposition 14.4], $ {\mathrm {Pic}}^0(W)$ embeds into $ {\mathrm {Pic}}(U_{\overline {k}})$ , and thus, we deduce it is a torsion group.

Suppose now by contradiction that X is not rational. By the classification of log canonical singularities [Reference KollárKol13, Corollary 3.39], the exceptional divisor E is either an elliptic curve, a nodal curve or a circle of smooth rational curves. In the first case, $ {\mathrm {Pic}}^0(E)\simeq E(k)$ , while in the latter cases, $ {\mathrm {Pic}}^0(E) \simeq k^*$ by [Reference Bosch, Lütkebohmert and RaynaudBLR90, Chapter 9.3, Corollary 11 and 12] and since $h^1(E,\mathcal {O}_E) = 1$ . By [Reference LipmanLip69, Lemma 14.3], the restriction map $ {\mathrm {Pic}}(W) \to {\mathrm {Pic}}(E)$ is surjective. Considering the exact sequence $0 \to {\mathrm {Pic}}^0(E) \to {\mathrm {Pic}}(E) \to \mathbb {Z}^{n} \to 0$ , we can deduce that the map $ {\mathrm {Pic}}^0(W) \to {\mathrm {Pic}}^0(E) $ is surjective. This is a contradiction, as $ {\mathrm {Pic}}^0(W)$ is torsion while $k^*$ and $E(k)$ are not.

Proof. By Proposition 2.9, X is geometrically log canonical. As X has rational singularities, X is geometrically rational by Lemma 2.10. As X is Gorenstein, we conclude that X is geometrically canonical by Corollary 2.7.

Proof. The integrality of the conductor follows from [Reference ReidRei94, Lemma, page 718] for integral del Pezzo surfaces. Then, (1) follows from the proof of [Reference ReidRei94, Corollary 4.10], as D is irreducible. (2) is proved in [Reference ReidRei94, 4.11]

Proof. We only prove the case where X is a regular weak del Pezzo surface, as the others are similar. As $-K_X$ is $\pi $ -big and $\pi $ -nef, we conclude that Y has canonical singularities by the negativity lemma [Reference TanakaTan18b, Lemma 2.11]. As $-K_Y=\pi _*(-K_X)$ , we conclude by projection formula that $-K_Y$ is big and nef.

Proof. By Serre duality, we have $H^2(X, \mathcal {O}_X)=H^0(X, \omega _X)=0$ , and therefore, by [Reference Fantechi, Göttsche, Illusie, Kleiman, Nitsure and VistoliFGI05, Corollary 9.4.18.3, Corollary 9.5.13 and Remark 9.5.15], the group scheme $ {\mathrm {Pic}}^0_{X/k}$ is smooth of dimension $h^1(\mathcal {O}_X)$ . We are left to show that $ {\mathrm {Pic}}^0_{X/k}$ is unipotent. For this, we can suppose k is separably closed. By [Reference Bernasconi and TanakaBT22, Theorem 1.3], there exists $n>0$ such that for every $L \in {\mathrm {Pic}}^0(X)$ , we have $L^{\otimes p^n} \simeq \mathcal {O}_X$ . In other words, multiplication by $p^n$ on $ {\mathrm {Pic}}^0_{X/k}$ coincides with the zero homomorphism on k-rational points. By density of rational points [Reference PoonenPoo17, Proposition 3.5.70] and since $ {\mathrm {Pic}}^0_{X/k}$ is reduced, we conclude that taking $p^n$ -powers on $ {\mathrm {Pic}}^0_{X/k}$ coincides with the zero homomorphism as a morphism of schemes, and thus, $ {\mathrm {Pic}}^0_{X/k}$ is unipotent.

Proof. Recall that X is rational and $\mathbb {Q}$ -factorial by Proposition 2.6. Let D be a Weil integral divisor on X, and write $f^{*} D = f_*^{-1} D + \sum _{i=1}^n a_i E_i$ for some $a_i \in \mathbb {Q}$ .

We claim it is sufficient to show $d_xa_i$ is integral. Indeed, then $f^{*} (d_xD)$ is an integral divisor on a regular surface, and thus, $f^{*} (d_xD)$ is Cartier. If we write $K_Y+\Delta _Y=f^*(K_X+\Delta )$ , then $\Delta _Y$ is effective by the negativity lemma and $(Y,\Delta _Y)$ is klt. As $f^*(d_xD)-(K_Y+\Delta _Y)$ is f-nef and big, and $f^*(d_xD)$ is f-trivial, there exists $b_0>0$ such that for all $b \geq b_0$ , we have that $bf^*(d_xD)=f^*A_{b}$ for a Cartier divisor $A_b$ on X by the base point free theorem for excellent surfaces [Reference TanakaTan18b, Theorem 4.4]. Then $f^*(d_xD)=(b_0+1)f^*d_xD-b_0f^*d_xD=f^*(A_{b_0+1}-A_{b_0})$ , and thus, $d_xD$ is Cartier.

We denote by $(a_i)$ (resp. $f_*^{-1} D \cdot E_j$ ) the vector $(a_1, \dots , a_n)$ (resp. $(f_*^{-1} D \cdot E_1, \dots , f_*^{-1} D \cdot E_n)$ ). Given a closed point $x \in X$ , we denote by $k(x)$ the residue field of X at x. By the projection formula,

$$ \begin{align*}(a_i)=M^{-1} (-f_*^{-1} D \cdot E_j)=\frac{1}{d_x [k(x):k]} A (f_*^{-1} D \cdot E_j),\end{align*} $$

where A is a matrix with integer coefficients. We have $(-f_*^{-1} D \cdot E_j) = \sum _j m_j [k(y_j):k]$ for some $m_j \in \mathbb {Z}$ , where the $y_j$ are the intersection points of $f_*^{-1}D$ with $E_j$ . As $k(x) \subset k(y_j)$ , we conclude that $[k(x):k]$ divides $(f_*^{-1} D \cdot E_j)$ , thus showing $d_x a_i$ is an integer.

Proof. As $K_{\mathbb {P}(1,1,d)} \sim (-d-2)L$ and the Cartier index of L is d, the lemma is immediate.

Proof. We can assume k to be separably closed, and we will repeatedly use the fact that $\epsilon (X/k)$ is a k-birational invariant [Reference TanakaTan21, Proposition 7.10]. If X is geometrically normal, then we conclude by Proposition 2.11. So we suppose that X is not geometrically normal and $p =2,3$ by [Reference Bernasconi and TanakaBT22, Theorem 3.7.(1)]. The bounds on $\ell _F(X/k)$ are proven in [Reference Bernasconi and TanakaBT22, Theorem 3.7.(2)-(3)].

Let $Z \to X$ be the minimal resolution of X. As Z is a regular weak del Pezzo surface, by Lemma 2.14, a $K_Z$ -MMP will end with a regular weak del Pezzo surface Y admitting a Mori fibre space $f \colon Y \to B$ (i.e., f is a contraction where $-K_Y$ is f-ample and $\dim (B) \leq 1$ ). Note that $K_Y^2 \geq K_Z^2=K_X^2$ .

If $B= {\mathrm {Spec}}(k)$ , then Y is a regular del Pezzo surface, and we conclude by [Reference TanakaTan24, Theorem 4.7]. If B is a curve, as Y is weak del Pezzo, the cone theorem [Reference TanakaTan18b, Theorem 2.14] implies that the Mori cone of Y is

$$ \begin{align*}{\mathrm{NE}}(Y)=\mathbb{R}_{+}[F]+\mathbb{R}_{+}[\Gamma],\end{align*} $$

where F the class of a closed fibre of f and $\Gamma $ is the class of an integral curve with self-intersection $\Gamma ^2 \leq 0$ . If $K_Y \cdot _k \Gamma <0$ , then Y is a regular del Pezzo surface by Kleiman’s criterion, and we conclude again by [Reference TanakaTan24, Theorem 4.7].

