# Nullstellensatz for relative existentially closed groups

M. Shahryari  M. Shahryari: Department of Mathematics, College of Science,
Sultan Qaboos university, Muscat, Oman<EMAIL_ADDRESS>
###### Abstract.

We prove that in every variety of $G$-groups, every $G$-existentially closed
element satisfies nullstellensatz for finite consistent systems of equations.
This will generalize Theorem G of [1]. As a result we see that every pair of
$G$-existentially closed elements in an arbitrary variety of $G$-groups
generate the same quasi-variety and if both of them are $q_{\omega}$-compact,
they are geometrically equivalent.

AMS Subject Classification Primary 20F70, Secondary 08A99.
Keywords algebraic geometry over groups; nullstellensatz; existentially closed
groups; varieties of groups; quasi-varieties.

##  1\. Introduction

Let $K$ be an algebraically closed field and $K[x_{1},\ldots,x_{m}]$ be the
ring of $n$-variable polynomials with coefficients in $K$. Let $S\subseteq
K[x_{1},\ldots,x_{m}]$ be a system of equations and $\mathrm{Rad}_{K}(S)$
denotes the set of all consequences of $S$ over $K$. Note that a polynomial
$f$ is called a consequence of $S$, if every solution of the system $S=1$ is
also a solution of $f=1$. Hilbert’s classical nullstellensatz, describes the
ideal $\mathrm{Rad}_{K}(S)$ in terms of elementary algebraic operation. More
precisely by Hilbert’s nullstellensatz, we have

$\mathrm{Rad}_{K}(S)=\sqrt{\langle S\rangle_{\mathrm{id}}},$

where $\langle S\rangle_{\mathrm{id}}$ is the ideal generated by $S$ and
$\sqrt{-}$ shows the radical: the set of all polynomials $f$ such that
$f^{m}\in\langle S\rangle_{\mathrm{id}}$ for some power $m$.

In 1999, Baumslag, Myasnikov, and Remeslennikov published the article
Algebraic Geometry over Groups, I. Algebraic Sets and Ideal Theory, where the
foundations of algebraic geometry of groups is introduced, [1]. Among many
important notions discussed in their work is the notion of nullstellensatz for
group equations. Generally it is any effective (algebraic or algorithmic)
description of the radical of systems of equations over groups. The last
section of that article was devoted to nullstellensatz in groups. They proved
the following theorem (Theorem G of [1]):

######  Theorem 1.

Let $H$ be a $G$-group and suppose that $H$ is $G$-existentially closed. Then
every finite $H$-consistent system of group equations with coefficients from
$G$, satisfy nullstellensatz, i.e.

$\mathrm{Rad}_{H}(S)=\langle S^{G[x_{1},\ldots,x_{m}]}\rangle.$

In the above theorem $G$ is an arbitrary fixed group and a $G$-group is a
group which contains $G$ as a distinguished subgroup. An equation with
coefficients in $G$ is an element of the free product
$G[x_{1},\ldots,x_{m}]=G\ast F[x_{1},\ldots,x_{m}]$, where
$F[x_{1},\ldots,x_{m}]$ is the free group generated by the variables
$x_{1},\ldots,x_{n}$. Being $G$-existentially closed means that any finite
system of $G$-equations and $G$-inequations which has a solution in a super-
group of $H$, has already a solution in $H$. A system is $H$-consistent if it
has at least one solution in $H$. Note that the notation

$\langle S^{G[x_{1},\ldots,x_{m}]}\rangle$

is used for the normal closure of $S$ in the group $G[x_{1},\ldots,x_{m}]$.
There are many similarities between Theorem 1 above and the classical
nullstellensatz of Hilbert, because it can be shown that in the variety of
commutative rings with identity, algebraically closed fields paly the role of
existentially closed structures.

The aim of this article is to generalize the above theorem for the case of
existentially closed groups in an arbitrary variety of $G$-groups. Our main
theorem has the following formulation:

######  Theorem 2.

Let $\mathbf{V}$ be a variety of $G$-groups and $H\in\mathbf{V}$ be a relative
$G$-existentially closed element. Then for every finite
$\mathbf{V}$-consistent system $S\subseteq G[x_{1},\ldots,x_{m}]$, we have

$\mathrm{Rad}_{H}(S)=\langle
S^{G[x_{1},\ldots,x_{m}]}\rangle\mathrm{Id}_{\mathbf{V}}(x_{1},\ldots,x_{n}).$

Recall that, in this theorem,

$\mathrm{Id}_{\mathbf{V}}(x_{1},\ldots,x_{n})\unlhd G[x_{1},\ldots,x_{m}]$

consists of all $G$-identities of the variety $\mathbf{V}$, having variables
$x_{1},\ldots,x_{n}$.

##  2\. The proof

In this section, we prove Theorem 2, but before, we collect some necessary
notions from algebraic geometry over groups. For detailed discussion, one can
consult [1] and [2].

We assume that $G$ is an arbitrary group. A $G$-group is a pair $(H,\lambda)$,
where $H$ is a group and $\lambda:G\to H$ is an embedding. If there is no risk
of confusion, we will say that $H$ is a $G$-group, and so it contains a
distinguished copy of $G$. Let $\mathcal{L}=(\cdot,^{-1},1)$ be the language
of groups and for any $g\in G$ attach a constant symbol $g$ to $\mathcal{L}$.
We denote the extended language by $\mathcal{L}(G)$ and so every $G$-group $H$
becomes an algebra of type $\mathcal{L}(G)$, if we interpret $g$ as
$\lambda(g)$. Note that any congruence of $H$ is in fact a normal subgroup $K$
with the property $G\cap K=1$. Through this section, we will call such a
normal subgroup an $G$-ideal.

For a set $X=\\{x_{1},\ldots,x_{n}\\}$ the free $G$-group generated by $X$ is
the free product $G[X]=G\ast F[X]$, where $F[X]$ is the ordinary free group on
$X$. We will assume that the embedding $G\hookrightarrow G[X]$ is the
inclusion map. Any subset $S\subseteq G[X]$ corresponds to a system of
$G$-equations, and if $w=w(x_{1},\ldots,x_{n},g_{1},\ldots,g_{m})\in G[X]$,
then the expression $w\approx 1$ is a $G$-equation with coefficient
$g_{1},\ldots,g_{m}\in G$. Let $(H,\lambda)$ be a $G$-group. We say that
$\overline{a}=(a_{1},\ldots,a_{n})\in H^{n}$ is a solution for this
$G$-equation if

$w(a_{1},\ldots,a_{n},\lambda(g_{1}),\ldots,\lambda(g_{m}))=1.$

We denote by $V_{H}(S)$ the algebraic set corresponding to a system of
$G$-equations $S$, which is the set of all common solutions of the elements of
$S$. Define the radical of $S$, with respect to $H$, as

$\mathrm{Rad}_{H}(S)=\\{w\in G[X]:V_{H}(S)\subseteq V_{H}(w\approx 1)\\}.$

This is clearly a $G$-ideal of $G[X]$, which contains the normal closure
$\langle S^{G[X]}\rangle$. The quotient group
$\Gamma_{H}(S)=G[X]/\mathrm{Rad}_{H}(S)$ is called the corresponding
coordinate $G$-group of $S$ over $H$. Many important problems of algebraic
geometry over groups transform to the study of these coordinate groups, and
hence it is important to have an algebraic description of the radical
$\mathrm{Rad}_{H}(S)$.

Generally there is no such an algebraic description for every system $S$,
because it is equivalent to axiomatizablity of the prevariety generated by $H$
in the class of $G$-groups, see [2]. In some cases, such a description exists,
and in this situation we say that a $G$-group $H$ satisfies the generalized
nullstellensatz for a system $S$. Recall that a $G$-group $H$ is
$G$-existentially closed, if every finite set of $G$-equations and
$G$-inequations with a solution in a larger $G$-group $H^{\prime}$ has already
a solution in $H$. As we said in the introduction, Baumslag, Myasnikov, and
Remeslennikov, proved that $G$-existentially closed $G$-groups satisfy
nullstellensatz for finite consistent systems. We can generalize their result
for the case of relatively existentially closed elements of varieties. If
$\mathbf{V}$ is a variety of $G$-groups, then an element $H\in\mathbf{V}$ is
relative $G$-existentially closed, if every finite set of $G$-equations and
$G$-inequations with a solution in a larger $G$-group
$H^{\prime}\in\mathbf{V}$ has already a solution in $H$. Clearly, this is
equivalent to the existence of solution in $H$ for every
$\mathbf{V}$-consistent finite system of $G$-equations and $G$-inequations
(systems with at least one solution in some element of $\mathbf{V}$). It is
proved that every element of $\mathbf{V}$ embeds in some relative
$G$-existentially closed $G$-group $H\in\mathbf{V}$, and this $H$ is not too
large (see for example [3]). Now, we are ready to give the proof of Theorem 2.

###### Proof.

Suppose $X=\\{x_{1},\ldots,x_{n}\\}$. We denote the free $G$-group of the
variety $\mathbf{V}$ over the set $X$ by $F_{\mathbf{V}}(X)$. Let $T$ be the
image of $S$ in $F_{\mathbf{V}}(X)$, that is the set $S$ modulo $G$-identities
of the variety $\mathbf{V}$. Suppose also $\mathrm{Rad}_{H}^{\mathbf{V}}(T)$
is relative radical of $T$ in $\mathbf{V}$. Then we have
$\mathrm{Rad}_{H}^{\mathbf{V}}(T)$ is a $G$-ideal of $F_{\mathbf{V}}(X)$. Let
$Q=\langle T^{F_{\mathbf{V}}(X)}\rangle$ be the normal closure of $T$ in
$F_{\mathbf{V}}(X)$. Note that $V_{H}(Q)\neq\emptyset$, so $Q\cap G=1$. Hence
$K=F_{\mathbf{V}}(X)/Q$ is a $G$-group and it belongs to $\mathbf{V}$. Let
$T=\\{w_{1},\ldots,w_{k}\\}$ and suppose $f$ does not belong to $Q$. Consider
the finite system of $G$-equations and $G$-inequations $T^{\prime}$:

$w_{1}\approx 1,\ldots,w_{k}\approx 1,f\not\approx 1.$

Let $H^{\prime}=H\times K$ and $b_{1}=x_{1}Q,\ldots,b_{n}=x_{n}Q$. The for all
$1\leq i\leq k$, we have

$w_{i}(b_{1},\ldots,b_{n})=w_{i}(x_{1},\ldots,x_{n})Q=Q(=1\ in\ K),$

and on the other hand

$f(b_{1},\ldots,b_{n})=fQ\neq Q.$

This shows that $(b_{1},\ldots,b_{n})$ is a solution of $T^{\prime}$ in
$H^{\prime}$, therefore $T^{\prime}$ has a solution in $H$, say
$a_{1},\ldots,a_{n}\in H$. This means that $(a_{1},\ldots,a_{n})\in V_{H}(T)$,
but since $f(a_{1},\ldots,a_{n})\neq 1$, so $f$ does not belong to
$\mathrm{Rad}_{H}^{\mathbf{V}}(T)$. This shows that
$\mathrm{Rad}_{H}^{\mathbf{V}}(T)\subseteq Q$ and hence
$Q=\mathrm{Rad}_{H}^{\mathbf{V}}(T)$. Now, we claim that

$\mathrm{Rad}_{H}^{\mathbf{V}}(T)=\frac{\mathrm{Rad}_{H}(S).\mathrm{Id}_{\mathbf{V}}(X)}{\mathrm{Id}_{\mathbf{V}}(X)}.$

Note that $F_{\mathbf{V}}(X)=G[X]/I$, where $I$ stands for
$\mathrm{Id}_{\mathbf{V}}(X)$. Note that also

$T=\\{wI:\ w\in S\\}.$

Since for all $(h_{1},\ldots,h_{n})\in H^{n}$, we have

$(wI)(h_{1},\ldots,h_{n})=w(h_{1},\ldots,h_{n}),$

so, we have $V_{H}(T)=V_{H}(S)$. Therefore

$\displaystyle\mathrm{Rad}_{H}^{\mathbf{V}}(T)$ $\displaystyle=$
$\displaystyle\mathrm{Rad}_{H}^{\mathbf{V}}(V_{H}(T))$ $\displaystyle=$
$\displaystyle\mathrm{Rad}_{H}^{\mathbf{V}}(V_{H}(S))$ $\displaystyle=$
$\displaystyle\\{wI\in F_{\mathbf{V}}(X):V_{H}(S)\subseteq V_{H}(wI\approx
1)\\}$ $\displaystyle=$ $\displaystyle\\{wI\in
F_{\mathbf{V}}(X):V_{H}(S)\subseteq V_{H}(w\approx 1)\\}$ $\displaystyle=$
$\displaystyle\\{wI\in F_{\mathbf{V}}(X):w\in\mathrm{Rad}_{H}(S)\\}$
$\displaystyle=$ $\displaystyle\frac{\mathrm{Rad}_{H}(S).I}{I}.$

Note that the normal closure $\langle T^{F_{\mathbf{V}}(X)}\rangle$ is
generated by the elements of the form $(uI)(wI)^{\pm 1}(u^{-1}I)$, with $u\in
G[X]$ and $w\in S$. Therefore, we have

$\langle T^{F_{\mathbf{V}}(X)}\rangle=\frac{\langle S^{G[X]}\rangle.I}{I}.$

Comparing these two last equalities, we obtain

$\mathrm{Rad}_{H}(S)=\langle S^{G[X]}\rangle.\mathrm{Id}_{\mathbf{V}}(X).$

∎

A $G$-group $H$ is called $q_{\omega}$-compact, if for every system $S$, we
have

$\mathrm{Rad}_{H}(S)=\bigcup\\{\mathrm{Rad}_{H}(S_{0}):S_{0}\subseteq S,\
S_{0}=finite\\}.$

Being $q_{\omega}$-compact is a generalization of the property of being
$G$-equational Noetherian (see [2] for its various equivalents and logical
implications). We say that two $G$-groups $H$ and $H^{\prime}$ are
geometrically equivalent, if
$\mathrm{Rad}_{H}(S)=\mathrm{Rad}_{H^{\prime}}(S)$, for all system $S$. In
this case the algebraic geometry of $H$ and $H^{\prime}$ are essentially the
same. As a result, we have

######  Corollary 1.

Let $\mathbf{V}$ be a variety of $G$-groups and $H$ and $H^{\prime}$ be two
relative $G$-existentially closed elements of $\mathbf{V}$. If $H$ and
$H^{\prime}$ are $q_{\omega}$-compact, then they are geometrically equivalent.

###### Proof.

First suppose that $S\subseteq G[X]$ is a consistent system of $G$-equations.
Then, we have

$\displaystyle\mathrm{Rad}_{H}(S)$ $\displaystyle=$
$\displaystyle\bigcup\\{\mathrm{Rad}_{H}(S_{0}):S_{0}\subseteq S,\
S_{0}=finite\\}$ $\displaystyle=$ $\displaystyle\bigcup\\{\langle
S_{0}^{G[X]}\rangle.\mathrm{Id}_{\mathbf{V}}(X):S_{0}\subseteq S,\
S_{0}=finite\\}$ $\displaystyle=$
$\displaystyle\bigcup\\{\mathrm{Rad}_{H^{\prime}}(S_{0}):S_{0}\subseteq S,\
S_{0}=finite\\}$ $\displaystyle=$ $\displaystyle\mathrm{Rad}_{H^{\prime}}(S).$

If $S$ is not consistent, then we have $V_{H}(S)=V_{H^{\prime}}(S)=\emptyset$,
and hence $\mathrm{Rad}_{H}(S)=\mathrm{Rad}_{H^{\prime}}(S)$. ∎

In the special case of $G=1$, every element of the variety of metabelian
groups or the variety of nilpotent groups of a fixed class is equationally
Noetherian and hence is $q_{\omega}$-compact. Therefore, we have the next
result:

######  Corollary 2.

Let $H$ and $H^{\prime}$ be two relative existentially closed groups in the
variety of metabelian groups or in the variety of nilpotent groups of the
fixed class $c$. Then $H$ and $H^{\prime}$ are geometrically equivalent as
$1$-groups.

