2 Log Noether–Fano inequalities and $\unicode[STIX]{x1D6FC}$ -invariant

In this section, we will establish Theorems A and B. For this goal we generalise the classical Noether–Fano inequality for affine Fano varieties $X\setminus S_{X}$ in the context of purely log terminal pairs $(X,S_{X})$ .

First, we suppose that there is an affine Fano variety $Y\setminus S_{Y}$ and a birational map $\unicode[STIX]{x1D719}:X{\dashrightarrow}Y$ such that

1. (1) the map $\unicode[STIX]{x1D719}$ is not an isomorphism in codimension $1$ ;

2. (2) the map $\unicode[STIX]{x1D719}$ induces an isomorphism of $X\setminus S_{X}$ onto $Y\setminus S_{Y}$ .

Let ${\mathcal{M}}_{Y}$ be a very ample complete system on $Y$ , and let ${\mathcal{M}}_{X}$ be the proper transform of ${\mathcal{M}}_{Y}$ by  $\unicode[STIX]{x1D719}$ . We consider the following resolution of indeterminacy of $\unicode[STIX]{x1D719}$ :

where $f$ and $g$ are birational morphisms and $W$ is a smooth projective variety. Let $\widetilde{S}_{X}$ and $\widetilde{S}_{Y}$ be the proper transforms of $S_{X}$ and $S_{Y}$ by $f$ and $g$ , respectively. Due to the conditions above, the divisor $\widetilde{S}_{X}$ is $g$ -exceptional, the divisor $\widetilde{S}_{Y}$ is $f$ -exceptional, and $f(\widetilde{S}_{Y})$ is contained in $S_{X}$ .

By analogy with the notion of $\unicode[STIX]{x1D716}$ -Kawamata log terminal singularities, we say that a pair $(Y,S_{Y})$ with irreducible and reduced boundary $S_{Y}$ has $\unicode[STIX]{x1D716}$ -purely log terminal singularities, if for every resolution of singularities $h:Z\rightarrow Y$ , where $Z$ is a smooth projective variety, the log discrepancy of every exceptional divisor of $h$ is bigger than $\unicode[STIX]{x1D716}$ .

Now let $Y$ be a quasi-projective variety, let $S_{Y}$ be a prime divisor on the variety $Y$ , and let  $\unicode[STIX]{x1D70C}:Y\rightarrow B$ be a dominant non-birational projective morphism with connected fibers such that the complement $Y_{\unicode[STIX]{x1D702}}\setminus S_{Y_{\unicode[STIX]{x1D702}}}$ is an affine Fano variety over the function field of the variety  $B$ . Here $Y_{\unicode[STIX]{x1D702}}$ is a generic fiber of $\unicode[STIX]{x1D70C}$ , and $S_{Y_{\unicode[STIX]{x1D702}}}=S_{Y}|_{Y_{\unicode[STIX]{x1D702}}}$ . We projectivize $Y$ and $B$ into projective varieties $\overline{Y}$ and  $\overline{B}$ respectively. We may assume that $\unicode[STIX]{x1D70C}:Y\rightarrow B$ extends to a projective morphism  $\bar{\unicode[STIX]{x1D70C}}:\overline{Y}\rightarrow \overline{B}$ .

Suppose that $\dim (B)>0$ , and that there exists a birational map $\unicode[STIX]{x1D713}:\overline{Y}{\dashrightarrow}X$ that induces an embedding $Y\setminus S_{Y}$ into $X\setminus S_{X}$ . As above, we consider the following resolution of indeterminacy of $\unicode[STIX]{x1D713}$ :