If $K_Y \cdot _k \Gamma =0$ , by the Hodge index theorem $\Gamma ^2<0$ and, if we denote $k_{\Gamma }=H^0(\Gamma , \mathcal {O}_{\Gamma })$ , by adjunction, the equality $\Gamma ^2=\deg _k \omega _{\Gamma /k}=-2[k_{\Gamma }:k]$ holds. Then there exists a birational contraction $Y \to T$ where T is a canonical del Pezzo surface of Picard rank 1 with a unique singular point x and $K_T^2=K_Y^2$ . As $k_\Gamma =k(x)$ by [Reference KollárKol13, Corollary 10.10], we have $\Gamma ^2=-2[k(x):k]$ , which implies that the Cartier index of T divides $2$ by Lemma 3.1. If T is geometrically normal, it is geometrically canonical by [Reference Bernasconi and TanakaBT22, Theorem 3.7]. Moreover, as $T_{\overline {k}}$ has Picard rank 1 and a singular point, we conclude $K_X^{2} \leq 8$ . If T is not geometrically normal and $g \colon V=(T \times _k \overline {k})_{ {\mathrm {red}}}^{ {\mathrm {norm}}} \to T$ is the normalised base change where $K_V+(p-1)C=g^*K_T$ , we deduce that $2p^{\ell _F(T/k)} K_V$ is Cartier by [Reference TanakaTan21, Theorem 5.12]. By the classification of the normalised base changes of canonical del Pezzo surfaces with Picard rank 1 [Reference Patakfalvi and WaldronPW22, Theorem 4.1], the bounds on the Frobenius length [Reference Bernasconi and TanakaBT22, Theorem 3.7] and Lemma 3.3, we deduce the following:

• ○ if $p=3$ , then $6K_V$ is Cartier, and thus, $V \simeq \mathbb {P}(1,1,d)$ for $d \in \left \{1, 2, 3, 4, 6, 12\right \}$ and $C =L$ ;

• ○ if $p=2$ , then $8K_V$ is Cartier, and thus, $V \simeq \mathbb {P}(1,1,d)$ for $d \in \left \{1, 2, 4, 8, 16\right \}$ and $C=L$ or $2L$ by [Reference TanakaTan24, Proposition 4.1].

Using [Reference TanakaTan24, Lemma 4.5], we have

$$ \begin{align*}p^{\epsilon(X/k)}(g^*K_T)^2=p^{\epsilon(X/k)} (K_V+(p-1)C)^2 = K_T^2.\end{align*} $$

If $p=3$ , we have $V=\mathbb {P}(1,1,d)$ , $C=L$ , and thus, $K_T^2 \leq (dL)^2 \cdot 3^{\epsilon (X/k)}= d \cdot 3^{\epsilon (X/k)} \leq 12 \cdot 3^{\epsilon (X/k)} $ . Similarly, in the case where $p=2$ , we obtain that $K_T^2 \leq 16 \cdot 2^{\epsilon (X/k)}$ .

Proof. By Proposition 3.4, we are only left to prove the classification in (3) and (4). Suppose $p=3$ . The only possible normalised base change is $\mathbb {P}(1,1,d)$ by [Reference TanakaTan24, Proposition 4.1 and Remark 4.3]. However, by Proposition 3.4, we have $K_X^2=p^{\epsilon (X/k)} d \leq 12 \cdot p^{\epsilon (X/k)} $ .

Suppose $p=2$ . The list of possibilities without the bounds on d is proved in [Reference TanakaTan24, Proposition 4.1]. It is now sufficient to note that in Case (4f), $K_X^2=p^{\epsilon (X/k)}(d+2)$ ; in Case (4g), $K_X^2=p^{\epsilon (X/k)}(d+4)$ ; and in Case (4c), $K_X^2=p^{\epsilon (X/k)}d$ . Using Proposition 3.4, we deduce the desired bounds on d.

Proof. We let $\mathcal {A}_m=\mathcal {O}_X(mA)$ for $m \in \mathbb {Z}$ . We fix $d>0$ . We show that $H^1(X, \mathcal {A}_{-dn})=0$ for n sufficiently large. If A is ample, we conclude by Serre duality and Serre vanishing. If A is only big and nef and X is a canonical del Pezzo surface, by the base point free theorem [Reference BernasconiBer21b, Proposition 2.1], there is a birational contraction $\pi \colon X \to Y$ such that $A=\pi ^*H$ , where H is an ample Cartier divisor and by Lemma 2.14, Y is a del Pezzo surface with canonical singularities. Thus, the singularities of Y are rational by Proposition 2.6, and the projection formula implies $H^1(X, \mathcal {A}_{-dn})=H^1(Y, \mathcal {H}_{-dn})$ . As Y is a normal, we can apply Serre duality to deduce $H^1(Y, \mathcal {H}_{-dn}) \simeq H^1(Y, \omega _X \otimes \mathcal {H}_{dn})$ , which vanishes for n large enough by Serre vanishing.

Suppose $H^1(X,\mathcal {A}_{-d}) \neq 0$ . Without loss of generality by the previous paragraph, we can assume $F^*\colon H^1(X,\mathcal {A}_{-d}) \to H^1(X,\mathcal {A}_{-pd})$ has a nontrivial element $\zeta $ in the kernel $H^1_{ {\mathrm {fppf}}}(X, \alpha _{\mathcal {A}_{-d}})$ . By [Reference Patakfalvi and WaldronPW22, Theorem 2.11], associated to $\zeta $ there exists a degree p purely inseparable morphism $\pi \colon Z \to X$ such that Z is an integral Gorenstein surface with $\omega _Z=\pi ^*(\omega _X(-(p-1)dA))$ . Let $\nu \colon Z^{ {\mathrm {norm}}} \to Z$ be the normalisation and let $\mu \colon Y:=(Z^{ {\mathrm {norm}}} \times _k \overline {k})_{ {\mathrm {red}}}^{ {\mathrm {norm}}} \to Z^{ {\mathrm {norm}}}$ be the normalised base change to the algebraic closure. We denote by $\Gamma $ the divisorial part of the ramification locus. We have $\mathcal {O}_{Z^{ {\mathrm {norm}}}}(K_{Z^{ {\mathrm {norm}}}}+\Gamma )=\nu ^*(\omega _Z)$ , and there exists an effective Weil divisor $D \geq 0$ such that $K_Y+(p-1)D = \mu ^*K_{Z^{ {\mathrm {norm}}}}$ by [Reference Patakfalvi and WaldronPW22, Theorem 1.1], and we conclude

$$ \begin{align*}K_Y+(p-1)D+\mu^*\Gamma= f^*(K_X-(p-1)dA),\end{align*} $$

where $f =\pi \circ \nu \circ \mu $ . Consider a general curve C on Y of genus $g \geq 1$ so that C is contained in the smooth locus of Y and $C \cdot ((p-1)D+\mu ^*\Gamma ) \geq 0$ . Therefore, $K_Y \cdot C<0$ , and the bend and break lemma [Reference KollárKol96, Chapter II, Theorem 5.8] shows that for every point $x \in C$ , there exists a rational curve $L_x$ such that

$$ \begin{align*}-(K_Y+(p-1)D+\mu^*\Gamma) \cdot L_x \leq 4 \frac{-(K_Y+(p-1)D+\mu^*\Gamma)\cdot C}{-K_Y \cdot C} \leq 4,\end{align*} $$

as $-(K_Y+(p-1)D+\mu ^*\Gamma )$ is big and nef. Since $-K_X$ is ample, we infer the inequality

$$ \begin{align*}f^*((p-1)dA) \cdot L_x < f^*(-K_X+(p-1)dA) \cdot L_x \leq 4.\end{align*} $$

As $A=\pi ^*H$ where H is an ample Cartier divisor and x is a general point on C, we have $f^*A \cdot L_x \geq 1$ , and thus, we have $(p-1)d \leq 3$ , which concludes the proof.