## References

  * [1] Baumslag G., Myasnikov A., Remeslennikov V. Algebraic geometry over groups, I. Algebraic sets and ideal theory. J. Algebra, 1999, 219, pp. 16-79.
  * [2] Myasnikov A., Remeslennikov V. Algebraic geometry over groups, II. Logical Fundations. J. Algebra, 2000, 234, pp. 225-276.
  * [3] Shahryari M. Existentially closed structures and some emebedding theorems. Math. Notes, 2017, 101 (6), pp. 1023-1032.


# Quasi-étale covers of Du Val del Pezzo surfaces and Zariski dense
exceptional sets in Manin’s conjecture

Runxuan Gao  Graduate School of Mathematics, Nagoya University, Furocho
Chikusa-ku, Nagoya, 464-8602, Japan<EMAIL_ADDRESS>
###### Abstract.

We construct first examples of singular del Pezzo surfaces with Zariski dense
exceptional sets in Manin’s conjecture, varying in degrees $1,2$ and $3$. To
systematically study these examples, we classify all quasi-étale covers of Du
Val del Pezzo surfaces up to singularity types and study their equivariant
geometry.

###### Contents

  1. 1 Introduction
  2. 2 Construction
    1. 2.1 Du Val del Pezzo surface
    2. 2.2 classification of Du Val del Pezzo surfaces
    3. 2.3 fundamental groups
    4. 2.4 negative curves on quasi-étale covers
  3. 3 Classification
    1. 3.1 quasi-étale covers
    2. 3.2 quasi-étale covers with higher $b$-values
  4. 4 Examples
    1. 4.1 degree $3$, type $4A_{1}$
    2. 4.2 degree $2$, type $D_{4}+3A_{1}$
    3. 4.3 degree $2$, type $3A_{2}$
    4. 4.4 degree $1$, type $E_{6}+A_{2}$
  5. 5 Proof of Theorem C

##  1\. Introduction

This paper concerns the distribution of rational points on algebraic
varieties. Let $X$ be a geometrically rationally connected variety over a
number field $k$. A well-known conjecture of Colliot-Thélène implies that the
set $X(k)$ of rational points is Zariski dense in $X$ as soon as there exists
a rational point on the smooth locus of $X$. So it makes sense to study the
distribution of them with respect to some height. Let ${\mathcal{L}}$ be an
adelically metrized line bundle whose underlying line bundle $L$ is big and
nef. There is a unique height function
$H_{\mathcal{L}}:X(k)\rightarrow{\mathbb{R}}$ associated to ${\mathcal{L}}$.
We define the counting function

$N(U,{\mathcal{L}},B):=\\#\\{x\in U(k)\mid H_{\mathcal{L}}(x)\leq B\\}$

for any $B\geq 0$ and any subset $U\subseteq X$.

The closed-set version of Manin’s conjecture, first formulated in [FMT89,
BM90], predicts that there exists a proper closed subset $Z$ of $X$ such that

$N(X\backslash Z,{\mathcal{L}},B)\sim c\cdot B^{a}(\log B)^{b-1},$

where $a=a(X,L)$ and $b=b(X,L)$ are birational invariants and
$c=c(X,{\mathcal{L}})$ is the Peyre’s constant introduced in [Pey95, BT98b].

The closed-set version of Manin’s conjecture is not true in general. In
[BT98b], Batyrev and Tschinkel showed that for some cubic surface bundles $X$,
it is not enough to remove a proper closed subset $Z$, or the $b$-invariant
will be violated. These counterexamples appear in each dimension $\geq 3$. See
[Gao23a] for an analysis of Batyrev-Tschinkel’s example in the spirit of
[LST22].

To overcome such counterexamples, it was first suggested in [Pey03] that the
exceptional set $Z$ should be assumed to be a thin set (i.e. a subset of a
finite union of $Z_{i}(F)$ and $f_{j}(Y_{j}(F))$ where $Z_{i}$ is a proper
closed subset of $X$ and $f_{j}:Y_{j}\rightarrow X$ is a generically finite
morphism of degree $\geq 2$). Nowadays, there is increasing evidence of the
thin-set version of Manin’s conjecture, such as the established cases [LR19,
BHB20], the geometry consistency of the conjecture [LT17, LST22], and its
compatibility with Manin’s conjectures for stacks [DY22, ESZB23].

Manin’s conjecture has been widely studied in dimension $2$, especially for Du
Val del Pezzo surfaces (i.e. surfaces that have an ample anticanonical divisor
and admit only canonical singularities). In the later case, the conjecture
reads as follows.

######  Conjecture 1.1.

Let $X$ be a Du Val del Pezzo surface over a number field $F$ and let
${\mathcal{L}}=-{\mathcal{K}}_{X}$ be the anticanonical line bundle of $X$
with an adelic metric. Suppose $X(F)$ is not a thin set. Then there exists a
thin subset $Z$ of $X(F)$ such that

$N(X\backslash Z,{\mathcal{L}},B)\sim c\cdot B(\log
B)^{\rho({\widetilde{X}})-1},$

where ${\widetilde{X}}\rightarrow X$ is the minimal resolution of $X$.

Conjecture 1.1 has been established for all toric Du Val del Pezzo surfaces
and numerous specific surfaces of degree $\geq 3$. The only established case
of degree $\leq 2$ is a Du Val del Pezzo surface of degree $2$ with a $E_{7}$
singularity [BB13]. See [Der13] for a list of established cases in dimension
$2$. There are also many new cases established after the publication of
[Der13], such as [FP16, LWZ17, dlBDL+19].

In all the established cases in dimension $2$, the exceptional set is the
union of the negative curves, which is always a proper closed subset. However,
there do exist smooth del Pezzo surfaces of degree $1$ for which the closed-
set version of Manin’s conjecture does not hold [Gao23b], where the
$c$-constant will be violated if one merely removes a proper closed
exceptional set. This pathological example is caused by a family of conics on
the surface parameterized by an elliptic curve, which can not lead a
counterexample of a singular del Pezzo surface.

In this paper, we provide many more counterexamples to the closed-set version
of Manin’s conjecture in dimension $2$, caused by quasi-étale covers (i.e.
finite and étale-in-codimension-$1$ morphisms, see Definition 2.3). In
contrast to the previous one, this pathological phenomenon exists only for
singular del Pezzo surfaces, and furthermore, even the $b$-invariant is
violated in these counterexamples. This together with [Gao23b] is conjectured
to be the only possible obstructions to the closed-set version of Manin’s
conjecture in dimension $2$; see [LT17, Section 6] and [LT19b, Section 9]. To
systematically study such examples, we give a classification of all quasi-
étale covers of Du Val del Pezzo surfaces up to singularity types (Proposition
3.1), which do not seem to exist in the existing literature.

Let $S$ be a Du Val del Pezzo surface over a field of characteristic $0$ and
let $T\rightarrow S$ be a quasi-étale cover. We denote by ${\widetilde{S}}$
and ${\widetilde{T}}$ the minimal resolutions of $S$ and $T$. Our first result
shows that such pathological examples can only exist in low degrees.

######  Theorem A.

(=Corollary 3.4) Over any field of characteristic $0$, we have

  1. (1)

$\rho({\widetilde{T}})\leq\rho({\widetilde{S}})$ when $\deg S\geq 3$;

  2. (2)

$\rho({\widetilde{T}})<\rho({\widetilde{S}})$ when $\deg S\geq 4$.

To achieve a classification, we study the possible group actions on the
pseudo-effective cones. The Galois action on $\pi$ induces a map

$\rho:\mathrm{Gal}(\overline{k}/k)\rightarrow\mathrm{Cris}(\pi),$

where $\mathrm{Cris}(\pi)$ is a group that acts on the pseudo-effective cones
of the base change of ${\widetilde{S}}$ and ${\widetilde{T}}$ to an algebraic
closed field, and depends only on the singularity types of $T$ and $S$ in most
cases; see Definition 2.17. It is not always true and difficult to determine
whether, for any subgroup $H$ of $\mathrm{Cris}(\pi)$, there exist surfaces
$T$ and $S$ such that the image of $\rho$ equals $H$ (see [Vir23] for some
results in this direction). We circumvent this realization problem for the
moment. When a group $G$ acts on the Néron-Severi space $N^{1}(X)$, we write
$\rho(X,G):=\dim N^{1}(X)^{G}$. Our second result provides all possibilities
of such pathological examples in degrees $\geq 2$, and the data we provided on
[Gao24] can be used to complete the cases of degrees $1$.

######  Theorem B.

(=Proposition 3.3) Let $\pi:T\rightarrow S$ be a nontrivial quasi-étale cover
of Du Val del Pezzo surfaces over an algebraically closed field $k$ of
characteristic $0$. Assume $S$ is of degree $\geq 2$. Then there exists a
subgroup $H\subseteq\mathrm{Cris}(\pi)$ such that
$\rho({\widetilde{T}},H)\geq\rho({\widetilde{S}},H)$ if and only if
$\mathrm{Type}(S)$ is one of the following types.

  * •

degree $3$: $4A_{1};$

  * •

degree $2$: $4A_{1}$ $(\text{with }20\text{ lines})$, $5A_{1}$,
$A_{3}+3A_{1}$, $3A_{2}$, $6A_{1}$, $D_{4}+2A_{1}$, $2A_{3}$, $D_{4}+3A_{1}$,
$2A_{3}+A_{1}$.

There exists a subgroup $H\subseteq\mathrm{Cris}(\pi)$ such that
$\rho(T,H)>\rho(S,H)$ if and only if $\mathrm{Type}(S)$ is one of the
following types.

  * •

degree $2$: $4A_{1}$ $(\text{with }20\text{ lines})$, $5A_{1}$, $3A_{2}$,
$6A_{1}$.

Based on the above analysis, we provide new counterexamples to the closed-set
version of Manin’s conjecture in dimension $2$. These counterexamples are Du
Val del Pezzo surfaces of degrees $3,2,2$ and $1$, respectively, and are all
rational over the base field (Proposition 5.3). In particular, the assumption
that $X(F)$ is not a thin set in Conjecture 1.1 is satisfied for all these
surfaces.

######  Theorem C.

The following assertions hold true.

  1. (1)

Let $S$ be either

    1. (a)

(degree $3$, type $4A_{1}$) the singular cubic surface in
${\mathbb{P}}^{3}_{\mathbb{Q}}$ defined by the equation

$X^{3}+2XYW+XZ^{2}-Y^{2}Z+ZW^{2}=0,$

    2. (b)

(degree $2$, type $D_{4}+3A_{1}$) the hypersurface in
${\mathbb{P}}_{\mathbb{Q}}(1,1,1,2)$ defined by the equation

$W^{2}-XY(Z^{2}+Y^{2})=0.$

Then for any open subset $U$ of $S$, there exists an adelically metrized
anticanonical line bundle $-{\mathcal{K}}_{S}$ on $S$ such that the equation

$N(U,-{\mathcal{K}}_{S},B)\sim c(S,-{\mathcal{K}}_{S})B(\log
B)^{\rho({\widetilde{S}})-1}$

does not hold true.

  2. (2)

Let $S$ be either

    1. (a)

(degree $2$, type $3A_{2}$) the hypersurface in
${\mathbb{P}}_{\mathbb{Q}}(1,1,1,2)$ defined by the equation

$W^{2}+(3X^{2}-YZ)W+9X^{4}-6X^{2}YZ+X(Y^{3}+Z^{3})=0,$

    2. (b)

(degree $1$, type $E_{6}+A_{2}$) the hypersurface in
${\mathbb{P}}_{\mathbb{Q}}(1,1,2,3)$ defined by the equation

$W^{2}+Z^{3}+X^{4}Y^{2}=0.$

Let $U$ be an dense open subset of $S$ and let $-{\mathcal{K}}_{S}$ be an
adelically metrized anticanonical line bundle. Then for any constant $C>0$,
there exists $B_{0}$ such that for any $B>B_{0}$, we have

$N(U,-{\mathcal{K}}_{S},B)>CB(\log B)^{\rho({\widetilde{S}})-1}.$

In other words, Manin’s conjecture with Peyre’s constant for $S$ in (1) does
not hold for some anticanonical height for any proper closed exceptional set,
and Manin’s conjecture with the correct $b$-constant for $S$ in (2) does not
hold for any anticanonical height and any proper closed exceptional set.

Finally, we restate Manin’s conjecture with the conjectural exceptional set
proposed in [LST22] for Du Val del Pezzo surfaces (see [LT19b, Section 9]).

######  Notation 1.2.

(the geometric exceptional set) Let $S$ be a Du Val del Pezzo surface of
degree $d$ defined over a number field $k$ and $-{\mathcal{K}}_{S}$ be an
adelically metrized anticanonical line bundle on $X$.

  1. (1)

Let $Z_{1}$ be the union of $(-1)$-curves on $S$.

  2. (2)

Let $Z_{2}$ be the union of $\pi(T(k))$ where $\pi:T\rightarrow S$ is a quasi-
étale cover of degree $\geq 2$ such that
$\rho({\widetilde{T}})\geq\rho({\widetilde{S}})$ and $\pi$ is face-
contracting.

  3. (3)

Let $Z_{3}$ be

    1. (a)

the empty set if either $d\geq 3$ or $\rho({\widetilde{S}})\geq 2$;

    2. (b)

the union of rational curves in $\left|-K_{S}\right|$ when $d=2$ and
$\rho({\widetilde{S}})=1$;

    3. (c)

the union of rational curves in $\left|-2K_{S}\right|$ and rational curves $C$
on $S$ with $C^{2}=2$ and $K_{S}\cdot C=-2$, when $d=1$ and
$\rho({\widetilde{S}})=1$.

######  Conjecture 1.3.

Under Notation 1.2, Conjecture 1.1 holds true for $Z:=Z_{1}\cup Z_{2}\cup
Z_{3}$.

The study of quasi-étale covers of singular del Pezzo surfaces is of
independent interest. For instance, cubic surfaces which admit a double quasi-
étale cover have been classically studied as cubic symmetroids. Moreover,
quasi-étale covers of Du Val del Pezzo surfaces of degrees $3$ and $4$ are key
tools in the study of Enriques surfaces [CD89]. For relatively minimal Du Val
del Pezzo surfaces, the universal quasi-étale covers were determined in [MZ88]
and [MZ93]. We generalize these results to all Du Val del Pezzo surfaces using
a more direct method (see Proposition 2.10). There is also an intersecting
connection between the fundamental group and the Brauer group for these
surfaces (see Remark 2.11).

Notations. Throughout this paper, we will use the following notations:

  * •

$C_{n}$ is a cyclic group of order $n$.

  * •

${\mathfrak{S}}_{n}$ is a symmetric group on $n$ letters.

  * •

${\mathfrak{D}}_{n}$ is a dihedral group of order $2n$.

  * •

$H\wr S$ is the wreath product of groups $H$ and
$S\subseteq{\mathfrak{S}}_{n}$.

  * •

$A_{\mathrm{tors}}$ is the torsion subgroup of an abelian group $A$.

  * •

$\rho(X)$ is the Picard rank of a variety $X$.

  * •

$K_{X}$ is the canonical line bundle of a variety $X$.

  * •

${\mathcal{L}}$ is an adelically metrized line bundle whose underlying line
bundle is $L$.

  * •

$f(B)\sim g(B)$ means $\lim_{B\rightarrow\infty}({f(B)}/{g(B)})=1$.

  * •

$f(B)\ll g(B)$ means $\limsup_{B\rightarrow\infty}({f(B)}/{g(B)})<+\infty$.

Acknowledgments. The author would thank his advisor Sho Tanimoto for constant
support, and Tim Browning, Ulrich Derenthal, Daniel Loughran and Yuri
Tschinkel for useful conversations and suggestions. The author was partially
supported by JST FOREST program Grant Number JPMJFR212Z, JSPS Bilateral Joint
Research Projects Grant Number JPJSBP120219935 and Grant-in-Aid for JSPS
Fellows Number 24KJ1234.

##  2\. Construction

###  2.1. Du Val del Pezzo surface

We work over an arbitrary field $k$ of characteristic $0$. A _Du Val del Pezzo
surface_ $S$ is a projective variety of dimension $2$ with ample anticanonical
divisor $-K_{S}$ and at worst canonical singularities. When the surface is
smooth, we call it a _smooth del Pezzo surface_ or simply a _del Pezzo
surface_.

A _weak del Pezzo surface_ is a smooth projective surface with a big and nef
anticanonical divisor.

Let $X$ be a weak del Pezzo surface. We call $d:=K_{X}^{2}$ the degree of $X$.
It turns out that $1\leq d\leq 9$ and $X_{\overline{k}}$ is isomorphic to
${\mathbb{P}}^{1}_{\overline{k}}\times{\mathbb{P}}^{1}_{\overline{k}}$, the
Hirzebruch surface ${\mathbb{F}}_{2}$, or the blow up of
${\mathbb{P}}^{2}_{\overline{k}}$ at $9-d$ points in almost general positions,
where $\overline{k}$ is an algebraic closure of $k$. The Proj construction of
the graded ring

$\bigoplus_{m\geq 0}H^{0}(X,-mK_{X})$

defines a birational morphism $X\rightarrow S$ which is known as the
anticanonical model map. It turns out that $S$ is a Du Val del Pezzo surface
and $X\rightarrow S$ coincides with the minimal resolution of $S$. Conversely,
the minimal resolution of any Du Val del Pezzo surface is a weak del Pezzo
surface [Dol12, Theorem 8.3.2]. So we may abuse the notation of a Du Val del
Pezzo surface $S$ and its associated weak del Pezzo surface $X$.

###  2.2. classification of Du Val del Pezzo surfaces

In this section, we give a quick review of classification of Du Val del Pezzo
surfaces; see [Dol12] for more details. We work over an algebraically closed
field $k$ of characteristic $0$. An integral rational curve $C$ on a normal
surface with $(C\cdot C)=n$ is called a _$(n)$ -curve_. We call $C$ a
_negative curve_ if $n<0$. A $(-1)$-curve is also called a line. Let $X$ be a
weak del Pezzo surface. Then the anticanonical model map $X\rightarrow S$
exactly contracts the $(-2)$-curves on $X$.

We call a numerical class of an integral curve in $N^{1}(X)$ an extremal curve
if it generates an extremal ray of $\overline{\mathrm{Eff}}^{1}(X)$. The _dual
graph_ of $X$, denoted by $\Gamma(X)$, is a graph where each vertex represents
an extremal curve and the number of edges between two nodes equals the
intersection number of the two integral curves. For $X$ of degree $d\leq 7$,
the extremal curves on $X$ are exactly the negative curves and $\Gamma(X)$ is
the dual graph of negative curves on $X$.

An _isometry_ between two lattice is a linear map that preserve the inner
product. The following notion of a Cremona isometry is the same as that
defined in [Dol12, Section 8.2.8].

######  Definition 2.1.

Let $X$ and $X^{\prime}$ be two weak del Pezzo surfaces. Then an isometry
$\sigma:N^{1}(X)\rightarrow N^{1}(X^{\prime})$ is called a _Cremona isometry_
if $\sigma$ restricts to a bijection between $\overline{\mathrm{Eff}}^{1}(X)$
and $\overline{\mathrm{Eff}}^{1}(X^{\prime})$ which sends $K_{X}$ to
$K_{X^{\prime}}$. We say $X$ and $X^{\prime}$ have the same _type_ whenever
there exists a Cremona isometry between them. We write $\mathrm{Cris}(X)$ for
the group of Cremona isometries of $X$ to itself.

The group of isometries of $N^{1}(X)$ that preserve $K_{X}$ is isomorphic to
the Weyl group $W(R_{d})$ of the root system

$R_{d}:=\\{D\in N^{1}(X)\mid(D\cdot D)=-2,(K_{X}\cdot D)=0\\},$

which only depends on $d$ when $d\leq 7$. The types of weak del Pezzo surfaces
are in bijection with the root subsystems $R$ of $R_{d}$ up to automorphisms,
except for four types ($7A_{1}$ in degree $1$ and $7A_{1},8A_{1}$ and
$D_{4}+4A_{1}$ in degree $1$) which only occur over fields of characteristic
$2$. There do exist isomorphic root systems which are not isomorphic as root
subsystems of $R_{d}$. Nonetheless, it turns out that the type of a weak del
Pezzo surface $X$ is determined by its degree $\deg(X)$, its ADE type $R$, and
the number of $(-1)$-curves by scanning the classification, which has been
done in degree $3$ by Schläfli [Sch63] and Cayley [Cay69], and in degrees $1$
and $2$ by Du Val [Val34]. When the degree and the ADE type are not enough to
describe the type, we add the number of lines. For example,
$S_{4}(2A_{1}(8l))$ means a Du Val del Pezzo surface of degree $4$ with two
$A_{1}$ singularities and $8$ lines. We use $S_{d}$ to denote the type of a
(smooth) del Pezzo surface of degree $d$.

For a weighted graph $\Gamma$, we denote by $\mathrm{Aut}(\Gamma)$ the group
of permutations of vertices that preserve the weights.

######  Proposition 2.2.

Let $X$ be a weak del Pezzo surface. Then the groups $\mathrm{Cris}(X)$ and
$\mathrm{Aut}(\Gamma(X))$ are isomorphic.

###### Proof.

When $X$ is of degree $d\leq 6$, by [Man86, Theorem 23.9], the following two
groups are isomorphic:

  * •

the group of isometries of $N^{1}(X)$ that preserve $K_{X}$, and

  * •

the group of permutations of $(-1)$-classes preserving the pairwise
intersection numbers between them.

Then the assertion follows from the fact that any $(-1)$-class is numerically
equivalent to a linear combination of $(-1)$-curves and $(-2)$-curves with
rational coefficients [DJT08, Proposition 3.6 and 3.7].

For weak del Pezzo surfaces $X$ of degree $\geq 7$, a complete list of
$\mathrm{Cris}(X)$ is as follows:

  * •

$\mathrm{Cris}(X)\cong C_{2}$ when $X$ is ${\mathbb{P}}^{2}$ or
${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$;

  * •

$\mathrm{Cris}(X)\cong C_{1}$ when $X$ is the Hirzebruch surface
${\mathbb{F}}_{1}$ or ${\mathbb{F}}_{2}$.

It is readily to check that $\mathrm{Cris}(X)\cong\mathrm{Aut}(\Gamma(X))$ in
each case. ∎

###  2.3. fundamental groups

######  Definition 2.3.

Let $X$ be a normal variety. A _cover_ of $X$ is a morphism $\pi:Y\rightarrow
X$ where $Y$ is a normal variety, and $\pi$ is finite and surjective. A
_morphism of covers_ $\pi:Y\rightarrow X$ and
$\pi^{\prime}:Y^{\prime}\rightarrow X$ is a morphism $f:Y\rightarrow
Y^{\prime}$ such that $\pi^{\prime}\circ f=\pi$. A morphism $f:Y\rightarrow X$
between normal varieties is called _quasi-étale_ if $f$ is quasi-finite and
étale in codimension $1$.

######  Definition 2.4.

A cover $\pi:Y\rightarrow X$ of varieties is called _Galois_ if there exists a
finite group $G\subset\mathrm{Aut}(Y)$ such that $\pi$ is isomorphic to the
quotient map $Y\rightarrow Y/G$. In this case, we say $\pi$ is Galois with
group $G$.

By the following lemma, étale and unramified covers are the same thing in our
case.

######  Lemma 2.5.

([CDJ+22, Lemma 2.3]) Let $X$ be an integral normal noetherian scheme and let
$\pi:Y\rightarrow X$ be a finite surjective morphism of integral schemes. Then
$\pi$ is either ramified or étale.

To study quasi-étale covers of $X$, it is enough to study the fundamental
group of the smooth part of $X$.

######  Lemma 2.6.

Let $X$ be a normal projective variety and let $X^{\circ}$ be any open
subvariety of $X\backslash\mathrm{Sing}(X)$ that is isomorphic in codimension
$2$. Then there is a one-to-one correspondence between isomorphic classes of
quasi-étale covers of $X$ and that of étale covers of $X^{\circ}$.

###### Proof.

Since $X$ is normal, $\mathrm{Sing}(X)$ has codimension $\geq 2$ in $X$. Let
$\pi:Y\rightarrow X$ be an quasi-étale cover. By Zariski-Nagata purity, $\pi$
is étale over $S\backslash\mathrm{Sing}(X)$. Conversely, let
$\pi^{\circ}:Y^{\circ}\rightarrow X^{\circ}$ be an étale cover. By Zariski’s
main theorem, $\pi^{\circ}$ can be uniquely extended to a finite morphism
$\pi:Y\rightarrow X$. We have $Y$ is projective since $X$ is. So $\pi$ is
surjective and in particular a quasi-étale cover. ∎

######  Remark 2.7.

By Zariski’s main Theorem in the equivariant setting [GKP16, Theorem 3.8], if
the cover $\pi^{\circ}:Y^{\circ}\rightarrow X^{\circ}$ is Galois with group
$G$, then $\pi:Y\rightarrow X$ is also Galois with group $G$, i.e. the group
action of $G$ on $X^{\circ}$ can be extended to $X$ uniquely.

######  Lemma 2.8.

Let $S$ be a Du Val del Pezzo surface and $\pi:T\rightarrow S$ be a quasi-
étale cover. Then $T$ is also a Du Val del Pezzo surface.

###### Proof.

Since $\pi$ is finite, the Hurwitz formula $K_{T}\sim\pi^{\ast}K_{S}$ implies
that $-K_{T}$ is ample. Since $\mathrm{discrep}(T)\geq\mathrm{discrep}(S)=0$
[KM98, Proposition 5.20 (3)], $T$ has at worst Du Val singularities. ∎

Now let $S$ be a Du Val del Pezzo surface. It has been shown in [MZ93] that
the fundamental group of the smooth part of $S$ is a finite abelian group.

######  Remark 2.9.

In [MZ88, MZ93], the authors compute $\pi_{1}(S_{\mathrm{sm}})$ and the
singularities of the universal cover for relative minimal Du Val del Pezzo
surface $S$. They show that the fundamental group is preserved by the MMP
steps. However, the type of singularities of the covers in non-minimal cases
is unclear from this perspective.

We provide a more direct method than that in [MZ88, MZ93] to compute
$\pi_{1}(S_{\mathrm{sm}})$, though its proof relies on [MZ88].

######  Proposition 2.10.

Let $S$ be a Du Val del Pezzo surface over an algebraically closed field of
characteristic $0$ and $\rho:X\rightarrow S$ be the minimal resolution. Let
${\mathbf{E}}$ be the sublattice of $\mathrm{Pic}(X)$ generated by the
$(-2)$-curves on $X$. Then

$\pi_{1}^{\mathrm{\acute{e}t}}(S_{\mathrm{sm}})\cong(\mathrm{Pic}(X)/{\mathbf{E}})_{\mathrm{tors}}.$

###### Proof.

A standard argument reduces us to the case of $S$ over the complex field
${\mathbb{C}}$. Working over ${\mathbb{C}}$, Let $E$ be the exceptional
divisor of $\rho$. Consider the long exact sequence:

${\cdots}$${{H_{2}(E;{\mathbb{Z}})}}$${{H_{2}(X;{\mathbb{Z}})}}$${{H_{2}(X,E;{\mathbb{Z}})}}$${{H_{1}(E;{\mathbb{Z}})}}$${\cdots}$$\scriptstyle{i}$

Write $E=\sum_{i=1}^{r}E_{i}$ for the irreducible decomposition of $E$. Then
$H_{2}(E;{\mathbb{Z}})$ is isomorphic to the free ${\mathbb{Z}}$-module
generated by the class of $E_{i}$. Since the matrix $(E_{i}\cdot E_{j})$ is
negative definite [KM98, Lemma 3.40], $E_{i}$ are linearly independent in
$H_{2}(X;{\mathbb{Z}})$. So the map $i$ is injective. Note that
$H_{1}(E;{\mathbb{Z}})=0$ and $H_{2}(X,E;{\mathbb{Z}})\cong
H^{2}(S_{\mathrm{sm}};{\mathbb{Z}})$ by Lefschetz duality. We conclude that
$H^{2}(S_{\mathrm{sm}};{\mathbb{Z}})\cong\mathrm{Pic}(X)/{\mathbf{E}}$, where
$\mathrm{Pic}(X)\cong H^{2}(X;{\mathbb{Z}})$ since $X$ is a rational surface.
By the universal coefficient theorem, we have
$H^{2}(S_{\mathrm{sm}};{\mathbb{Z}})_{\mathrm{tors}}\cong
H_{1}(S_{\mathrm{sm}},{\mathbb{Z}})_{\mathrm{tors}}$. Then,
$\pi_{1}(S_{\mathrm{sm}})$ is isomorphic to
$H_{1}(S_{\mathrm{sm}},{\mathbb{Z}})_{\mathrm{tors}}$ since
$\pi_{1}(S_{\mathrm{sm}})$ is a finite abelian group [MZ88]. The assertion
follows. ∎

######  Remark 2.11.

In [Bri13], the author shows that the torsion subgroup of
$\mathrm{Pic}(X)/\mathbf{E}$ is isomorphic to the Brauer group
$\mathrm{Br}(S)$ and computes $\mathrm{Br}(S)$ for each type of $S$. Combining
with Proposition 2.10, we see that
$\pi_{1}(S_{\mathrm{sm}})\cong\mathrm{Br}(S)$ for any Du Val del Pezzo surface
$S$ over an algebraically closed field of characteristic $0$.

######  Remark 2.12.

There is a version of Proposition 2.10 for weak Fano pairs with klt
singularities, see [Bra21, Corollary 4].