Proof. The proof is the same as [Reference TanakaTan24, Theorem 3.5]. There is a factorisation of the iterated Frobenius morphism by [Reference TanakaTan21, Theorem 5.9]:

$$ \begin{align*}F^{\ell_F(X/k)}_{X \times_k \overline{k}} \colon X \times_k \overline{k} \to (X \times_k \overline{k})_{{\mathrm{red}}}^{{\mathrm{norm}}} \xrightarrow{\mu} X \times_k \overline{k},\end{align*} $$

where $(X \times _k \overline {k})_{ {\mathrm {red}}}^{ {\mathrm {norm}}}$ is a toric variety by Theorem 3.5. Thus, $\mu ^*A$ is globally generated and also $(F^\ell _F(X/k)_{X \times _k \overline {k}})^*A=A^{p^{\ell _F(X/k)}}.$

Proof. By Lemma 3.7 and Proposition 3.8, it is sufficient to verify that for $p=3$ (resp. $p=2, \ell _F(X/k)\,{=}\,1$ and $p=2, \ell _F(X/k)=2$ ), the line bundle $\omega _X^{ -6}$ (resp. $\omega _X^{ -5}$ and $\omega _X^{ -8}$ ) is $0$ -regular with respect to $\omega _X^{ -3}$ (resp. $\omega _X^{ -2}$ and $\omega _X^{ -4}$ ) to show the statement. We prove only the case $p=2$ and $\ell _F(X/k)=2$ , as the others are analogous. In this case, $H^1(X, \omega _X^{ -8} \otimes \omega _X^{ 4})=H^1(X, \omega _X^{ -4})=H^1(X, \omega _X^{ 5})=0$ by Proposition 3.6 and $H^2(X, \mathcal {O}_X)=H^0(X, \omega _X)=0$ .

Proof. If X is geometrically normal, then it is geometrically canonical by Proposition 2.11 and $\omega _X^{\otimes -6}$ is very ample by [Reference Bernasconi and TanakaBT22, Proposition 2.14]. If X is not geometrically normal, we apply Proposition 3.9.

Proof. The inequality $\ell _F(X/k) \leq \gamma (X/k)$ is shown in [Reference TanakaTan21, Proposition 8.7], so we are left to show $\gamma (X/k) \leq \delta (X_{\bar {k}}/\bar {k})$ . As the statement can be checked on an open covering of X, we can assume that the conductor D of $X_{\bar {k}}$ is irreducible, with generic point $\eta $ .

By definition of $\gamma (X/k)$ [Reference TanakaTan21, Definition 4.1], we can find a sequence of purely inseparable field extensions such that, if we inductively define $X_0 := X$ and $X_i := (X_{i-1,k_i})^{ {\mathrm {norm}}}$ , then $X_{i,{k_{i+1}}}$ is not normal and there is no longer sequence of fields with this property. In particular, $X_{\gamma (X/k)}$ is geometrically normal, and the normalization $\nu : Y \to X_{\bar {k}}$ of $X_{\bar {k}}$ factors as

Note that each $X_{i,{\bar {k}}}$ has the property $(S_2)$ , being the base change of a normal variety along a field extension.

Now, after localizing at $\eta $ , the factorization of $\nu $ corresponds to an ascending chain of subrings Each inclusion $\mathcal {O}_{X_{i-1,{\bar {k}}},\eta } \subseteq \mathcal {O}_{X_{i,{\bar {k}}},\eta }$ is strict: otherwise, $\nu _i$ would be an isomorphism in codimension $1$ , and hence so would be $X_i \to X_{i-1,k_i}$ . Since $X_i$ is normal and $X_{i-1,k_i}$ has property $(S_2)$ , this would imply that $X_{i-1,k_i}$ is normal as well, contradicting our choice of $k_i$ .

By definition, we have isomorphisms of $(\mathcal {O}_{X_{\bar k},\eta })$ -modules

$$ \begin{align*}\mathcal{O}_{Y,\eta}/\mathcal{O}_{X_{\bar k},\eta} \cong (\mathcal{O}_{Y,\eta}/\mathcal{C}_{\eta})/(\mathcal{O}_{X_{\bar k},\eta}/\mathcal{C}_{\eta}) \cong \mathcal{O}_{C,\eta}/\mathcal{O}_{D,\eta.} \end{align*} $$

Note that both sides are annihilated by the conductor ideal $\mathcal {C}_{\eta }$ ; hence, this is also an isomorphism of $(\mathcal {O}_{D,\eta })$ -modules. Therefore, by strictness of $\mathcal {O}_{X_{i-1,{\bar {k}}},\eta } \subseteq \mathcal {O}_{X_{i,{\bar {k}}},\eta }$ for every $i \leq \gamma (X/k)$ , we have

$$ \begin{align*}\gamma(X/k) \leq \mathrm{length}_{\mathcal{O}_{X_{\bar k},\eta}}(\mathcal{O}_{Y,\eta}/\mathcal{O}_{X_{\bar k},\eta}) = \mathrm{length}_{\mathcal{O}_{D,\eta}}(\mathcal{O}_{C,\eta}/\mathcal{O}_{D,\eta}) = \delta(X_{\bar{k}}/\bar{k}),\end{align*} $$

as claimed.

Proof. By Proposition 4.2, we only have to show the last equality. Let $\eta $ be the generic point of an irreducible component of the conductor $D \subset X$ . The Gorenstein condition implies $ {\mathrm {length}}_{\mathcal {O}_{D,\eta }}{\mathcal {O}_{C, \eta }}=2 {\mathrm {length}}_{\mathcal {O}_{D,\eta }}{\mathcal {O}_{D, \eta }}$ by [Reference Fanelli and SchröerFS20, Proposition A.2], which shows that $\delta (X_{\overline {k}}/\overline {k})= \max _{\eta \in D} {\mathrm {length}}_{\mathcal {O}_{D,\eta }}{\mathcal {O}_{D, \eta }}$ by [Sta, Tag 00IV], as claimed.

The last statement is immediate as $ {\mathrm {length}}_{\mathcal {O}_{D,\eta }}{\mathcal {O}_{D, \eta }}=1$ if D is reduced.

Proof. By Theorem 2.13, the conductor D is reduced, and thus, we can apply Proposition 4.3 to conclude. The proof of the assertion on torsion line bundles follows as in [Reference Bernasconi and TanakaBT22, Theorem 4.1]. For the last statement, by Proposition 2.16, the Picard scheme $ {\mathrm {Pic}}^0_{X_{\bar {k}}/\bar {k}}$ is a smooth commutative unipotent algebraic group of dimension $h^1(X,\mathcal {O}_X)$ . As it is annihilated by p, we conclude by [Reference SerreSer88, Proposition VII.11].

Proof. By Corollary 4.4, $\ell _F(X/k) = 1$ , and we conclude by combining Lemma 3.7 and Proposition 3.9.

Proof. By the natural identifiation $\omega _Y(C)|_C \cong \omega _C$ given by adjunction, the space V is equal to the kernel of the homomorphism $H^0( {\mathrm {Res}} \otimes \omega _X^{-n-1})$ ; hence, it is contained in the kernel of $H^0(( {\mathrm {Tr}} \circ {\mathrm {Res}}) \otimes \omega _X^{-n-1}) = \nu ^* H^0(X,\omega _X^{-n})$ by Lemma 4.6.

If $\omega _{X}^{-n}$ is globally generated, then the linear system $|\nu ^* H^0(X,\omega _X^{-n})|$ has no base points on C. Since all sections in V vanish on C and $\omega _X^{-1}$ is ample, there are at least two more linearly independent sections of $\nu ^* H^0(X,\omega _X^{-n})$ that are nonzero when restricted to C, thus concluding the inequality.

Proof. If X is geometrically normal, then X is geometrically canonical by Proposition 2.11. By Serre duality, it is sufficient to show that $h^1(X_{\overline {k}}, \omega _{X_{\overline {k}}}^{\otimes n})=0$ for $n> 0$ . This follows from [Reference BernasconiBer21a, Theorem 5.6.a]. If X is not geometrically normal and the ramification divisor contains a reduced component, then $X_{\overline {k}}$ is tame and $h^1(X,\omega _X^{\otimes n}) = 0$ for all $n \in \mathbb {Z}$ by [Reference ReidRei94, Corollary 4.10]. Therefore, by [Reference Patakfalvi and WaldronPW22, Theorem 4.1], we may assume that $p \in \{2,3\}$ , $h^1(X,\mathcal {O}_X)> 0$ and $(Y,C) = (\mathbb {P}(1,1,d),2L)$ for some $d \geq 1$ where L is a line through the vertex of the cone.

Choose weighted coordinates $x,y,z$ of degree $1,1,d$ on Y such that $L=\left \{x = 0 \right \}$ , hence ${2L=\left \{x^2=0 \right \}}$ in weighted coordinates. Let $n \geq 1$ and $V_{n,d} \subseteq H^0(\mathbb {P}(1,1,d),\mathcal {O}(nd))$ be the subspace of sections vanishing along $2L$ . Then, $V_{n,d}$ consists of weighted homogeneous polynomials of the form $x^2f_{nd-2}(x,y,z)$ ; hence, $\dim V_{n,d} = \sum _{j=1}^{n} (jd - 1) = \frac {n^2 + n}{2}d - n.$ As $\nu ^* \omega _{X_{\bar {k}}} \cong \mathcal {O}(-dL)$ , we have $\nu ^* \omega _{X_{\bar {k}}}^{-n} \cong \mathcal {O}(ndL)$ . By Corollary 4.7, we have

(4.1) $$ \begin{align} h^0(X,\omega_X^{- n}) \geq \begin{cases} \frac{n^2 + n}{2}d - n. \\ \frac{n^2 + n}{2}d - n+2 \text{ if, additionally}, \omega_X^{-n} \text{ is globally generated.} \end{cases} \end{align} $$

By the Riemann–Roch formula [Reference TanakaTan18b, Theorem 2.10], we have

$$ \begin{align*}h^0(X,\omega_X^{- n})-h^1(X, \omega_X^{-n})=1-h^1(X,\mathcal{O}_X)+\frac{n^2 + n}{2}K_X^2=1-h^1(X,\mathcal{O}_X)+ \frac{n^2 + n}{2}d.\end{align*} $$

Thus, if we assume $h^1(X, \omega _X^{- n})=0$ , we deduce from Equation (4.1) that

(4.2) $$ \begin{align} h^1(X,\mathcal{O}_X) = 1 - h^0(X,\omega_X^{- n}) + \frac{n^2 + n}{2}d \leq \begin{cases} n + 1 \\ n - 1 \text{ if, additionally}, \omega_X^{- n} \text{is g.g. } \end{cases} \end{align} $$

We also recall Maddock’s bound [Reference MaddockMad16, Corollary 1.2.6]: if $h^1(X,\omega _X^{ n}) \neq 0$ but $h^1(X,\omega _X^{pn}) = 0$ , then

(4.3) $$ \begin{align} h^1(X,\mathcal{O}_X) \geq \frac{nd(p-1)(3 + n(2p-1))}{12}. \end{align} $$

Now assume $p = 3$ . By Serre vanishing and $h^1(\mathcal {O}_X) \neq 0$ , there exists a largest $N \geq 0$ such that $h^1(X,\omega _X^{- N}) = h^1(X,\omega _X^{ (N + 1)}) \neq 0$ . By (4.2) and (4.3), we have the following chain:

$$ \begin{align*}N + 2 \geq h^1(X,\mathcal{O}_X) \geq \frac{(N+1)d(p-1)(3 + (N+1)(2p-1))}{12} = \frac{(N+1)d(8 + 5N)}{6}. \end{align*} $$

Hence, $N = 0$ , $d = 1$ , showing $h^1(X, \mathcal {O}_X) \leq 2$ . Finally, $h^1(X,\mathcal {O}_X) = 2$ by Theorem 2.13.

Now assume $p = 2$ . Then, the argument of the previous paragraph yields

(4.4) $$ \begin{align} N + 2 \geq h^1(X,\mathcal{O}_X) \geq \frac{(N+1)d(p-1)(3 + (N+1)(2p-1))}{12} = \frac{(N+1)d(N+2)}{4}. \end{align} $$

Hence, $N \leq 3$ . Therefore, $h^1(X,\omega _X^{- 4}) = 0$ , and, by Corollary 4.5, $\omega _X^{- 4}$ is globally generated, so $h^1(X,\mathcal {O}_X) \leq 3$ by (4.2). If $h^1(X,\mathcal {O}_X) = 1$ , then $N = 0$ and $d \in \{1,2\}$ by (4.4), and we get Cases (3) and (4).

So, it remains to exclude the possibility $h^1(X,\mathcal {O}_X) \geq 2$ in characteristic $p = 2$ . By Corollary 4.5, $\omega _X^{- 2}$ is globally generated, so by (4.4), the inequality $h^1(X,\mathcal {O}_X) \geq 2$ implies $N = 2$ , $d = 1$ , and $h^1(X,\mathcal {O}_X) = 3$ .

Seeking a contradiction, assume that there exists an X with these invariants. Since $N = 2$ , we have $H^1(X, \omega _X^{3}) \neq 0$ and $H^1(X, \omega _X^6)=0$ . Let $Z \to X$ be a nontrivial $\alpha _{\omega _X^3}$ -torsor and let $k_Z:=H^0(Z, \mathcal {O}_Z)$ . Note that Z is an l.c.i. del Pezzo surface by [Reference MaddockMad16, Theorem 1.2.3], and by [Reference MaddockMad16, Equation (1.2.4)], we have $[k_Z :k] (1-h^1(Z, \mathcal {O}_Z)) = 2$ so that we have $[k_Z :k]=2$ and $H^1(Z,\mathcal {O}_Z) = 0$ . By [Reference MaddockMad16, Equation (1.2.5)], we then conclude that $K_Z^2=16$ . Now, we consider the following diagram:

where $Z_{\bar {k}} = Z \times _{ {\mathrm {Spec}} k_Z} {\mathrm {Spec}} \bar {k}$ . Since f and $X_{k_Z} \to X$ are finite of degree $2$ , the morphism $\pi $ is finite and birational. In particular, Z, considered as a $k_Z$ -scheme, is geometrically integral and the induced map $(\pi _{\bar {k}})^{ {\mathrm {norm}}}$ of the normalisations is an isomorphism. As $(X_{\bar {k}})^{ {\mathrm {norm}}} \cong \mathbb {P}^2$ , $16=K_Z^2 \leq K_{(Z_{\bar {k}})^{ {\mathrm {norm}}}}^2=9$ , reaching a contradiction.

Proof. Claims (1), (2) and (3) are a consequence of [Reference Bernasconi and TanakaBT22, Proposition 2.14] if X is geometrically normal, and hence geometrically canonical, and [Reference ReidRei94, Corollary 4.10] if X is not geometrically normal.

Let us prove Claim (4). By Proposition 4.8, we have $p=2$ and the desired classification of $(Y,C)$ . By Lemma 3.7, $\omega _X^{-2}$ is globally generated. Using Proposition 4.8, it is easy to check that $\omega _X^{-4}$ is $0$ -regular with respect to $\omega _X^{-2}$ . Claim (5) is proven similarly.

Proof. Without loss of generality, we may assume that k is separably closed.