###  2.4. negative curves on quasi-étale covers

Work over an arbitrary field $k$ of characteristic $0$. Let $\pi:T\rightarrow
S$ be a cyclic quasi-étale cover of degree $d$ of Du Val del Pezzo surfaces
and let $Y\rightarrow T$ and $X\rightarrow S$ be the minimal resolutions. Let
$f:T^{\prime}\rightarrow X$ be the normalization of $X$ in the function field
$K(T)$ of $T$. Let $f:T^{\prime}\dashrightarrow Y$ be the rational map defined
by composition.

The following proposition can be seen as a global version of [Art77,
Proposition 1.5].

######  Proposition 2.13.

The birational map $g:T^{\prime}\dashrightarrow Y$ is a morphism, which
exactly contracts the ramification divisor of $f$.

${{T^{\prime}}}$${Y}$${T}$${X}$${S}$$\scriptstyle{g}$$\scriptstyle{f}$$\scriptstyle{\rho_{T}}$$\scriptstyle{\widetilde{\pi}}$$\scriptstyle{\pi}$$\scriptstyle{\rho_{S}}$

###### Proof.

By Zariski’s main theorem, $T^{\prime}$ factors through $T$ by a morphism, and
a divisor contracted by $T^{\prime}\rightarrow T$ must lie above the singular
locus of $S$. So let $D$ be a prime divisor above $T$ and write $C$ for its
image above $S$. Let $r$ be the ramification index along $D$. Then
$a(D,T)+1=r(a(C,S)+1)$, where $a(D,T)$ means the discrepancy of $D$ with
respect to $T$; see the proof of [KM98, Proposition 5.20]. Since $S$ has only
canonical singularities, we have $a(C,S)\geq 0$. Thus $a(D,T)=0$ if and only
if $r=1$ and $a(C,S)=0$. The prime divisors above $S$ with discrepancy $0$ are
exactly the exceptional divisors of $X\rightarrow S$. And similarly, the prime
divisors above $T$ with discrepancy $0$ are exactly the exceptional divisors
of $Y\rightarrow T$. The assertion follows immediately. ∎

To study the equivariant geometry of $\pi:T\rightarrow S$, we first want to
understand how external curves on $Y$ and $X$ correspond under this morphism.
It should be noted that this correspondence is not completely determined by
the type of $\pi$. See Section 3.2.12 for such an example.

It turns out that in each cases where $S$ is of degree $\geq 2$, the rational
map $\widetilde{\pi}$ sends each $(-1)$-curve on $Y$ to a $(-1)$-curve on $X$,
but this is not true in degree $1$. Section 3.2.12 provides such an example. A
more straightforward example is the quasi-étale cover of type
$S_{3}(A_{2})\rightarrow S_{1}(A_{8})$. In this case, there are $15$
$(-1)$-curves on $S_{3}(A_{2})$ but only $2$ $(-1)$-curves on $S_{1}(A_{8})$:
there are some $(-1)$-curves map to $(1)$-curves. This kind of phenomenon is
summarized in the following lemma.

######  Lemma 2.14.

Let $X$ be a surface and let $\pi:Y\rightarrow X$ be a cyclic cover of degree
$d$. Let $C$ be a $(-s)$-curve on the étale locus of $Y$ for some integer $s$.
Then $\pi^{\ast}\pi_{\ast}C=\sum_{i=1}^{d}C_{i}$ where $C_{i}$ are integral
rational curves whose intersection matrix $(C_{i}\cdot C_{j})$ is

$\left(\begin{array}[]{ccccc}-s&r&&&r\\\ r&-s&r&&\\\ &\ddots&\ddots&\ddots&\\\
&&r&-s&r\\\ r&&&r&-s\end{array}\right)$

for some integer $r\geq 0$. Moreover, $\pi(C)$ is a $(-s)$-curve if and only
if $r=0$.

###### Proof.

Let $\pi^{\circ}:Y^{\circ}\rightarrow X^{\circ}$ be the étale locus of $\pi$.
Since $\pi^{\circ}$ is Galois by definition, we have
$\mathrm{Aut}(Y^{\circ}/X^{\circ})\cong{\mathbb{Z}}/d{\mathbb{Z}}$ and let
$\sigma$ be a generator of it. Since an automorphism does not affect
intersection numbers, $\sigma^{i}(C)$ is still a $(-s)$-curve for each $i$.
Suppose $C\cdot\sigma(C)=r$, then $\sigma^{i}(C)\cdot\sigma^{i+1}(C)=r$ for
each $i$. Then the first assertion follows. The class $\pi_{\ast}C$ is reduced
since $C$ is on the étale locus of $\pi$. By the projection formulas, we have

$(\pi_{\ast}C)^{2}=-s+(d-1)r,\text{ and}\quad K_{X}\cdot C=s-2.$

The last assertion follows immediately. ∎

On the other hand, the pullback $\pi^{\ast}$ also may not send $(-1)$-curves
to $(-1)$-curves, which is common for each degree. This is summarized in the
following lemma.

######  Lemma 2.15.

Let $T\rightarrow S$ be a quasi-étale cover of Du Val del Pezzo surfaces which
is cyclic of order $d=2,3,$ or $5$. Let $B$ be the branch divisor $f$ and let
$B_{\mathrm{red}}$ be the branch locus of $f$.

  1. (1)

Let $C$ be a $(-2)$-curve on $X$ which does not contained in $B$.

     * •

If $C\cdot B=0$, then $g_{\ast}f^{\ast}C$ is a disjoint union of $d$
$(-2)$-curves.

     * •

If $C\cdot B>0$, then $C\cdot B_{\mathrm{red}}=2$ and $C$ intersects
$B_{\mathrm{red}}$ transversally at $2$ points. In this case,
$g_{\ast}f^{\ast}C$ is a $(-2)$-curve.

  2. (2)

Let $C$ be a $(-1)$-curve on $X$.

     * •

If $C\cdot B=0$, then $g_{\ast}f^{\ast}C$ is a disjoint union of $d$
$(-1)$-curves.

     * •

If $C\cdot B>0$, then $C\cdot B$ is divided by $d$. In particular, we have
$C\cdot B_{\mathrm{red}}\geq 2$. Suppose the intersections of $C$ and
$B_{\mathrm{red}}$ are transversal, then $g_{\ast}f^{\ast}C$ is an integral
curve with non-negative self-intersection.

###### Proof.

If $C\cdot B=0$, then Let $C$ be a $(-n)$-curve on $X$. The condition $C\cdot
B=0$ implies that $C$ lies in the étale locus of $f$. Then the proof of Lemma
2.14 shows that each component of $f^{\ast}C$ is a rational curve with self-
intersection $-2-(d-1)r$. When $r>0$, $f^{\ast}C$ can not be contracted to Du
Val singularities. So we must have $r=0$. This proves the first assertion of
(1) and (2).

The second assertion of (1) can be seen from the classification of quasi-étale
covers of Du Val singularities; see Example 2.16.

It remains to prove the second assertion of (2). By general theory of cyclic
covers, we have $B\equiv dL$ for some $L\in\mathrm{Pic}(X)$. Thus $C\cdot
B=dL$ is divided by $d$. Since the multiplicity of each component of $B$ is at
most $d-1$, if $C\cdot B>0$, we have $C\cdot B_{\mathrm{red}}\geq 2$.

Now suppose $C\cdot B_{\mathrm{red}}=r\geq 2$ and the intersections are
transversal. This in particular implies that they intersect at smooth points
of $B$. Then $f$ is locally ramified at the two common points of $C$ and $B$.
A topological argument shows that $f^{\ast}C$ must be irreducible. By
projection formula, we have $(f^{\ast}C)^{2}=-d$.

When $d=2$, the branch divisor $B$ is smooth and reduced. So the ramification
divisor $R=R_{\mathrm{red}}$ is smooth and reduced and each component of $R$
is a $(-1)$-curve. By projection formula, we have $f^{\ast}C\cdot
R_{\mathrm{red}}=C\cdot f_{\ast}R_{\mathrm{red}}=r$. Hence contracting
$R_{\mathrm{red}}$ will increase the self-intersection of $f^{\ast}C$ by $r$
and so $g_{\ast}f^{\ast}C$ has self-intersection $r-2\geq 0$.

When $d=3$, by Example 2.16, $T^{\prime}$ can only have
$\frac{1}{3}(1,1)$-singularities. So $g_{\ast}f^{\ast}C$ has self-intersection
$2r-3\geq 1$.

When $d=5$, the only possibility is an $A_{4}$-singularity being covered by a
smooth point. The exceptional divisor of the minimal resolution of an
$A_{4}$-singularity is a linear chain

$B_{1}$$B_{2}$$B_{3}$$B_{4}$

of $(-2)$-curves. The branch divisor is $B=B_{1}+2B_{2}+3B_{3}+4B_{4}$. So the
three singularities of the branch cover are

$\displaystyle z^{5}-xy^{2}=0,\quad x^{5}-x^{2}y^{3}=0,\quad
z^{5}-x^{3}y^{4}=0,$

respectively. They are étale-locally isomorphic to the singularities defined
by

$z^{5}-xy^{2}=0,\quad x^{5}-xy^{4}=0,\quad z^{5}-xy^{3}=0,$

respectively. These singularities can be resolved by Hirzebruch–Jung continued
fractions; see [BHPVdV15]. The dual graph of the resulting divisor $D$ is

$(-1)$$(-2)$$(-3)$$(-1)$$(-5)$$(-1)$$(-3)$$(-2)$$(-1),$

where the $(-1)$-curves are the pullback of the $(-2)$-curves. Suppose $F$ is
a curve intersects $D$ transversally at one points on any of the four
$(-1)$-curves of $D$. It is readily to check that contracting $D$ will
increase the self-intersection of $F$ by $4$. So $g_{\ast}f^{\ast}C$ is a
rational curve with self-intersection $4r-5\geq 3$. ∎

######  Example 2.16.

Let $S$ be the henselization of the local ring of a Du Val singularity of type
$A_{r},D_{s},E_{6},E_{7},$ or $E_{8}$ $(1\leq r\leq 8,4\leq s\leq 8)$. All the
cyclic quasi-étale covers of $S$ of prime order $d$ are listed below. The dual
graph of the exceptional divisors is provided, with dark vertices representing
curves in the ramification locus, and the above numbers denoting the
multiplicities of the curves in the branch divisors. The $d=2$ cases are also
listed in [Art77]. This classification can be achieved by scanning the normal
subgroups of the groups defining these quotient singularities. Each of the
local covers appears globally in some quasi-étale covers of Du Val del Pezzo
surfaces, except for the case when $d=7$.

  * •

$d=2$

$A_{1}$ $\leftarrow$ smooth

$A_{3}\leftarrow A_{1}$

$A_{5}\leftarrow A_{2}$

$A_{7}$ $\leftarrow$ $A_{3}$

$D_{4}\leftarrow A_{3}$

$D_{5}\leftarrow A_{5}$

$D_{6}$ $\leftarrow$ $A_{7}$

$D_{6}\leftarrow D_{4}$

$D_{8}\leftarrow D_{5}$

$E_{7}\leftarrow E_{6}$

  * •

$d=3$

$1$$2$$A_{2}\leftarrow\text{smooth}$

$1$$2$$1$$2$$A_{5}\leftarrow A_{1}$

$1$$2$$1$$2$$1$$2$$A_{8}\leftarrow A_{2}$

$1$$2$$1$$2$$E_{6}\leftarrow D_{4}$

  * •

$d=5$

$1$$2$$3$$4$$A_{4}\leftarrow\text{smooth}$

  * •

$d=7$

$1$$2$$3$$4$$5$$6$$A_{6}\leftarrow\text{smooth}$

To understand the equivariant information of the quasi-étale covers, we
provide a relative version of Cremona isometry defined in [Dol12, Section
8.2.8] for quasi-étale covers.

######  Definition 2.17.

Let $\pi:T\rightarrow S$ be a quasi-étale cover of Du Val del Pezzo surfaces
and denote the minimal resolutions of $T$ and $S$ by $Y$ and $X$ respectively.
Define the _group of Cremona isometries_ of $\pi$ to be

$\mathrm{Cris}(\pi)=\left\\{(\sigma,\tau)\in\mathrm{Cris}(S)\times\mathrm{Cris}(T)\middle|\begin{array}[]{l}f_{\ast}g^{-1}_{\ast}(\tau(C))=\sigma(f_{\ast}g^{-1}_{\ast}C)\text{
for any}\text{ extremal curve }\\\ C\text{ on }Y\text{ such that
}f_{\ast}g^{-1}_{\ast}C\text{ is an extremal curve}\end{array}\right\\}.$

This definition depends only on the type of the quasi-étale cover
$T\rightarrow S$ in most cases. One might have other natural definitions of
equivariant data for $\pi$, but the one defined above is one of the easiest to
compute. Since each of the maps is $\mathrm{Gal}(\overline{k}/k)$-equivariant,
we have the following corollary.

######  Corollary 2.18.

Let $k$ be the base field of the morphism $\pi:T\rightarrow S$. Then the image
of
$\rho:\mathrm{Gal}(\overline{k}/k)\rightarrow\mathrm{Cris}(S)\times\mathrm{Cris}(T)$
is contained in $\mathrm{Cris}(\pi)$.

When the realization problem has an affirmative answer, we recover the Picard
rank over $k$.

######  Proposition 2.19.

([DJT08, Proposition 6.2]) Let $H_{S}\times H_{T}$ denote the image of $\rho$
defined in Corollary 2.18 and let $\overline{k}$ be an algebraic closure of
$k$. Suppose that there exists a rational point on the smooth locus of $T$,
then $\rho(X)=\rho(X_{\overline{k}},H_{S})$ and
$\rho(Y)=\rho(Y_{\overline{k}},H_{T})$.

It is enough to compute $\rho(X_{\overline{k}},H_{S})$ up to conjugacy classes
by the following result.

######  Proposition 2.20.

Work over an algebraically closed field of characteristic $0$. Let $H$ and
$H^{\prime}$ be conjugate subgroups of $\mathrm{Cris}(S)$. Then
$\rho(X,H)=\rho(X,H^{\prime})$.

###### Proof.

By definition, the group $H$ acts on the set of negative curves on $S$. Let
$P$ be the set of orbits under this action. Then $\rho(X,H)$ equals the
dimension of the subspace generated by the orbit sums $\sum_{D\in O}D$, where
$O\in P$. Let $\sigma\in\mathrm{Cris}(S)$ such that $\sigma
H\sigma^{-1}=H^{\prime}$. The orbits under the action of $H^{\prime}$ are
$\sigma O$ for $O\in P$. Thus, $\rho(X,H^{\prime})$ equals the dimension of
the subspace generated by $\sigma(\sum_{D\in O}D)$, where $O\in P$. Since an
invertible matrix preserves the rank of a set of vectors, the assertion
follows. ∎

##  3\. Classification

###  3.1. quasi-étale covers

By Proposition 2.10, the information of quasi-étale covers of a Du Val del
Pezzo surfaces is contained in its Picard group and effective classes. We
compute these for all Du Val del Pezzo surfaces by $\mathtt{Magma}$ [BCP97]
and the results are summarized as follows.

######  Proposition 3.1.

Let $S$ be a Du Val del Pezzo surface over a field of characteristic $0$. Then
there exists a quasi-étale cover $\pi:T\rightarrow S$ of degree $\geq 2$ if
and only if there exists an arrow
$\mathrm{Type}(T)\rightarrow\mathrm{Type}(S)$ in the following diagrams.
111For each group $G$, the first diagram presents the subgroups $H$ of $G$ up
to conjugacy, and the later diagrams present the types of the corresponding
quasi-étale covers $T$ of a surface $S$ with $\pi_{1}(T_{\mathrm{sm}})\cong
H$. We omit those diagrams which can be embedded into a larger one.

  1. (1)

$C_{3}^{2}$

$C_{1}$$C_{3}$$C_{3}$$C_{3}$$C_{3}$$C_{3}^{2}$

${\mathbb{P}}^{2}$$S_{3}(3A_{2})$$S_{3}(3A_{2})$$S_{3}(3A_{2})$$S_{3}(3A_{2})$$S_{1}(4A_{2})$

  2. (2)

$C_{2}\times C_{4}$

$C_{1}$$C_{2}$$C_{2}$$C_{2}$$C_{2}^{2}$$C_{4}$$C_{4}$$C_{2}\times C_{4}$

${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$$S_{4}(4A_{1})$$S_{4}(4A_{1})$$S_{4}(4A_{1})$$S_{2}(6A_{1})$$S_{2}(2A_{3}+A_{1})$$S_{2}(2A_{3}+A_{1})$$S_{1}(2A_{3}+2A_{1})$

  3. (3)

$C_{6}$

$C_{1}$$C_{2}$$C_{3}$$C_{6}$

$S_{6}$$S_{3}(4A_{1})$$S_{2}(3A_{2})$$S_{1}(A_{5}+A_{2}+A_{1})$

  4. (4)

$C_{5}$

$C_{1}\rightarrow C_{5}$ $S_{5}\rightarrow S_{1}(2A_{4})$
---

  5. (5)

$C_{2}^{2}$

$C_{1}$$C_{2}$$C_{2}$$C_{2}$$C_{2}^{2}$

$S_{4}(A_{3}(4l))$$S_{2}(A_{7})$$S_{2}(D_{4}+2A_{1})$$S_{2}(D_{4}+2A_{1})$$S_{1}(D_{6}+2A_{1})$

$S_{4}(A_{1})$$S_{2}(A_{3}+2A_{1}(12l))$$S_{2}(A_{3}+2A_{1}(12l))$$S_{2}(A_{3}+2A_{1}(12l))$$S_{1}(D_{4}+3A_{1})$

$S_{4}$$S_{2}(4A_{1}(20l))$$S_{2}(4A_{1}(20l))$$S_{2}(4A_{1}(20l))$$S_{1}(6A_{1})$

$S_{4}(2A_{1}(8l))$$S_{2}(2A_{3})$$S_{2}(5A_{1})$$S_{2}(5A_{1})$$S_{1}(A_{3}+4A_{1})$

$S_{4}(2A_{1}(8l))$$S_{2}(2A_{3})$$S_{2}(2A_{3})$$S_{2}(2A_{3})$$S_{1}(2D_{4})$

$S_{8}(A_{1})$$S_{4}(A_{3}+2A_{1})$$S_{4}(A_{3}+2A_{1})$$S_{4}(A_{3}+2A_{1})$$S_{2}(D_{4}+3A_{1})$

  6. (6)

$C_{4}$

$C_{1}\rightarrow C_{2}\rightarrow C_{4}$ | $S_{4}\rightarrow S_{2}(4A_{1}(20l))\rightarrow S_{1}(2A_{3}+A_{1})$
---|---
$S_{4}(A_{2})\rightarrow S_{2}(A_{5}+A_{1}(6l))\rightarrow S_{1}(D_{5}+A_{3})$ | $S_{4}\rightarrow S_{2}(A_{3}+2A_{1}(12l))\rightarrow A_{1}(A_{7}+A_{1})$
$S_{4}(4A_{1})\rightarrow S_{2}(6A_{1})\rightarrow S_{1}(2A_{3}+2A_{1})$ |
  7. (7)

$C_{3}$

$C_{1}\rightarrow C_{3}$ |  $S_{3}(3A_{1})\rightarrow S_{1}(3A_{2}+A_{1})$
---|---
$S_{3}(A_{1})\rightarrow S_{1}(A_{5}+A_{2})$ |  $S_{3}(D_{4})\rightarrow S_{1}(E_{6}+A_{2})$
$S_{3}\rightarrow S_{1}(3A_{2})$ |  $S_{3}(A_{2})\rightarrow S_{1}(A_{8})$
$S_{6}\rightarrow S_{2}(3A_{2})$ |  $S_{6}(A_{1})\rightarrow S_{2}(A_{5}+A_{2})$
  8. (8)

$C_{2}$

$C_{1}\rightarrow C_{2}$ |  $S_{2}(A_{2})\rightarrow S_{1}(A_{5}+A_{1}(21l))$
---|---
$S_{2}(E_{6})\rightarrow S_{1}(E_{7}+A_{1})$ |  $S_{2}(2A_{2}+A_{1})\rightarrow S_{1}(A_{3}+A_{2}+2A_{1})$
$S_{2}(A_{1})\rightarrow S_{1}(A_{3}+2A_{1}(44l))$ |  $S_{2}(A_{2}+2A_{1})\rightarrow S_{1}(A_{5}+2A_{1})$
$S_{2}(A_{3})\rightarrow S_{1}(D_{4}+2A_{1})$ |  $S_{2}(A_{5}(8l))\rightarrow S_{1}(D_{5}+2A_{1})$
$S_{2}(3A_{1}(26l))\rightarrow S_{1}(A_{3}+3A_{1})$ |  $S_{2}(2A_{2})\rightarrow S_{1}(A_{2}+4A_{1})$
$S_{2}(2A_{1})\rightarrow S_{1}(5A_{1})$ |  $S_{2}(A_{3}+A_{1}(16l))\rightarrow S_{1}(D_{4}+A_{3})$
$S_{2}(2A_{1})\rightarrow S_{1}(2A_{3}(23l))$ |  $S_{2}(A_{3})\rightarrow S_{1}(A_{7}(8l))$
$S_{2}(D_{5})\rightarrow S_{1}(D_{8})$ |  $S_{4}(3A_{1})\rightarrow S_{2}(A_{3}+3A_{1})$
$S_{4}(D_{4})\rightarrow S_{2}(D_{6}+A_{1})$ |  $S_{6}\rightarrow S_{3}(4A_{1})$
$S_{6}(A_{1}(4l))\rightarrow S_{3}(A_{3}+2A_{1})$ |  $S_{6}(A_{2})\rightarrow S_{3}(A_{5}+A_{1})$

###### Proof.

For each degree $d$, we determine all root subsystems of $R_{d}$ by using, for
instance, [Dol12]. Let $X$ be a weak del Pezzo surface of degree $d$
corresponding to the root subsystem $R^{\prime}$ of $R_{d}$ and let $S$ be the
contraction of $(-2)$-curves on $X$. The Picard group $\mathrm{Pic}(X)$ along
with the intersection numbers depends only on $d$. We then determine all
torsion elements $B$ of $\mathrm{Pic}(X)/\mathbf{E}$ where $\mathbf{E}$
denotes the sublattice of $\mathrm{Pic}(X)$ generated by the roots of
$R^{\prime}$. The source codes and results for degree $\leq 3$ are available
on GitHub [Gao24].

Each $B$ induces a branch cover $f:T^{\prime}\rightarrow X$ and let $R$ denote
its ramification divisor. Let $T^{\prime}\rightarrow Y$ be the contraction of
$R$. By Proposition 2.13, $Y$ is the weak del Pezzo surface that induces a
quasi-’etale cover of $S$. We then find (a part of) the dual graph of negative
curves on $Y$ by the method of Proposition 2.15. This is enough to determine
the type of $Y$ in practice. We then replace $X$ by $Y$ and repeat this
process until the group $\mathrm{Pic}(X)/\mathbf{E}$ is torsion-free. ∎

######  Remark 3.2.

By [LT17, Theorem 6.2], Proposition 3.1 completes the classification of
adjoint rigid $a$-covers of geometrically uniruled surfaces.

###  3.2. quasi-étale covers with higher $b$-values

We work over an algebraically closed field of characteristic $0$. We summarize
the results of this section as follows.

######  Proposition 3.3.

Let $\pi:T\rightarrow S$ be a quasi-étale cover of Du Val del Pezzo surfaces
such that $S$ is of degree $2$, or of degree $1$ and with at most $4$ lines.
There exists a subgroup $H=H_{T}\times H_{S}\subseteq\mathrm{Cris}(\pi)$ such
that $\rho({\widetilde{T}},H_{T})\geq\rho({\widetilde{S}},H_{S})$ (resp.
$\rho({\widetilde{T}},H_{T})>\rho({\widetilde{S}},H_{S})$) if and only if the
column $(\geq)$ (resp. $(>)$) is YES in Table 1.

###### Proof.

We adopt the notation in Proposition 2.13. By Proposition 2.2, computing
$\mathrm{Cris}(X)$ is equivalent to computing the automorphism group of the
graph $\Gamma(X)$. We use $\mathtt{Magma}$ [BCP97] to calculate
$\mathrm{Cris}(X)$ and all its subgroups up to conjugacy for all involved $X$.
The source codes and results for $X$ of degree $\leq 3$ are available on
GitHub [Gao24]. Lemma 2.15 provides a correspondence of $\Gamma(Y)$ and
$\Gamma(X)$ induced by $f_{\ast}g_{\ast}^{-1}$. For each subgroup
$H_{S}\subseteq\mathrm{Cris}(X)$, the Picard rank
$\rho({\widetilde{S}},H_{S})$ is also available on [Gao24]. We then find those
subgroups $H_{T}$ of $\mathrm{Cris}(Y)$ that are equivariant under the
correspondence and satisfying
$\rho({\widetilde{T}},H_{T})\geq\rho({\widetilde{S}},H_{S})$ by hands. ∎

In Table 1, we omit details of some cases since either there is no such $H$
for trivial reasons or the dual graphs are too complicated to draw. For each
quasi-étale cover $\pi:T\rightarrow S$ up to types, the graphs $\Gamma(T)$ and
$\Gamma(S)$ are presented. There are two types of vertices: a rectangle means
a $(-1)$-curve and a circle means a $(-2)$-curve. The number of edges between
two vertecies means the intersection number of the two curves. We denote a
$(-2)$-curve where $\pi$ is branched by a thick circle. The negative curves on
$S$ and $T$ are named $E_{i}$ and $F_{j}$ where $i$ and $j$ reflect the
correspondence defined by $f_{\ast}g^{-1}_{\ast}$.

In each case, we make a table that lists all subgroups $H_{T}\times
H_{S}\subseteq\mathrm{Cris}(\pi)$ up to conjugacy such that
$\rho({\widetilde{T}},H_{T})\geq\rho({\widetilde{S}},H_{S})$. For the
convenience of the reader, we also provide a diagram of subgroups of
$\mathrm{Cris}(S)$ up to conjugacy in some cases, which is made by Tim
Dokchitser and is available online [Dok24].

Table 1. A summary of Section 3.2, where $(\geq)$ (resp. $(>)$) means there exists an $H_{T}\times H_{S}\subseteq\mathrm{Cris}(\pi)$ such that $\rho({\widetilde{T}},H_{T})\geq\rho({\widetilde{S}},H_{S})$ (resp. $\rho({\widetilde{T}},H_{T})>\rho({\widetilde{S}},H_{S})$). reference | $\deg(S)$ | Lines | $\mathrm{Type}(S)$ | $\mathrm{Cris}(S)$ | $\deg(T)$ | $\mathrm{Type}(T)$ |  $(\geq)$? |  $(>)$?
---|---|---|---|---|---|---|---|---
omit | $4$ | $2$ | $A_{3}+2A_{1}$ | $C_{2}$ | $8$ | $A_{1}$ | NO | NO
(3.2.1) | $4$ | $4$ | $4A_{1}$ | ${\mathfrak{D}}_{4}$ | $8$ | ${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$ | NO | NO
omit | $3$ | $2$ | $A_{5}+A_{1}$ | $C_{1}$ | $6$ | $A_{2}$ | NO | NO
omit | $3$ | $3$ | $3A_{2}$ | ${\mathfrak{S}}_{3}$ | $9$ | ${\mathbb{P}}^{2}$ | NO | NO
omit | $3$ | $5$ | $A_{3}+2A_{1}$ | $C_{2}$ | $6$ | $A_{1}(4l)$ | NO | NO
(3.2.2) | $3$ | $9$ | $4A_{1}$ | ${\mathfrak{S}}_{4}$ | $6$ | smooth | YES | NO
(3.2.3) | $2$ | $2$ | $A_{7}$ | $C_{2}$ | $4$ | $A_{3}(4l)$ | NO | NO
omit | $2$ | $2$ | $D_{6}+A_{1}$ | $C_{1}$ | $4$ | $D_{4}$ | NO | NO
omit | $2$ | $3$ | $A_{5}+A_{2}$ | $C_{2}$ | $6$ | $A_{1}$ | NO | NO
(3.2.4) | $2$ | $4$ | $2A_{3}+A_{1}$ | $C_{2}^{2}$ | $4$ | $4A_{1}$ | YES | NO
(3.2.5) | $2$ | $4$ | $D_{4}+3A_{1}$ | ${\mathfrak{S}}_{3}$ | $4$ | $A_{3}+2A_{1}$ | YES | NO
(3.2.6) | $2$ | $6$ | $2A_{3}$ | $C_{2}^{3}$ | $4$ | $2A_{1}(8l)$ | YES | NO
(3.2.7) | $2$ | $6$ | $A_{5}+A_{1}(6l)$ | $C_{2}$ | $4$ | $A_{2}$ | NO | NO
(3.2.8) | $2$ | $6$ | $D_{4}+2A_{1}$ | $C_{2}^{2}$ | $4$ | $A_{3}(4l)$ | YES | NO
(3.2.9) | $2$ | $8$ | $3A_{2}$ | ${\mathfrak{D}}_{6}$ | $6$ | smooth | YES | YES
(3.2.10) | $2$ | $8$ | $A_{3}+3A_{1}$ | $C_{2}^{2}$ | $4$ | $3A_{1}$ | YES | NO
(3.2.11) | $2$ | $10$ | $6A_{1}$ | $C_{2}\times{\mathfrak{S}}_{4}$ | $4$ | $4A_{1}$ | YES | YES
omit | $2$ | $12$ | $A_{3}+2A_{1}(12l)$ | $C_{2}^{3}$ | $4$ | $A_{1}$ | NO | NO
omit | $2$ | $14$ | $5A_{1}$ | $C_{2}^{2}\wr C_{2}$ | $4$ | $2A_{1}(8l)$ | YES | YES
omit | $2$ | $20$ | $4A_{1}(20l)$ | $C_{2}\times C_{2}^{2}\rtimes{\mathfrak{S}}_{4}$ | $4$ | smooth | YES | YES
omit | $1$ | $3$ | $E_{7}+A_{1}$ | $C_{1}$ | $2$ | $E_{6}$ | NO | NO
(3.2.12) | $1$ | $4$ | $E_{6}+A_{2}$ | $C_{2}$ | $3$ | $D_{4}$ | YES | YES

The results in Table 1, together with Corollary 2.18, imply the following.

######  Corollary 3.4.

(=Theorem A) Let $S$ be a Du Val del Pezzo surface over a field of
characteristic $0$.