First, assume that X is regular. Seeking a contradiction, we assume that $h^1(X, \mathcal {O}_X) \neq 0$ . Let ${\nu \colon Y \to X_{\bar {k}}}$ be the normalisation of $X_{\bar {k}}$ and let $C \subseteq Y$ be the ramification of $\nu $ . By Proposition 4.8 and Serre duality, we know that $h^1(X,\mathcal {O}_X) = 2$ , $K_X^2 = 1$ , $h^1(X,\omega _X^{-n}) = 0$ for $n> 0$ , and $(Y,C) = (\mathbb {P}^2,2L)$ .

First, we claim that $h^0(X,\omega _X^{-n} \otimes \mathcal {L}) = 0$ for all nontrivial torsion line bundles $\mathcal {L}$ and for $n \in \left \{0, 1\right \}$ . Since X is reduced, this holds if $n = 0$ . For the case $n=1$ , by the Riemann–Roch theorem, we have

$$ \begin{align*}\chi(\omega_X^{-1} \otimes \mathcal{L}) = 0, \end{align*} $$

so if $h^1(X,\omega _X^{-1} \otimes \mathcal {L}) \neq 0$ , then $h^1(X,\omega _X^{2} \otimes \mathcal {L}^{-1}) \neq 0$ by Serre duality. Since $\omega _X^{6} \otimes \mathcal {L}^{-3} \cong \omega _X^{6}$ by Corollary 4.4 and $h^1(X,\omega _X^{6}) = 0$ by Proposition 4.8, there exists a nontrivial $\alpha _{(\omega _X^{2} \otimes \mathcal {L}^{-1})}$ -torsor $Z \to X$ such that Z is an l.c.i. del Pezzo surface. Moreover, by [Reference MaddockMad16, Equation (1.2.5)], we have $ 2^e(1-q_Z)= 10 $ for some integers $0 \leq e \leq 1$ and $q_Z \geq 0$ , contradicting our assumption.

By Riemann–Roch, we have $h^0(X,\omega _X^{-2}) = 2$ . Write

$$ \begin{align*}|-2K_X| = F + |M|, \end{align*} $$

where F is the fixed part and M is the movable part of the linear system. Since $M \neq 0$ , we have $F \in |-nK_X + E|$ for some $0 \leq n \leq 1$ and a divisor E such that $\mathcal {O}_X(E)$ is torsion. By the previous paragraph, we have $h^0(X,\omega _X^{-n}(E)) = 0$ ; hence, $F = 0$ .

Since the linear system $|-2K_X|$ does not have fixed components, its base locus Z is $0$ -dimensional, and we denote by A the ring of global section $H^0(Z, \mathcal {O}_Z)$ . Since $(-2K_X)^2 = 4$ , we have $\mathrm {length}_k(A) = 4$ , so A is an Artinian k-algebra of length $4$ . As k is separably closed, we can write $A = \prod _{i=1}^s A_i$ where each $A_i$ is a local Artinian k-algebra of dimension $n_i$ over its residue field $k_i$ and $k_i$ is a purely inseparable extension of k. Since

$$ \begin{align*}4 = \mathrm{length}_k(A) = \sum_{i=1}^s \mathrm{length}_k(A_i) = \sum_{i=1}^s n_i [k_i:k] \end{align*} $$

and $p = 3$ , we have $k_i = k$ for at least one i. In other words, at least one of the base points of $|-2K_X|$ is a k-rational point P.

If P is in the image of the conductor D under the natural map $X_{\bar {k}} \to X$ , then P lies in the non-smooth locus of $X \to {\mathrm {Spec}} k$ , so X cannot be regular at P by [Reference Fanelli and SchröerFS20, Corollary 2.6]. Hence, in this case, the proof is finished.

So, seeking a contradiction, assume that P is not in the image of D. Let $P'$ be the unique preimage of P under the map $Y \to X_{\bar {k}} \to X$ . Since $\nu $ is an isomorphism around $P'$ , the point $P'$ lies in the base locus of $\nu ^* |-2K_{X_{\bar {k}}}|$ . By Corollary 4.7, we know that $C = 2L \in \nu ^* |-2K_{X_{\bar {k}}}|$ hence, $P' \in C$ , and thus, P is in the image of D under $X_{\bar {k}} \to X$ . This contradicts our choice of P.

Finally, assume that X is canonical. By Proposition 2.6, we can replace X with its minimal resolution, which is a regular weak del Pezzo surface. By running a $K_X$ -MMP, we can suppose X is a weak del Pezzo surface admitting a Mori fibre space structure $\pi \colon X \to B$ . If X is a regular del Pezzo surface, we conclude by the previous case. If B is a curve and X is a weak del Pezzo surface, then the generic fibre F is a regular conic. As $p = 3$ , F is smooth by [Reference Bernasconi and TanakaBT22, Lemma 2.17], and thus, $-K_B$ is ample by [Reference EjiriEji19, Corollary 4.10.c]. Therefore, $H^1(B, \mathcal {O}_B)=0$ and, as $H^1(X, \mathcal {O}_X)=H^1(B, \mathcal {O}_B)=0$ by the relative Kawamata–Viehweg vanishing theorem [Reference TanakaTan18b, Theorem 4.2], we conclude.

Proof. If X is not tame, then $\rho (X)=1$ by Proposition 4.8. If $ {\mathrm {p-deg}}(k)=1$ , then X is geometrically canonical by [Reference Fanelli and SchröerFS20], contradicting $h^1(X, \mathcal {O}_X) \neq 0$ .

By Proposition 4.8, we have $h^1(X,\mathcal {O}_X) = h^1(X,\omega _X) = 1$ , $h^1(X,\omega _X^{n}) = 0$ for $n \geq 2$ , and $K_X^2 \in \{1,2\}$ . In particular, there exists a nontrivial $\alpha _{\omega _X}$ -torsor $f \colon Z \to X$ . In the following, we treat the cases $K_X^2 = 2$ and $K_X^2 = 1$ separately. We set . Note that, by the same proof as for the second paragraph of the proof of Proposition 4.10, we have $h^1(X,\omega _X^{n} \otimes \mathcal {L}) = 0$ for all torsion line bundles $\mathcal {L}$ and all $n \geq 2$ .

Assume $K_X^2 = 2$ . In this case, by [Reference MaddockMad16, Equation (1.2.5)], we have $[k_Z:k] = 2$ , $h^1(Z,\mathcal {O}_Z) = 0$ , and $K_Z^2 = 8$ , where we compute the self-intersection over $k_Z$ . Now, consider the commutative diagram

where $Z_{\bar {k}} = Z \times _{ {\mathrm {Spec}} k_Z} {\mathrm {Spec}} \bar {k}$ . As in the end of the proof of Proposition 4.8, the diagram shows that Z is geometrically integral when considered as a $k_Z$ -scheme and $(\pi _{\bar {k}})^{ {\mathrm {norm}}}$ is an isomorphism; hence, $(Z_{\bar {k}})^{ {\mathrm {norm}}} \cong (X_{\bar {k}})^{ {\mathrm {norm}}} \cong \mathbb {P}(1, 1, 2)$ by Proposition 4.8. In particular, we have $K_Z^2 = 8 = K^2_{(Z_{\bar {k}})^{ {\mathrm {norm}}}}$ , so Z is in fact geometrically normal. Therefore, $Z_{\bar {k}} \cong \mathbb {P}(1,1,2)$ , so Z is a twisted form of $\mathbb {P}(1,1,2)$ over $k_Z$ .