  1. (1)

If $\deg S\geq 3$, then there does not exist a quasi-étale cover $T\rightarrow
S$ with $\rho({\widetilde{T}})>\rho({\widetilde{S}})$;

  2. (2)

If $\deg S\geq 4$, then there does not exist a quasi-étale cover $T\rightarrow
S$ with $\rho({\widetilde{T}})\geq\rho({\widetilde{S}})$.

######  Remark 3.5.

Let $S$ be a Du Val del Pezzo surface that admits a quasi-étale cover of
degree $\geq 2$. Then $S$ is toric if and only if $\mathrm{Type}(S)$ is
$A_{3}+2A_{1}$ or $4A_{1}$ in degree $4$, or $3A_{2}$ in degree $3$ (see
[Der13, Remark 6]).

See [LT19a, Definition 3.5] for the definition of face-contracting morphisms.
We also observe the following interesting phenomena from the computation.

######  Proposition 3.6.

Let $\pi:T\rightarrow S$ be a quasi-étale cover of Du Val del Pezzo surfaces.
Suppose $S$ is of degree $\geq 2$, or of degree $1$ with at most $4$ lines.
Then for any subgroup $H_{S}\subseteq\mathrm{Cris}(S)$, there exists at most
one subgroup $H_{S}\times H_{T}\subseteq\mathrm{Cris}(\pi)$ such that
$\rho({\widetilde{T}},H_{T})\geq\rho({\widetilde{S}},H_{S})$. Moreover, when
$\rho({\widetilde{T}},H_{T})=\rho({\widetilde{S}},H_{S})$, the morphism $\pi$
is face-contracting.

Proposition 3.6 suggests the following questions.

######  Question 3.7.

For any quasi-étale cover $\pi:T\rightarrow S$ of Du Val del Pezzo surfaces
over a field of characteristic $0$, does there exist at most one $\sigma\in
H^{1}(\mathrm{Gal}(\overline{k}/k),\mathbf{Aut}(T/S))$ such that
$\rho(T^{\sigma})\geq\rho(S)$?

######  Question 3.8.

Let $\pi:T\rightarrow S$ be a quasi-étale cover of Du Val del Pezzo surfaces
over a field of characteristic $0$ such that $S$ is not toric. Is $\pi$ face-
contracting whenever $\rho({\widetilde{T}})=\rho({\widetilde{S}})$?

It would be interesting to study these questions in more general cases.

######  Remark 3.9.

By Table 1, there is no toric Du Val del Pezzo surface $S$ which admits a
quasi-étale cover $T\rightarrow S$ with $b(T,-K_{T})\geq b(S,-K_{S})$. This is
not the case in higher dimensions; see [LT17, Example 8.7]. This example does
not contradict to the conjectural exceptional set of [LST22]. Indeed, let $X$
be a toric variety and let $L$ be a big and nef divisor on $X$. It has been
shown in [LT19b, Example 8.3] that there does not exist dominant thin map
$f:Y\rightarrow X$ with $a(Y,f^{\ast}L)=a(X,L)$ that is face-contracting.

######  Remark 3.10.

It should also be noted that any adjoint rigid covers $Y\rightarrow X$ of
geometrically uniruled surfaces with $a(Y,L)=a(X,L)$ and $b(Y,L)\geq b(X,L)$
can be reduced to the case of quasi-étale covers of Du Val del Pezzo surfaces
by [LT17]. Thus, our study essentially completes the classification of these
covers in dimension $2$.

####  3.2.1. Degree $4$, type $4A_{1}$

$F_{5,7}$$F_{6,8}$$T={\mathbb{P}}_{1}\times{\mathbb{P}}_{1}$$\longrightarrow$$E_{4}$$E_{1}$$E_{3}$$E_{2}$$E_{7}$$E_{6}$$E_{5}$$E_{8}$$S:$
degree $4$, type $4A_{1}$

$\mathrm{Cris}(S)\cong{\mathfrak{D}}_{4}.$

There is no $H_{\widetilde{S}}$ and $H_{\widetilde{T}}$ such that
$\rho({\widetilde{T}},H_{T})\geq\rho({\widetilde{S}},H_{S}).$

####  3.2.2. Degree $3$, type $4A_{1}$

$F_{12}$$F_{5}^{\prime}$$F_{9}^{\prime}$$F_{12}^{\prime}$$F_{5}$$F_{9}$$T:$
degree $6$,
smooth$\longrightarrow$$E_{4}$$E_{1}$$E_{13}$$E_{3}$$E_{11}$$E_{2}$$E_{8}$$E_{12}$$E_{7}$$E_{9}$$E_{10}$$E_{5}$$E_{6}$$S:$
degree $3$, type $4A_{1}$
$\mathrm{Cris}(S)=\langle\sigma_{1},\sigma_{2},\sigma_{3},\sigma_{4}\rangle\cong{\mathfrak{S}}_{4}$

where

$\displaystyle\sigma_{1}$ $\displaystyle=(1,4)(2,3)(6,13)(7,11),$
$\displaystyle\sigma_{2}$ $\displaystyle=(1,2)(3,4)(7,11)(8,10),$
$\displaystyle\sigma_{3}$ $\displaystyle=(1,3)(5,12)(6,10)(8,13),$
$\displaystyle\sigma_{4}$ $\displaystyle=(1,2)(5,9)(7,10)(8,11).$

The diagram of subgroups up to conjugacy of $\mathrm{Cris}(S)$ is

$C_{1}$$C_{2}(2)$$C_{2}(1)$$C_{3}$$C_{2}^{2}(2)$$C_{2}^{2}(1)$$C_{4}$${\mathfrak{S}}_{3}$${\mathfrak{D}}_{4}$${\mathfrak{A}}_{4}$${\mathfrak{S}}_{4}$3643343

The subgroups $H_{T}\times H_{S}\subseteq\mathrm{Cris}(\pi)$ up to conjugacy
satisfying $\rho({\widetilde{T}},H_{T})\geq\rho({\widetilde{S}},H_{S})$ are
listed in Table 2.

Table 2. Degree $3$, type $4A_{1}$ $H_{\widetilde{S}}$ | $H_{\widetilde{T}}$ | $\rho({\widetilde{S}},H_{S})$ | $\rho({\widetilde{T}},H_{T})$
---|---|---|---
${\mathfrak{S}}_{4}$ | ${\mathfrak{S}}_{3}$ | $2$ | $2$
${\mathfrak{A}}_{4}$ | $C_{3}$ | $2$ | $2$
${\mathfrak{D}}_{4}$ | $C_{2}$ | $3$ | $3$
$C_{2}^{2}$(2) | $C_{1}$ | $4$ | $4$
$C_{4}$ | $C_{2}$ | $3$ | $3$

####  3.2.3. Degree $2$, type $A_{7}$

$F_{2}$$F_{4}$$F_{6}$$F_{8}$$F_{8}^{\prime}$$F_{9}$$F_{9}^{\prime}$$\longrightarrow$$E_{1}$$E_{2}$$E_{3}$$E_{4}$$E_{5}$$E_{6}$$E_{7}$$E_{8}$$E_{9}$$T:$
degree $4$, type $A_{3}(4l)$$S:$ degree $2$, type $A_{7}$
$\mathrm{Cris}(S)\cong C_{2}.$

There is no $H_{\widetilde{S}}$ and $H_{\widetilde{T}}$ such that
$\rho({\widetilde{T}},H_{T})\geq\rho({\widetilde{S}},H_{S}).$

####  3.2.4. Degree $2$, type $2A_{3}+A_{1}$

$F_{2}$$F_{9}$$F_{7}$$F_{10}$$F_{5}$$F_{9}^{\prime}$$F_{7}^{\prime}$$F_{10}^{\prime}$$\longrightarrow$$E_{2}$$E_{9}$$E_{7}$$E_{8}$$E_{1}$$E_{4}$$E_{11}$$E_{10}$$E_{5}$$E_{3}$$E_{6}$$\longrightarrow$${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$$T:$
degree $4$, type $4A_{1}$$S:$ degree $2$, type $2A_{3}+A_{1}$
$\mathrm{Cris}(S)\cong C_{2}^{2}.$

The subgroups $H_{T}\times H_{S}\subseteq\mathrm{Cris}(\pi)$ up to conjugacy
satisfying $\rho({\widetilde{T}},H_{T})\geq\rho({\widetilde{S}},H_{S})$ are
listed in Table 3.

Table 3. Degree $2$, type $2A_{3}+A_{1}$. $H_{\widetilde{S}}$ | $H_{\widetilde{T}}$ | $\rho({\widetilde{S}},H_{S})$ | $\rho({\widetilde{T}},H_{T})$
---|---|---|---
$C_{2}^{2}$ | $C_{2}$ | $4$ | $4$
$C_{2}=\langle(1,3)(4,6)(8,11)\rangle$ | $C_{1}$ | $6$ | $6$

####  3.2.5. Degree $2$, type $D_{4}+3A_{1}$

$F_{5}$$F_{9}$$F_{2}$$F_{3}$$F_{2}^{\prime}$$F_{9}^{\prime}$$F_{5}^{\prime}$$\longrightarrow$$\longrightarrow$$G_{9}$$G_{3}$$E_{11}$$E_{5}$$E_{9}$$E_{2}$$E_{3}$$E_{8}$$E_{7}$$E_{1}$$E_{10}$$E_{6}$$E_{4}$$T:$
degree $4$, type $A_{3}+2A_{1}$$S:$ degree $2$, type $D_{4}+3A_{1}$degree 8,
type $A_{1}$ $\mathrm{Cris}(S)\cong{\mathfrak{S}}_{3}.$

The subgroups $H_{T}\times H_{S}\subseteq\mathrm{Cris}(\pi)$ up to conjugacy
satisfying $\rho({\widetilde{T}},H_{T})\geq\rho({\widetilde{S}},H_{S})$ are
listed in Table 4.

Table 4. Degree $2$, type $D_{4}+3A_{1}$. $H_{\widetilde{S}}$ | $H_{\widetilde{T}}$ | $\rho({\widetilde{S}},H_{S})$ | $\rho({\widetilde{T}},H_{T})$
---|---|---|---
$C_{2}$ | $C_{1}$ | $6$ | $6$

####  3.2.6. Degree $2$, type $2A_{3}$

$F_{2}$$F_{7}^{\prime}$$F_{9}$$F_{7}$$F_{9}^{\prime}$$F_{10}^{\prime}$$F_{11}$$F_{10}$$F_{11}^{\prime}$$F_{5}$$\longrightarrow$$E_{2}$$E_{7}$$E_{10}$$E_{5}$$E_{9}$$E_{11}$$E_{1}$$E_{3}$$E_{4}$$E_{6}$$E_{8}$$E_{12}$$S:$
degree $2$, type $2A_{3}$$T:$ degree $4$, type $2A_{1}(8l)$
$\mathrm{Cris}(S)=\langle a,b,c\rangle\cong C_{2}^{3},$

where

$\displaystyle a$ $\displaystyle=(7,9)(10,11),$ $\displaystyle b$
$\displaystyle=(1,4)(2,5)(3,6)(7,10)(9,11),$ $\displaystyle c$
$\displaystyle=(1,3)(4,6)(8,12).$

The subgroups $H_{T}\times H_{S}\subseteq\mathrm{Cris}(\pi)$ up to conjugacy
satisfying $\rho({\widetilde{T}},H_{T})\geq\rho({\widetilde{S}},H_{S})$ are
listed in Table 5.

Table 5. Degree $2$, type $2A_{3}$. $H_{\widetilde{S}}$ | $H_{\widetilde{T}}$ | $\rho({\widetilde{S}},H_{S})$ | $\rho({\widetilde{T}},H_{T})$
---|---|---|---
$C_{2}=\langle c\rangle$ | $C_{1}$ | $6$ | $6$

####  3.2.7. Degree $2$, type $A_{5}+A_{1}(6l)$

$F_{2}$$F_{4}$$F_{10}^{\prime}$$F_{11}$$F_{8}^{\prime}$$F_{7}$$F_{10}$$F_{11}^{\prime}$$F_{7}^{\prime}$$F_{8}$$\longrightarrow$$E_{2}$$E_{3}$$E_{4}$$E_{1}$$E_{5}$$E_{9}$$E_{6}$$E_{12}$$E_{10}$$E_{11}$$E_{7}$$E_{8}$$S:$
degree $2$, type $A_{5}+A_{1}(6l)$$T:$ degree $4$, type $A_{2}$
$\mathrm{Cris}(S)\cong C_{2}.$

There is no $H_{\widetilde{S}}$ and $H_{\widetilde{T}}$ such that
$\rho({\widetilde{T}},H_{T})\geq\rho({\widetilde{S}},H_{S}).$

####  3.2.8. Degree $2$, type $D_{4}+2A_{1}$

$F_{1}$$F_{3}$$F_{1}^{\prime}$$F_{7}$$F_{11}$$F_{7}^{\prime}$$F_{11}^{\prime}$$\longrightarrow$$E_{1}$$E_{2}$$E_{3}$$E_{4}$$E_{5}$$E_{6}$$E_{7}$$E_{8}$$E_{9}$$E_{10}$$E_{11}$$E_{12}$$S:$
degree $2$, type $D_{4}+2A_{1}$$T:$ degree $4$, type $A_{3}(4l)$
$\mathrm{Cris}(S)=\langle a,b\rangle\cong C_{2}^{2},$

where

$\displaystyle a$ $\displaystyle=(7,11)(8,12),$ $\displaystyle b$
$\displaystyle=(2,4)(5,6)(9,10).$

The subgroups $H_{T}\times H_{S}\subseteq\mathrm{Cris}(\pi)$ up to conjugacy
satisfying $\rho({\widetilde{T}},H_{T})\geq\rho({\widetilde{S}},H_{S})$ are
listed in Table 6.

Table 6. Degree $2$, type $D_{4}+2A_{1}$

.  $H_{\widetilde{S}}$ $H_{\widetilde{T}}$ $\rho({\widetilde{S}},H_{S})$
$\rho({\widetilde{T}},H_{T})$ $C_{2}=\langle b\rangle$ $C_{1}$ $6$ $6$

####  3.2.9. Degree $2$, type $3A_{2}$

$F_{9}$$F_{10}$$F_{9}^{\prime}$$F_{10}^{\prime}$$F_{9}^{\prime\prime}$$F_{10}^{\prime\prime}$$\longrightarrow$$E_{9}$$E_{13}$$E_{12}$$E_{7}$$E_{14}$$E_{11}$$E_{8}$$E_{1}$$E_{2}$$E_{3}$$E_{4}$$E_{5}$$E_{6}$$E_{10}$$S:$
degree $2$, type $3A_{2}$$T:$ degree $6$, smooth

$\mathrm{Cris}(S)\cong{\mathfrak{D}}_{6}.$

The subgroups of $\mathrm{Cris}(S)$ up to conjugacy are

$C_{1}$$C_{2}(3)$$C_{2}(1)$$C_{2}(2)$$C_{3}$$C_{2}^{2}$${\mathfrak{S}}_{3}$(1)${\mathfrak{S}}_{3}(2)$$C_{6}$${\mathfrak{D}}_{6}$333

The subgroups $H_{T}\times H_{S}\subseteq\mathrm{Cris}(\pi)$ up to conjugacy
satisfying $\rho({\widetilde{T}},H_{T})\geq\rho({\widetilde{S}},H_{S})$ are
listed in Table 7.