Assume $K_X^2=1$ . In this case, by [Reference MaddockMad16, Equation (1.2.5)], we have $k_Z = k$ and $h^1(Z, \mathcal {O}_Z)=0$ . We first claim that Z is not geometrically reduced. Indeed, let $\psi \colon \mathbb {P}^2 \to X$ be the normalised base change and consider the $\alpha _{\psi ^*\omega _X}$ -torsor $T:=Z \times _{X} \mathbb {P}^2 \to \mathbb {P}^2$ obtained by base changing along $\psi $ . As $\psi ^*\omega _X$ is anti-ample, considering the exact sequence

$$ \begin{align*}0 = H^0(\mathbb{P}^2, \psi^*\omega_X^p) \to H^1_{{{\mathrm{fppf}}}}(\mathbb{P}^2, \alpha_{\psi^*\omega_X}) \to H^1(\mathbb{P}^2, \psi^*\omega_X) = 0,\end{align*} $$

we have that $H^1_{{ {\mathrm {fppf}}}}(\mathbb {P}^2, \alpha _{\psi ^*\omega _X})=0$ , so that $T \to \mathbb {P}^2$ is a trivial torsor, and thus, T is not reduced. Since $T \to Z_{\bar {k}}$ is generically an isomorphism, Z is not geometrically reduced.

Next, we claim that Z is normal. Suppose by contradiction it is not and let $\nu \colon Z^{ {\mathrm {norm}}} \to Z$ be the normalisation. In this case, as $\nu $ is not an isomorphism, we have that $K_{Z^{ {\mathrm {norm}}}}^2>8$ , where we calculate the self-intersection number over k. Let $g \colon Z^{ {\mathrm {norm}}} \to X$ be the composition $f \circ \nu $ . As X is regular and $Z^{ {\mathrm {norm}}}$ is integral, by [Reference MaddockMad16, Proposition 2.2.1], there exists a line bundle $\mathcal {M}$ numerically equivalent to $mK_X$ for some integer m such that g is a nontrivial $\alpha _{\mathcal {M}}$ -torsor. In particular, $Z^{ {\mathrm {norm}}}$ is Gorenstein, and thus, as it is the normalization of a del Pezzo surface, $Z^{ {\mathrm {norm}}}$ is also a del Pezzo surface. Therefore, $m \geq 0$ . We now distinguish two cases:

1. 1. if $H^0(Z^{ {\mathrm {norm}}}, \mathcal {O}_{Z^{ {\mathrm {norm}}}}) =k$ , by [Reference MaddockMad16, Equation (1.2.4)], we have $2(1+m)^2 = K_{Z^{ {\mathrm {norm}}}}^2>8$ , which implies that $m>1$ , contradicting Proposition 4.8;

2. 2. if $[H^0(Z^{ {\mathrm {norm}}}, \mathcal {O}_{Z^{ {\mathrm {norm}}}}):k ]=2$ , we have that $(1+m)^2=K_{Z^{ {\mathrm {norm}}}}^2>4$ , which also implies that $m>1$ , contradicting Proposition 4.8 as well. Note that, here, we calculate the self-intersection $K_{Z^{ {\mathrm {norm}}}}^2$ over $H^0(Z^{ {\mathrm {norm}}}, \mathcal {O}_{Z^{ {\mathrm {norm}}}})$ .

Thus, Z must be normal.

By [Reference KleimanKle66, Example 1], we have that $8=K_Z^2=2^{\epsilon (Z/k)}(K_{(Z_{\bar {k}})_{ {\mathrm {red}}}^{ {\mathrm {norm}}}} + E)^2 $ . Since Z is geometrically non-reduced, we have $\epsilon (Z/k) \geq 1$ , and since f is finite flat of degree $2$ and $\epsilon (X/k) = 0$ , we have $\epsilon (Z/k) \leq 1$ ; hence, $\epsilon (Z/k) = 1$ . As Z is normal and Gorenstein, we can apply [Reference Patakfalvi and WaldronPW22, Theorem 4.1] to conclude that $Z=\mathbb {P}^2$ and E is a line.

Proof of Theorem 1.2

Combine Proposition 4.8, Proposition 4.10 and Proposition 4.11.

Proof. As X is geometrically integral, $\epsilon (X/k)=0$ , and thus, $K_X^2 \leq 16$ by Proposition 3.4. Since X is canonical, $K_X$ is Cartier and $K_X^2$ is an integer. By Proposition 4.8, Theorem 3.10 and Riemann–Roch, there exists an $N> 0$ such that $\omega _X^{-12}$ embeds X into $\mathbb {P}^n_k$ for some $n \leq N$ . Again by Proposition 4.8, and since $K_X^2 \leq 16$ , the possibilities for the Hilbert polynomial $\chi (X, \omega _X^{ -12})$ are finite. Therefore, all X arise via pullback from a universal family over a suitable finite union of Hilbert scheme of finite type over $ {\mathrm {Spec}} \mathbb {Z}$ [Reference KollárKol96, Theorem 1.4].

Proof. The existence of $\Gamma $ is a consequence of the cone theorem [Reference TanakaTan18b, Theorem 2.14], while Equation (5.1) is proved in [Reference Bernasconi and TanakaBT22, Lemmas 4.3, 4.6]. To prove Equation (5.2), we write $K_X \equiv xF_b+y\Gamma $ for some $x,y \in \mathbb {Q}$ . Set $d_b = [k(b):k]$ . As $K_X\cdot F_b=-2d_b$ , we conclude that $y=-\frac {2}{m_\Gamma }.$ Therefore,

$$ \begin{align*}d_\Gamma(n-2) = K_X \cdot \Gamma = xm_\Gamma d_b+\frac{2n d_\Gamma}{m_\Gamma},\end{align*} $$

which implies

$$ \begin{align*}K_X \equiv \frac{d_\Gamma(m_\Gamma(n-2)-2n)}{m_\Gamma^2d_b} F_b - \frac{2}{m_\Gamma}\Gamma.\end{align*} $$

A straightforward computation with intersection numbers then shows Equation (5.2). Finally, as ${(1-m_\Gamma ) \leq 0}$ , we have $K_X^2 \leq \frac {d_\Gamma }{m_\Gamma } 8$ and, as $d_\Gamma \leq m_\Gamma ,$ we conclude.

Proof. Let $f\colon Y \to X$ be the minimal resolution, and write $K_Y+\sum _i b_i E_i = f^*K_X$ . By the $\varepsilon $ -klt hypothesis and minimality of f, we have $0 < b_i < 1-\varepsilon .$ We run a $K_Y$ -MMP which ends with $\psi \colon Y \to Z$ , where Z is a regular projective surface admitting a Mori fibre space structure ${\pi \colon Z \to B}$ . Since $-(K_Y+\sum _i b_i E_i)$ is big and nef, so is $-(K_Z+\Delta _Z)$ , where $\Delta _Z=\psi _* \left ( \sum b_i E_i \right )$ . Moreover, ${K_X^2 = (K_Y+\sum _i b_i B_i)^2 \leq (K_Z+\Delta _Z)^2.}$

Suppose $\dim (B)=0$ . Then Z is a regular del Pezzo surface of Picard rank 1. As $-(K_Z+\Delta _Z)$ is ample, there exists $0 \leq \lambda <1$ such that $\Delta _Z \equiv - \lambda K_Z$ . Therefore, we deduce $(K_Z+\Delta _Z)^2=(1-\lambda )^2K_Z^2 \leq K_Z^2 \leq 9$ , where the last inequality follows by [Reference TanakaTan24, Theorem 1.2].