Table 7. Degree $2$, type $3A_{2}$. $H_{\widetilde{S}}$ | $H_{\widetilde{T}}$ | $\rho({\widetilde{S}},H_{S})$ | $\rho({\widetilde{T}},H_{T})$
---|---|---|---
${\mathfrak{D}}_{6}$ | $C_{2}$ | $2$ | $2$
${\mathfrak{S}}_{3}(1)$ | $C_{1}$ | $\bf 3$ | $\bf 6$
$C_{6}$ | $C_{2}$ | $2$ | $2$
$C_{3}$ | $C_{1}$ | $\bf 3$ | $\bf 6$
$C_{2}(1)$ | $C_{1}$ | $\bf 5$ | $\bf 6$

In Table 7, we have

$\displaystyle{\mathfrak{S}}_{3}(1)=\langle(1,3)(2,4)(5,6)(7,13)(11,14),(1,6,4)(2,5,3)(7,13,12)(8,14,11)\rangle,$
$\displaystyle C_{2}(1)=\langle(1,3)(2,4)(5,6)(7,13)(11,14)\rangle.$

####  3.2.10. Degree $2$, type $A_{3}+3A_{1}$

$F_{4}$$F_{13}$$F_{2}$$F_{13}^{\prime}$$F_{4}^{\prime}$$F_{7}$$F_{9}^{\prime}$$F_{9}$$F_{7}^{\prime}$$\longrightarrow$$E_{1}$$E_{2}$$E_{3}$$E_{13}$$E_{10}$$E_{12}$$E_{8}$$E_{7}$$E_{9}$$E_{4}$$E_{14}$$E_{6}$$E_{11}$$E_{5}$$S:$
degree $2$, type $A_{3}+3A_{1}$$T:$ degree $4$, type $3A_{1}$
$\mathrm{Cris}(S)=\langle a,b\rangle\cong C_{2}^{2},$

where

$\displaystyle a$ $\displaystyle=(1,3)(7,9)(8,14)(10,12),$ $\displaystyle b$
$\displaystyle=(5,6)(7,9)(8,10)(12,14).$

The subgroups $H_{T}\times H_{S}\subseteq\mathrm{Cris}(\pi)$ up to conjugacy
satisfying $\rho({\widetilde{T}},H_{T})\geq\rho({\widetilde{S}},H_{S})$ are
listed in Table 8.

Table 8. Degree $2$, type $A_{3}+3A_{1}$. $H_{\widetilde{S}}$ | $H_{\widetilde{T}}$ | $\rho({\widetilde{S}},H_{S})$ | $\rho({\widetilde{T}},H_{T})$
---|---|---|---
$C_{2}^{2}$ | $C_{2}=\langle(7,9^{\prime})(7^{\prime},9)\rangle$ | $5$ | $5$
$C_{2}=\langle ab\rangle$ | $C_{1}$ | $6$ | $6$

####  3.2.11. Degree $2$, type $6A_{1}$

${\mathbb{P}}^{1}\times{\mathbb{P}}^{1}$$\longrightarrow$$F_{5}^{\prime}$$F_{3}$$F_{3}^{\prime}$$F_{5}$$F_{16}^{\prime}$$F_{8}^{\prime}$$F_{16}$$F_{8}$$\longrightarrow$$E_{10}$$E_{9}$$E_{16}$$E_{7}$$E_{1}$$E_{2}$$E_{3}$$E_{5}$$E_{6}$$E_{4}$$E_{12}$$E_{13}$$E_{8}$$E_{14}$$E_{15}$$E_{11}$$S:$
degree $2$, type $6A_{1}$$T:$ degree $4$, type $4A_{1}$

$\mathrm{Cris}(S)\cong C_{2}\times{\mathfrak{S}}_{4}.$

The subgroups $H_{T}\times H_{S}\subseteq\mathrm{Cris}(\pi)$ up to conjugacy
satisfying $\rho({\widetilde{T}},H_{T})\geq\rho({\widetilde{S}},H_{S})$ are
listed in Table 9. We omit the diagram of subgroups of $\mathrm{Cris}(S)$
since it is too complected.

Table 9. Degree $2$, type $6A_{1}$ $H_{\widetilde{S}}$ | $H_{\widetilde{T}}$ | $\rho({\widetilde{S}},H_{S})$ | $\rho({\widetilde{T}},H_{T})$
---|---|---|---
${\mathfrak{D}}_{4}$ | $C_{2}$ | $4$ | $4$
$C_{2}^{3}$ | $C_{2}$ | $4$ | $4$
$C_{4}$ | $C_{2}$ | $4$ | $4$
$C_{2}^{2}(1)$ | $C_{1}$ | $\bf 5$ | $\bf 6$
$C_{2}^{2}(2)$ | $C_{2}$ | $4$ | $4$
$C_{2}^{2}(3)$ | $C_{2}$ | $4$ | $4$
$C_{2}$ | $C_{1}$ | $6$ | $6$

In Table 9, we have

$\displaystyle
C_{2}^{2}(1)=\langle(1,2)(4,6)(10,13)(11,14),(1,6)(2,4)(7,9)(11,14)(12,15)\rangle.$
$\displaystyle
C_{2}^{2}(2)=\langle(1,6)(2,4)(7,12)(8,16)(9,15)(11,14),(1,2)(4,6)(10,13)(11,14)\rangle,$
$\displaystyle
C_{2}^{2}(3)=\langle(1,2)(4,6)(7,15)(8,16)(9,12)(10,13)(11,14),(1,6)(2,4)(7,9)(11,14)(12,15)\rangle.$

####  3.2.12. Degree $1$, type $E_{6}+A_{2}$

$F_{12}$$F_{12}^{\prime}$$F_{12}^{\prime\prime}$$F_{1}$$F_{1}^{\prime}$$F_{1}^{\prime\prime}$$F_{4}$$F_{a}^{\prime}$$F_{a}$$F_{a}^{\prime\prime}$$\longrightarrow$$E_{11}$$E_{10}$$E_{9}$$E_{7}$$E_{8}$$E_{6}$$E_{12}$$E_{2}$$E_{5}$$E_{1}$$E_{3}$$E_{4}$$T:$
degree $3$, type $D_{4}$$S:$ degree $1$, type $E_{6}+A_{2}$

There are $2$ isomorphic classes of a Du Val del Pezzo surfaces of type
$S_{3}(D_{4})$ and $S_{1}(E_{6}+A_{2})$ respectively.

  1. (1)

If $E_{7}\cap E_{8}\cap E_{11}\neq\emptyset$, then $g_{\ast}f^{\ast}E_{11}$
equals to $F_{a}+F_{a}^{\prime}+F_{a}^{\prime\prime}$ and $F_{a}\cap
F_{a}^{\prime}\cap F_{a}^{\prime\prime}\neq\emptyset$.

  2. (2)

If $E_{7}\cap E_{8}\cap E_{11}=\emptyset$, then $g_{\ast}f^{\ast}E_{11}$
equals to a $(1)$-curve and, in particular, is not contained in $\Gamma(T)$.
In this case, the three $(-1)$-curves $F_{a},F_{a}^{\prime}$ and
$F_{a}^{\prime\prime}$ map to a $(1)$-curve on $S$.

We have

$\mathrm{Cris}(S)\cong C_{2}.$

In both cases, the subgroups $H_{T}\times H_{S}\subseteq\mathrm{Cris}(\pi)$ up
to conjugacy satisfying
$\rho({\widetilde{T}},H_{T})\geq\rho({\widetilde{S}},H_{S})$ are listed in
Table 10.

Table 10. Degree $1$, type $E_{6}+A_{2}$. $H_{\widetilde{S}}$ | $H_{\widetilde{T}}$ | $\rho({\widetilde{S}},H_{S})$ | $\rho({\widetilde{T}},H_{T})$
---|---|---|---
$C_{2}$ | $C_{1}$ | $\bf 6$ | $\bf 7$

##  4\. Examples

In this section, we present some examples which realize group actions in
Section 3. These examples are interesting in different aspects. The surface
$S$ in Section 4.1 is the unique singular cubic surface with a Zariski dense
exceptional set up to a twist over ${\mathbb{C}}$. The Cox rings of the
surfaces $S$ in Section 4.2 and 3.2.12 are defined by only one relation (see
[Der13]). Section 4.3 and 3.2.12 provide examples where the $b$-invariants are
violated by the quasi-étale covers.

###  4.1. degree $3$, type $4A_{1}$

######  Notation 4.1.

Let $S$ be the singular cubic surface in ${\mathbb{P}}^{3}$ defined by the
equation

$X^{3}+2XYW+XZ^{2}-Y^{2}Z+ZW^{2}=0$

over ${\mathbb{Q}}$.

######  Proposition 4.2.

Under Notation 4.1, the following assertions hold.

  1. (1)

$S$ is a cubic surface with four $A_{1}$ singularities (known as a Cayley
cubic surface).

  2. (2)

We have $\rho({\widetilde{S}})=4$.

  3. (3)

There exists a quasi-étale $T\rightarrow S$ defined over ${\mathbb{Q}}$ such
that $\rho({\widetilde{T}})=4$.

###### Proof.

The four singular points on $S$ are

$\displaystyle\mathrm{Sing}(S)=\\{(1:\zeta_{8}:\zeta_{8}^{2}:\zeta_{8}^{3}),(1:\zeta_{8}^{3}:\zeta_{8}^{6}:\zeta_{8}),(1:\zeta_{8}^{5}:\zeta_{8}^{2}:\zeta_{8}^{7}),(1:\zeta_{8}^{7}:\zeta_{8}^{6}:\zeta_{8}^{5})\\}.$

The Galois group $\mathrm{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ acts on
the set $\mathrm{Sing}(S)$ by $C_{2}^{2}(2)\subset{\mathfrak{S}}_{4}$ in the
notation of Section 3.2.2 (i.e. the unique normal subgroup of
${\mathfrak{S}}_{4}$ which is isomorphic to $C_{2}^{2}$). Thus $\rho(S)=4$ by
Table 2. (Note that if the action was $C_{2}^{2}(1)$, then the induced action
on $\Gamma({\widetilde{S}})$ would not fix the three lines $E_{5},E_{9}$, and
$E_{12}$.) Since there exists a smooth rational point on $S$, the homotopy
exact sequence of étale fundamental groups implies that the induced quasi-
étale cover $\overline{T}\rightarrow\overline{S}$ descend to a cover
$T\rightarrow S$ over ${\mathbb{Q}}$. After possibly a twisting, we may assume
that the Galois action on $\Gamma(T)$ is trivial since any automorphisms on
$\Gamma(T)$ can be realized as $\mathrm{Aut}(T)$ [Cor05]. Thus we have
$\rho(T)=4$ by Table 2. ∎

######  Example 4.3 (Construction of the example).

It is well-known that the Cayley cubic surface can be written as a
hypersurface $S_{0}$ in ${\mathbb{P}}^{3}_{\mathbb{Q}}$ defined by the
equation

$\frac{1}{x}+\frac{1}{y}+\frac{1}{y}+\frac{1}{z}=1.$

The surface $S_{0}$ is split over ${\mathbb{Q}}$, and the singular points are

$\mathrm{Sing}(S_{0})=\\{(1:0:0:0),(0:1:0:0),(0:0:1:0),(0:0:0:1)\\}.$

Pick a number field $K$ such that $[K:{\mathbb{Q}}]=4$ and
$\mathrm{Gal}(K/{\mathbb{Q}})$ acts on the four roots
$\\{a_{0},a_{1},a_{2},a_{3}\\}$ of the minimal polynomial by
$C_{2}^{2}(2)\subset{\mathfrak{S}}_{4}$. Let

$M:=\left(\begin{array}[]{cccc}1&1&1&1\\\ a_{0}&a_{1}&a_{2}&a_{3}\\\
a_{0}^{2}&a_{1}^{2}&a_{2}^{2}&a_{3}^{2}\\\
a_{0}^{3}&a_{1}^{3}&a_{2}^{3}&a_{3}^{3}\end{array}\right)$

be the Vandermonde matrix associated to $(a_{0},a_{1},a_{2},a_{3})$. Then the
change of variables by $M$ defines a surface $S_{1}$ with the desired Galois
action on $\mathrm{Sing}(S_{1})$. Since $S_{0}$ is defined by a symmetric
polynomial, the coefficients of the equation of $S_{1}$ are symmetric
polynomials in $a_{i}$, and thus are in ${\mathbb{Q}}$. The surface $S$ in
Notation 4.1 is obtained in this way by inputting the number field
${\mathbb{Q}}(\zeta_{8})$.

###  4.2. degree $2$, type $D_{4}+3A_{1}$

######  Notation 4.4.

Over field ${\mathbb{Q}}$, define a morphism of varieties as follows.

$\displaystyle
T=\mathrm{Proj}\left(\frac{k_{(1,1,2,2)}[x,y,z,w]}{y^{4}-16zw}\right)$
$\displaystyle\ \overset{\pi}{\rightarrow}\
S=\mathrm{Proj}\left(\frac{k_{(1,1,1,2)}[X,Y,Z,W]}{W^{2}-XY(Z^{2}+Y^{2})}\right),$
$\displaystyle(x:y:z:w)$ $\displaystyle\ \mapsto\
(x^{2}:y^{2}:2(z-w):2xy(z+w)).$

It is readily to confirm that $\pi$ is well-defined.

######  Proposition 4.5.

Under Notation 4.4, the following statements hold.

  1. (1)

$S$ is a Du Val del Pezzo surface of type $S_{2}(D_{4}+3A_{1})$, and $T$ is a
Du Val del Pezzo surface of type $S_{4}(A_{3}+2A_{1})$.

  2. (2)

We have $\rho({\widetilde{T}})=\rho({\widetilde{S}})=6$.

###### Proof.

For (1), note that we have an isomorphism

$\displaystyle
T=\mathrm{Proj}\left(\frac{{\mathbb{Q}}_{(1,1,2,2)}[x,y,z,w]}{y^{4}-zw}\right)$
$\displaystyle\ {\rightarrow}\
T^{\prime}=\mathrm{Proj}\left(\frac{{\mathbb{Q}}_{(1,1,1,1,1)}[X,Y,Z,S,T]}{(Y^{2}-XZ,Z^{2}-ST)}\right),$
$\displaystyle(x:y:z:w)$ $\displaystyle\ \mapsto\ (x^{2}:xy:y^{2}:z:w).$

For (2), write ${\widetilde{T}}$ and ${\widetilde{S}}$ for the minimal
resolution of $T$ and $S$. Note that the singular points on $T$ and $S$ are

$\mathrm{Sing}~{}T=\\{A_{3}:(1:0:0:0),A_{1}:(0:0:1:0),A_{1}:(0:0:0:1)\\},$
$\mathrm{Sing}~{}S=\\{D_{4}:(1:0:0:0),A_{1}:(0:0:1:0),A_{1}:(0:-i:2:0),A_{1}:(0:i:2:0)\\}.$

So $\mathrm{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$ acts on
$\Gamma({\widetilde{S}})$ so that $\rho({\widetilde{S}})=6$ as presented in
Section 3.2.5. The Galois action on $\Gamma({\widetilde{T}})$ is trivial since
the two lines on $T$ are not interchanged by the action. ∎

We present in the following example how $T\rightarrow S$ is constructed.