Suppose $\dim (B)=1$ . Let $\Gamma $ be the extremal curve described in Lemma 5.5. We write $\Delta _Z=\alpha \Gamma + G$ , where $\text {Supp}(G)$ does not contain $\Gamma $ . Since the Picard rank of Z is $2$ , G is a $\mathbb {Q}$ -Cartier nef divisor. As $-(K_Z+\Delta _Z)$ and $-(K_Z+\alpha \Gamma )$ are big and nef classes, their intersection with G is non-positive, and thus, we have

$$ \begin{align*}(K_Z+\Delta_Z)^2 = (K_Z+\alpha \Gamma)^2+ (K_Z+\alpha \Gamma)\cdot G+ (K_Z+\Delta_Z)\cdot G \leq (K_Z+\alpha \Gamma)^2.\end{align*} $$

Therefore, it is sufficient to bound the volume of del Pezzo surface pairs $(Z, \alpha \Gamma )$ , where $Z \to B$ is a Mori fibre space onto a curve, $\Gamma $ is the extremal curve of Lemma 5.5 and $0 \leq \alpha <1-\varepsilon $ . Note that

(5.3) $$ \begin{align} (K_Z+\alpha \Gamma)^2=K_Z^2+d_\Gamma \alpha \left( 2(n-2)-n\alpha \right) \text{ and } 0 \leq \alpha<1-\varepsilon. \end{align} $$

Claim 5.7. The self-intersection of $\Gamma $ is bounded:

$$ \begin{align*}n \leq \frac{2}{\varepsilon}.\end{align*} $$

Proof of Claim

By adjunction,

$$ \begin{align*}-2d_\Gamma=(K_Z+\Gamma) \cdot \Gamma= (K_Z+\alpha \Gamma) \cdot \Gamma +(1-\varepsilon -\alpha) \Gamma^2 +\varepsilon \Gamma^2.\end{align*} $$

As $-(K_Z+\alpha \Gamma )$ is big and nef and $1-\varepsilon> \alpha $ , we have $(K_Z+\alpha \Gamma )\cdot \Gamma +(1-\varepsilon -\alpha ) \Gamma ^2 \leq 0$ , and therefore, we deduce $2 d_\Gamma \geq \varepsilon d_\Gamma \cdot n$ .

If $K_Z \cdot \Gamma \leq 0$ , then $(K_Z+\alpha \Gamma )^2 \leq K_Z^2 \leq 8$ by Lemma 5.5, and we are done. So, assume $K_Z \cdot \Gamma>0$ , or, equivalently, $n>2$ . In this case, we have $d_\Gamma \leq m_\Gamma \leq 5$ by [Reference Bernasconi and TanakaBT22, Proposition 4.7]. Therefore, by Equation (5.3) and Claim 5.7, we deduce the following series of inequalities:

$$ \begin{align*} (K_Z+\alpha \Gamma)^2 & = K_Z^2+d_\Gamma \alpha \left( 2(n-2)-n\alpha \right) \\ & \leq K_Z^2+ 5 \alpha \left( 2n-4\right) \leq K_Z^2+ 5(1-\varepsilon)\left(\frac{4}{\varepsilon}-4\right) \\ & \leq 8+20 \frac{(1-\varepsilon)^2}{\varepsilon}.\\[-42pt] \end{align*} $$

Proof. By [Reference Bernasconi and TanakaBT22, Corollary 5.5] and [Reference Bernasconi and TanakaBT22, Theorem 5.7], klt del Pezzo surfaces in characteristic bigger than $5$ are geometrically integral and tame, so it suffices to show that $\mathcal {X}_{\text {dP}, n}^{\text {tame}}$ is bounded.

So, let $X \in \mathcal {X}_{\text {dP}, n}^{\text {tame}}$ . As X is $\left (\frac {1}{n} \right )$ -klt, Theorem 5.6 implies that $K_X^2$ is bounded. As the Cartier index of $K_X$ is fixed, the set of volumes $\left \{K_X^2 \mid X \in \mathcal {X}_{\text {dP}, n}^{\text {tame}} \right \}$ is a finite set. As X has rational singularities by Proposition 2.6, we can apply Riemann–Roch to compute for all $t \geq 1$ :

$$ \begin{align*}\chi(X, \mathcal{O}_X(-ntK_X))=\chi(X, \mathcal{O}_X)+\frac{nt(nt+1)K_X^2}{2}.\end{align*} $$

As X is tame, $\chi (X, \mathcal {O}_X)=1$ , and therefore, there are only a finite number of possibilities for the Hilbert polynomials $P_n(t):=\chi (X, \mathcal {O}_X(-ntK_X))$ . Finally, we apply [Reference KollárKol85, Theorem 2.1.2] to conclude that $X_{\overline {K}}$ form a bounded family over $ {\mathrm {Spec}}(\mathbb {Z}[1/30])$ . In particular, there exists $m:=m(n)$ such that $-mnK_{X_{\overline {K}}}$ is very ample, and thus, by faithfully flat descent, $-mnK_{X}$ is very ample. Therefore, there exists $N=N(n)>0$ such that $-mnK_{X}$ embeds X into some $\mathbb {P}^N$ with a finite number of possibilities for the Hilbert polynomial. This concludes that X belongs to a bounded family by a classical Hilbert scheme argument.

Proof. Without loss of generality, we assume that X is the spectrum of a local ring and that $\rho (Y) = n$ . Let $E= \sum _{i=1}^n E_i$ be the sum of the exceptional divisors of f, and write $K_Y + \sum _{i=1}^n b_i E_i =f^*K_X$ for some $0 \leq b_i< 1-\varepsilon $ . By the base point free theorem [Reference TanakaTan18b, Theorem 4.2], it suffices to find an effective $N=N(\varepsilon ,n)$ such that $Nb_i$ is integral. For each i, we write $E_i^2 = -d_{E_i} n_i$ for some integer $n_i>0$ , where $d_{E_i} = [H^0(E_i,\mathcal {O}_{E_i}):k]$ . Moreover, for each $i \neq j$ , we write $E_i \cdot E_j = d_{E_j} n_{ij}$ for some $n_{ij}>0$ . The $b_i$ are determined by the following system of equations:

(5.4) $$ \begin{align} (2 - n_j + b_jn_j) = \sum_{i\neq j} b_i n_{ij} \text{ for } j=1, \dots, n. \end{align} $$

Since X is $\varepsilon $ -klt, we have

(5.5) $$ \begin{align} n_j \leq \frac{2}{\varepsilon}. \end{align} $$

Indeed,

$$ \begin{align*} \begin{split} 0 & =(K_Y+\sum b_iE_i) \cdot E_j \geq (K_Y+b_j E_j) \cdot E_j \\ & \geq -2d_{E_j} + (b_j-1) \cdot (-d_{E_j}n_j) \geq -2d_{E_j}+\varepsilon n_jd_{E_j}. \\ \end{split} \end{align*} $$

Moreover, $n_{ij}$ is bounded from above by [Reference KollárKol13, Corollary 3.31 and Section 3.41] and [Reference SatoSat23, Appendix A]. As the $n_i$ and $n_{ij}$ are integers, we only have a finite number of possibilities for the coefficients in Equation 5.4, so we conclude that there are only finitely many possibilities for the solutions $b_i$ , thus showing the existence of a $N(\varepsilon , n)$ for which $Nb_i$ is integral.

Proof. Suppose $\dim (B)=0$ . Let H be an ample Cartier divisor generating $ {\mathrm {Num}}(X)$ , and let $d \geq 0$ such that $-K_X \equiv dH$ . As X is a regular del Pezzo surface, we have $K_X^2 \leq 9$ by [Reference TanakaTan24, Theorem 1.2], and therefore, $d \leq 3$ . Since X is regular, the $B_i$ are Cartier divisors, and thus, $\sum b_i \leq 3$ .

Suppose $\dim (B)=1$ . Let $F_b$ be a closed fibre of $\pi $ and $\Gamma $ the curve given by Lemma 5.5. Set . We write $\Delta =b_0\Gamma + \sum b_iE_i$ as a sum of pairwise distinct prime divisors. As $0 \geq (K_X+\Delta ) \cdot F_b$ , adjunction implies

(5.6) $$ \begin{align} 2d_b \geq \Delta \cdot F_b \geq b_0d_b+\sum_{(E_i \cdot F_b) \neq 0} b_i d_b. \end{align} $$

As $K_X \cdot \Gamma =d_\Gamma (n-2)$ , $0 \geq (K_X+\Delta ) \cdot \Gamma $ , and $b_0 \leq 1$ , adjunction implies

(5.7) $$ \begin{align} 2d_\Gamma \geq n d_\Gamma -b_0 n d_\Gamma +\sum_i b_i (E_i \cdot \Gamma) \geq \sum_{(E_i\cdot \Gamma ) \neq 0} b_i d_\Gamma. \end{align} $$

Summing the two equations and using that every curve on X intersects either $\Gamma $ or $F_b$ , we obtain $ 4 \geq \sum b_i, $ as desired.