######  Example 4.6 (Construction of the example).

Define a group action on ${\mathbb{P}}^{2}_{\mathbb{C}}$ by the quaternion
group $Q_{8}=\\{\pm 1,\pm I,\pm J,\pm K\\}$ by

$I:(X:Y:Z)\mapsto(iX,-iY:Z),\quad
J:(X:Y:Z)\mapsto(\zeta_{8}Y:\zeta_{8}^{3}X:Z),$

where $\zeta_{8}=\sqrt{2}(1+i)/2$. Then we have a tower of quasi-étale covers

$U:={\mathbb{P}}^{2}_{\mathbb{C}}/\langle
I^{2}\rangle\overset{\pi_{2}}{\rightarrow}T:={\mathbb{P}}^{2}_{\mathbb{C}}/\langle
I\rangle\overset{\pi_{1}}{\rightarrow}S:={\mathbb{P}}^{2}_{\mathbb{C}}/Q_{8}.$

The rings of invariants are

$\displaystyle{\mathbb{C}}[X,Y,Z,W]^{\langle I\rangle}$
$\displaystyle={\mathbb{C}}[Z,XY,X^{4},Y^{4}],$
$\displaystyle{\mathbb{C}}[X,Y,Z,W]^{Q_{8}}$
$\displaystyle={\mathbb{C}}[Z,X^{2}Y^{2},X^{4}-Y^{4},XY(X^{4}+Y^{4})]$
$\displaystyle={\mathbb{C}}[f_{1},f_{4},g_{4},f_{6}]/(f_{6}^{2}-f_{4}g_{4}^{2}-4f_{4}^{3}).$

Thus

$\displaystyle S$
$\displaystyle=\mathrm{Proj}\left(\frac{k_{(1,4,4,6)}[x,y,z,w]}{w^{2}-yz^{2}-4y^{3}}\right)\cong\mathrm{Proj}\left(\frac{k_{(1,1,1,2)}[X,Y,Z,W]}{W^{2}-XY(Z^{2}+Y^{2})}\right).$

The equations of $T$ and $\pi$ can be obtained in the same way.

###  4.3. degree $2$, type $3A_{2}$

######  Notation 4.7.

Let $S$ be the hypersurface in ${\mathbb{P}}_{\mathbb{Q}}(1,1,1,2)$ defined by
the equation

$W^{2}+(3X^{2}-YZ)W+9X^{4}-6X^{2}YZ+X(Y^{3}+Z^{3})=0.$

Let $T$ be the subvariety of ${\mathbb{P}}^{6}_{b,a_{1},\cdots,a_{6}}$ cutting
out by

$\\{ba_{i}=a_{i-1}a_{i+1},b^{2}=a_{i}a_{i+3}\mid i=1,\cdots,6\\}.$

Define the morphism $\pi:T\rightarrow S$ by

$\pi:(b:a_{1}:\cdots:a_{6})\mapsto(b:a_{1}+a_{3}+a_{5}:a_{2}+a_{4}+a_{6}:a_{1}a_{2}+a_{3}a_{4}+a_{5}a_{6})=(X:Y:Z:W).$

We will show that $\pi$ is well-defined.

######  Proposition 4.8.

The following statements hold.

  1. (1)

$S$ is a Du Val del Pezzo surface of type $S_{2}(3A_{2})$, and $T$ is a del
Pezzo surface of degree $6$.

  2. (2)

The morphism $\pi$ is a well-defined quasi-étale cover.

  3. (3)

We have $\rho({\widetilde{T}})=6>5=\rho({\widetilde{S}})$.

###### Proof.

We will show these assertions by constructing $S$ as a quotient variety of
$T$. It is well-known that $T$ is the anticanonical model of a del Pezzo
surface of degree $6$ which is split over ${\mathbb{Q}}$. Let $G=S_{3}$ acts
on $T$ by

$(b,a_{1},a_{2},a_{3},a_{4},a_{5},a_{6})\mapsto(b,a_{3},a_{4},a_{5},a_{6},a_{1},a_{2}).$

The action interchanges diagonal lines. Hence the quotient variety $T/G$ is of
type $S_{2}(3A_{2})$ by the diagram in Section 3.2.9. We obtain the equations
of $S$ and $\pi$ as in Notation 4.7 by computing the ring of invariants by
$\mathtt{Magma}$.

The singular locus of $S$ is

$\mathrm{Sing}(S)=\\{(1:3:3:3),(1:3\zeta_{3},3\zeta_{3}^{2}:3),(1:3\zeta_{3}^{2},3\zeta_{3}:3)\\}.$

We see that the Galois group acts on $\mathrm{Sing}(S)$ as $C_{2}$ and
trivially on $\Gamma(T)$. There are three conjugate classes of subgroups of
$\mathrm{Cris}(S)$ that act on $\mathrm{Sing}(S)$ as $C_{2}$. The only one
that fixes $\Gamma(T)$ is $C_{2}(1)$ (See Section 3.2.9). Thus, the action on
$\Gamma(S)$ is $C_{2}(1)$, and we have $\rho(T)=6>5=\rho(S)$. ∎

###  4.4. degree $1$, type $E_{6}+A_{2}$

######  Notation 4.9.

Let $S$ be the hypersurface in ${\mathbb{P}}_{\mathbb{Q}}(1,1,2,3)$ defined by
the equation

$W^{2}+Z^{3}+X^{4}Y^{2}=0.$

######  Proposition 4.10.

The following assertions hold.

  1. (1)

$S$ is a Du Val del Pezzo surfaces of type $S_{1}(E_{6}+A_{2})$.

  2. (2)

There exits a quasi-étale cover $\pi:T\rightarrow S$ where $T$ is a Du Val del
Pezzo surface of type $S_{3}(D_{4})$.

  3. (3)

We have $\rho({\widetilde{T}})=7>6=\rho({\widetilde{S}})$.

###### Proof.

$S$ is of type $S_{1}(E_{6}+A_{2})$ since the same equation is listed in
[CP21, Section 8]. The four lines on $S$ are defined by

$\displaystyle E_{9}$ $\displaystyle=\\{Z=W-iX^{2}Y=0\\},\quad$ $\displaystyle
E_{10}$ $\displaystyle=\\{Z=W+iX^{2}Y=0\\},$ $\displaystyle E_{11}$
$\displaystyle=\\{Y=W^{2}+Z^{3}=0\\},\quad$ $\displaystyle E_{12}$
$\displaystyle=\\{X=W^{2}+Z^{3}=0\\},$

which can be derived from those on a twist form of $S$ in [Der13]. So the
Galois group acts on $\Gamma(S)$ as $C_{2}$ and $\rho(S)=6$ by the analysis of
Section 3.2.12. Let $\pi:T\rightarrow S$ be the induced quasi-étale cover
where $T$ is of type $S_{3}(D_{4})$. All the actions on the dual graph of $T$
can be realized as automorphisms of $T$ by [Vir23]. The existence of a twist
$T^{\sigma}$ of $T$ above $S$ such that $\rho(T^{\sigma})=7$ follows from a
similar argument to the proof of Proposition 4.2. ∎

##  5\. Proof of Theorem C

Recall the following definition in [BL18].

######  Definition 5.1.

A smooth projective geometrically integral variety $X$ over a field k is
called _almost Fano_ if

  * •

$H^{i}(X,{\mathcal{O}}_{X})=0$ for $i=1,2$.

  * •

The geometric Picard group $\mathrm{Pic}(\overline{X})$ is torsion-free.

  * •

The anticanonical divisor $-K_{X}$ is big.

In particular, a weak del Pezzo surfaces is an almost Fano variety.

######  Lemma 5.2.

Let $\pi:T\rightarrow S$ be a quasi-étale cover of Du Val del Pezzo surfaces.

  1. (1)

Suppose that $\rho({\widetilde{T}})\geq\rho({\widetilde{S}})$ and that there
exists a dense open subset $V$ of $T$ such that the equation

$N(V,-{\mathcal{K}}_{T},B)\sim c(T,-{\mathcal{K}}_{T})B(\log
B)^{\rho({\widetilde{T}})-1}$

holds true for any adelically metrized anticanonical line bundle
$-{\mathcal{K}}_{T}$ on $T$. Then for any open subset $U$ of $S$, there exists
an adelically metrized anticanonical line bundle $-{\mathcal{K}}_{S}$ on $S$
such that the equation

$N(U,-{\mathcal{K}}_{S},B)\sim c(S,-{\mathcal{K}}_{S})B(\log
B)^{\rho({\widetilde{S}})-1}$ (5.1)

does not hold true.

  2. (2)

Suppose that $\rho({\widetilde{T}})>\rho({\widetilde{S}})$ and that for any
dense open subset $V$ of $T$ and any adelically metrized anticanonical line
bundle $-{\mathcal{K}}_{T}$ on $T$, we have

$N(V,-{\mathcal{K}}_{T},B)\gg B(\log B)^{\rho({\widetilde{T}})-1}.$ (5.2)

Then for any dense open subset $U$ of $S$ and any adelically metrized
anticanonical line bundle $-{\mathcal{K}}_{S}$ on $S$, the following assertion
holds true: For any constant $C>0$, there exists $B_{0}$ such that for any
$B\geq B_{0}$, we have

$N(U,-{\mathcal{K}}_{S},B)\geq CB(\log B)^{\rho({\widetilde{S}})-1}.$

###### Proof.

Since $\pi$ is quasi-étale, we have $\pi^{\ast}K_{S}\sim K_{T}$. We may also
pull back the adelic metric naturally. In the case of (1), fix a dense open
subset $U$ of $S$ and suppose equation (5.1) is true for any adelically
metrized anticanonical line bundle $-{\mathcal{K}}_{S}$. By [Pey95,
Proposition 5.0.1], the rational points are equidistributed on $U$. But then

$\displaystyle\lim_{B\rightarrow\infty}\frac{N(\pi(T({\mathbb{Q}}))\cap
U,-{\mathcal{K}}_{S},B)}{N(U,-{\mathcal{K}}_{S},B)}=\frac{c(T,-{\mathcal{K}}_{T})}{c(S,-{\mathcal{K}}_{S})}>0,$

which contradicts [BL18, Theorem 1.2].

In the case of (2), let $U_{0}$ be an dense open subset of $S$ over which
$\pi$ is étale. Let $U$ be any dense open subset of $X$. We have

$\displaystyle N(U,-{\mathcal{K}}_{S},B)\geq\frac{1}{\deg\pi}N(\pi^{-1}(U\cap
U_{0}),-{\mathcal{K}}_{T},B)$

Moreover, the assumption (5.2) implies that

$\liminf_{B\rightarrow\infty}\frac{N(\pi^{-1}(U\cap
U_{0}),-{\mathcal{K}}_{T},B)}{B(\log B)^{\rho({\widetilde{T}})-1}}>0.$

Hence

$\displaystyle\liminf_{B\rightarrow\infty}\frac{N(U,-{\mathcal{K}}_{S},B)}{B(\log
B)^{\rho({\widetilde{S}})-1}}$
$\displaystyle\geq\frac{1}{\deg\pi}\liminf_{B\rightarrow\infty}\left(\frac{N(\pi^{-1}(U\cap
U_{0}),-{\mathcal{K}}_{T},B)}{B(\log B)^{\rho({\widetilde{T}})-1}}\cdot(\log
B)^{\rho({\widetilde{T}})-\rho({\widetilde{S}})}\right)$
$\displaystyle=+\infty.\qed$

###### Proof of Theorem C.

Let $S$ be either of the surfaces in (1). By Proposition 4.2 and Proposition
4.5, there exists a quasi-étale cover $T\rightarrow S$ such that
$\rho({\widetilde{T}})=\rho({\widetilde{S}})$ and that $T$ is a toric Du Val
del Pezzo surface. The closed-set version of Manin’s conjecture with Peyre’s
constant is known for $T$ with any adelically metrized line bundle
$-{\mathcal{K}}_{T}$ [BT98a, CLT10]. Then, the assertion follows by (1) of
Lemma 5.2.

Let $S$ be the first surface in (2). By Proposition 4.8, there exists a quasi-
étale cover $T\rightarrow S$ such that
$\rho({\widetilde{T}})>\rho({\widetilde{S}})$ where $T$ is a del Pezzo surface
of degree $6$. The closed-set version of Manin’s conjecture with Peyre’s
constant is known for $T$ with any adelically metrized line bundle
$-{\mathcal{K}}_{T}$ [BT98a, CLT10]. Let $S$ be the second surface in (2). By
Proposition 4.10, there exists a quasi-étale cover $T\rightarrow S$ such that
$\rho({\widetilde{T}})=7>6=\rho({\widetilde{S}})$ where $T$ is of
$S_{3}(D_{4})$. By [FLS18, Theorem 1.1], we have

$N(V,-{\mathcal{K}}_{T},B)\gg B(\log B)^{\rho({\widetilde{T}})-1}$ (5.3)

for any dense open subset $V$ of $T$. In either case, the assertion follows by
(2) of Lemma 5.2. ∎

It is also worth noting that the four surfaces in Theorem C are all rational.
As an immediate conclusion, rational points on them do not form a thin set.

######  Proposition 5.3.

All the four surfaces defined in Theorem C are rational over the base field
${\mathbb{Q}}$.

###### Proof.

It is readily to check that each of the four surfaces defined in Theorem C has
a rational point on its smooth locus.

Let $S$ denote the surface defined in (1a) of Theorem C and $X$ be its minimal
resolution. We adopt the notations in Section 4.1. the curves $E_{6}$ and
$E_{13}$ on $X$ are disjoint to each other and form an orbit under the Galois
action. So we may contract them, and the result is a smooth del Pezzo surface
of degree $5$. A del Pezzo surface of degree $5$ always has a rational point
and is rational [Poo23, Section 9.4.12]. So we conclude that $S$ is rational
as well.

Let $S$ denote the surface defined in (1b) of Theorem C and $X$ be its minimal
resolution. Using the notations from Section 4.2, we may contract the Galois
orbit $\\{E_{8},E_{10}\\}$ of $(-1)$-curves on $X$, and then contract the
image of the Galois orbit $\\{E_{6},E_{7}\\}$. The result is a weak del Pezzo
surface of type $S_{6}(A_{1}+A_{2})$ for which the Galois action fixes each
negative curves. So we may contract it to a twist of ${\mathbb{P}}^{2}$ (see
[Der13, Figure 1]). The existence of a rational point on a twist of
${\mathbb{P}}^{2}$ implies that the twist is isomorphic to ${\mathbb{P}}^{2}$.
So we conclude that $S$ is rational over the base field.

Let $S$ denote the surface defined in (2a) of Theorem C and $X$ be its minimal
resolution. Using the notations from Section 4.3, we may contract
$E_{8},E_{10},E_{11}$ and $E_{13}$ on $X$ equivariantly. The result is a
smooth del Pezzo surface of degree $6$. A del Pezzo surface of degree $6$ with
a rational point is rational [Poo23, Section 9.4.12]. So we conclude that $S$
is rational over the base field.

Let $S$ denote the surface defined in (2b) of Theorem C and $X$ be its minimal
resolution. Using the notations from Section 3.2.12, we may contract the orbit
$\\{E_{9},E_{10}\\}$ to obtain a surface of type $S_{3}(D_{4})$, and then
contract the image of $\\{E_{2},E_{6}\\}$ to obtain a surface of type
$S_{5}(A_{2})$, and then contract the image of $\\{E_{3},E_{5}\\}$ to obtain a
surface of type $S_{7}(A_{1})$. This can be contracted to a twist of
${\mathbb{P}}^{2}$. So $S$ is rational over the base field. ∎

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