Proof. We can suppose k is separably closed. We follow the computations of [Reference Alexeev and MoriAM04, Theorem 1.8], verifying that the explicit classification of rational Mori fibre spaces over algebraically closed fields is not needed. Without loss of generality, we can suppose $\varepsilon < \frac {2}{3}$ . Since $-K_X$ is ample, by Bertini’s theorem, we can choose $H \sim _{\mathbb {Q}} -K_X$ to be an effective $\mathbb {Q}$ -divisor whose support is regular and contained in the regular locus of X such that $(X,H)$ is $\varepsilon $ -klt. Let $f \colon Y \to X$ be the minimal resolution and write $K_Y+\Gamma _Y = f^*(K_X+H) \sim _{\mathbb {Q}} 0$ , where and $b_i < 1- \varepsilon $ by hypothesis. We run a $K_Y$ -MMP which ends with $g \colon Y \to Z$ , where Z is a regular projective surface admitting a Mori fibre space structure $\pi \colon Z \to B$ . We summarise the situation in the following diagram:

We fix the following notation:

• ○ .

• ○ .

• ○

• ○ .

• ○

Note that, by construction, g is a $(K_Y+\Delta _Y)$ -non-positive (resp. $(K_Y+\Gamma _Y)$ -trivial) birational contraction and $(Z, \Delta _Z)$ (resp. $(Z, \Gamma _Z)$ ) is a log del Pezzo pair (resp. Calabi–Yau pair) with $\varepsilon $ -klt singularities.

The morphism g is a composition of blow-ups of closed points on regular surfaces by [Sta, Tag 0C5R]. We can decompose g as

$$ \begin{align*}g \colon Y \xrightarrow[]{\psi} W \xrightarrow[]{\varphi} Z ,\end{align*} $$

where $\psi $ and $\varphi $ are proper birational morphisms between regular surfaces such that

1. 1. $\varphi $ is a composition of blow-ups at closed points P such that $\text {mult}_P(\widetilde {\Gamma _Z})$ has multiplicity at least $\nu :=\frac {\varepsilon }{2}$ , where $\widetilde {\Gamma _Z}$ denotes the strict transform of $\Gamma _Z$ ;

2. 2. $\psi $ is a composition of blow-ups at closed points P, where $\text {mult}_P(\widetilde {\Gamma _Z})$ has multiplicity $< \nu $ .

We first bound $\rho (W/Z)$ in terms of $\varepsilon $ . On Y, as $f_*^{-1}H$ is big and nef, we can apply Equation (5.5) to obtain

(5.8) $$ \begin{align} (g_*^{-1}\Gamma_Z)^2 \geq \sum_{i \in I} b_i^2 (g_*^{-1} E_{i,Z})^2 \geq \sum_{i \in I_Z} b_{i} (1-\varepsilon)\left(\frac{-2}{\varepsilon}\right). \end{align} $$

If $\dim (B)=0$ (resp. $1$ ), we have $\sum _{i \in I_Z} b_i \leq 4$ (resp. $\leq 3$ ) by Lemma 5.10, and thus,

(5.9) $$ \begin{align} (g_{*}^{-1}\Gamma_Z)^2 \geq \begin{cases} 6-\frac{6}{\varepsilon} \text{ if } \dim(B)=0 \\ 8-\frac{8}{\varepsilon} \text{ if } \dim(B)=1. \end{cases} \end{align} $$

After each of the blow-ups in $\varphi $ , the self-intersection of $\widetilde {\Gamma _Z}$ decreases by at least $\nu ^2$ , and we deduce that $(g_*^{-1}\Gamma _Z)^2 \leq \Gamma _Z^2-\nu ^2 \rho (W/Z)$ . If $\dim (B)=0$ (resp. $1$ ), then $\Gamma _Z^2=K_Z^2 \leq 9$ (resp. $8$ ) by [Reference TanakaTan24, Theorem 1.2] (resp. Lemma 5.5). Therefore, $ (g_*^{-1}\Gamma _Z)^2 \leq 9-\nu ^2 \rho (W/Z)$ (resp. $8-\nu ^2 \rho (W/Z)$ ). Together with (5.9), we conclude

(5.10) $$ \begin{align} \rho(W/Z) \leq \begin{cases} \frac{(3\varepsilon + 6)}{\varepsilon \nu^2} \text{if } \dim(B)=0; \\ \frac{8}{\varepsilon \nu^2} \text{ if } \dim(B)=1. \end{cases} \end{align} $$

As we chose $\varepsilon < \frac {2}{3}$ , we have $\frac {(3\varepsilon + 6)}{\varepsilon \nu ^2}< \frac {8}{\varepsilon \nu ^2} $ .

We now prove a bound on $\rho (Y/W)$ depending only on $\varepsilon $ . Let $F=\sum _{i \in J} F_i$ be the sum of the exceptional divisors of $\varphi $ and let $f_i:=\text {coeff}_{F_i}(\Gamma _W)$ , where . As W and Z are regular surfaces, $\text {Supp}(F)$ is an snc divisor. Let $\psi \colon Y \xrightarrow {s} T \xrightarrow {t} W$ be a factorisation of $\psi $ , where t is a blow-up at a point P of W with exceptional divisor C. Write

$$ \begin{align*}K_T+\Gamma_T=K_T+\widetilde{\Gamma_Z} + \sum f_i\widetilde{F_i} +cC \sim_{\mathbb{Q}} t^*(K_W+\widetilde{\Gamma}_Z+\sum f_{i}F_{i}).\end{align*} $$

We claim that P must lie on the intersection of two components $F_1$ and $F_2$ of F. For this, let . Then, since $\text {mult}_P(\widetilde {\Gamma }_Z) \leq \nu < \varepsilon $ and $f_i < 1-\varepsilon $ , we have that

$$ \begin{align*}0 < c= (\text{mult}_P(\widetilde{\Gamma}_Z)+\sum_{i \in J_F} f_i-1) \leq \nu + |J_F|(1-\varepsilon)-1.\end{align*} $$

Hence, $|J_F| \geq 2$ . Since $\mathrm {Supp}(F)$ is an snc divisor, this implies $|J_F| = 2$ , as desired. This argument can be repeated for each blow-up $T \to W$ factorising $Y \to W$ .

As the number of nodes of F is bounded by $\rho (W/Z)-1$ , it remains to bound the number of times we are allowed to blow-up along a node to obtain a bound on $\rho (Y/W)$ . This follows from a straightforward induction as in [Reference Alexeev and MoriAM04, Lemma 1.9]. An explicit computation shows that

$$ \begin{align*}\rho(Y)=\rho(Y/W)+\rho(W/Z)+\rho(Z) \leq \left(\frac{1}{(\varepsilon-\nu)^2}-1 \right) \cdot \left( \frac{8}{\varepsilon \nu^2} -1 \right) + \left( \frac{8}{\varepsilon \nu^2} \right) +2.\end{align*} $$

As $\nu =\frac {\varepsilon }{2}$ , we deduce that

$$ \begin{align*} \rho(Y) \leq \frac{128}{\varepsilon^5}+\left(3-\frac{4}{\varepsilon^2}\right) \leq \frac{128}{\varepsilon^5}.\\[-42pt] \end{align*} $$

Proof. By Lemma 5.9 and Proposition 5.11, there exists $n=n(\varepsilon )>0$ such that $-nK_X$ is Cartier for all geometrically integral $\varepsilon $ -klt del Pezzo surfaces X. Hence, we can apply Corollary 5.8 to conclude that $\mathcal {X}_{\text {dP}, \varepsilon }^{\text {tame}} $ and $\mathcal {X}_{\text {dP}, \varepsilon }^{>5}$ are bounded.