where $p$ and $q$ are birational morphisms and $V$ is a smooth projective variety. Let $\overline{S}_{Y}$ be the closure of $S_{Y}$ in $\overline{Y}$ . Let $\widetilde{S}_{X}$ and $\widetilde{S}_{Y}$ be the proper transforms of $S_{X}$ and $\overline{S}_{Y}$ by $p$ and $q$ , respectively. Then $\widetilde{S}_{Y}$ is $p$ -exceptional, because $X$ has $\mathbb{Q}$ -factorial singularities, and its Picard group is of rank  $1$ . Moreover, since $\unicode[STIX]{x1D713}$ induces an embedding $Y\setminus S_{Y}$ into $X\setminus S_{X}$ , $q(\widetilde{S}_{X})$ is contained either in $\overline{S}_{Y}$ or in $\overline{Y}\setminus Y$ . Thus, the divisor $\widetilde{S}_{X}$ is either $q$ -exceptional or mapped by $\bar{\unicode[STIX]{x1D70C}}\circ q$ to a proper subvariety of $\overline{B}$ . Since $S_{X}$ is ample, the latter also implies that $p(\widetilde{S}_{Y})$ is contained in $S_{X}$ .

Let $H$ be a very ample Cartier divisor on $\overline{B}$ . Let ${\mathcal{H}}_{\overline{Y}}$ be the complete linear system $|\bar{\unicode[STIX]{x1D70C}}^{\ast }(H)|$ , and let ${\mathcal{H}}_{X}$ be the proper transform of ${\mathcal{H}}_{\overline{Y}}$ on the variety $X$ by $\unicode[STIX]{x1D713}$ .

As an application of Theorem A, we obtain the following positive solution to Conjecture 1.1 for smooth cubic surfaces without rational points.

Applying Theorem B to quasi-smooth well-formed (see [Reference Iano-FletcherIan00]) Fano hypersurfaces in well-formed weighted projective spaces, we are able to provide many examples of super-rigid affine Fano varieties. To be precise, let $X$ be a well-formed weighted projective space $\mathbb{P}(a_{0},a_{1},\ldots ,a_{n})$ , where $a_{0},a_{1},\ldots ,a_{n}$ are positive integers such that $a_{0}\leqslant a_{1}\leqslant \cdots \leqslant a_{n}$ . Let $S_{X}$ be a quasi-smooth well-formed hypersurface in $X$ of degree $d<\sum _{i=0}^{n}a_{i}.$ Then $X\setminus S_{X}$ is an affine Fano variety of dimension $n$ , and $\text{Diff}_{X}(0)=0$ .

The examples below are motivated by Conjecture 1.1.

Smooth cubic surfaces in $\mathbb{P}^{3}$ , smooth quartic surfaces in $\mathbb{P}(1,1,1,2)$ , and smooth sextic surfaces in $\mathbb{P}(1,1,2,3)$ form three special families of a much larger class of quasismooth well-formed two-dimensional del Pezzo hypersurfaces in three-dimensional weighted projective spaces. They provide a lot of examples of super-rigid affine Fano threefolds.

Smooth quartic threefolds in $\mathbb{P}^{4}$ form one family among the famous $95$ families of Reid and Iano-Fletcher in [Reference Iano-FletcherIan00]. They also provide many examples of super-rigid affine Fano fourfolds through Theorem B.

Meanwhile, Johnson and Kollár completely described the quintuples $(a_{0},a_{1},a_{2},a_{3},a_{4})$ that define quasi-smooth hypersurface $S_{X}$ of degree $\sum _{i=0}^{4}a_{i}-1$ in $\mathbb{P}(a_{0},a_{1},a_{2},a_{3},a_{4})$ in [Reference Johnson and KollárJK01]. They also show that $\unicode[STIX]{x1D6FC}(S_{X})\geqslant 1$ in many of these cases. In such cases, the corresponding affine Fano variety $X\setminus S_{X}$ is super-rigid by Theorem B.

3 Global finite quotients of affine spaces

As seen in the previous section, it is hardly expected that the $\unicode[STIX]{x1D6FC}$ -invariant plays a role of a criterion (a sufficient and necessary condition) for an affine Fano variety to be super-rigid. However, it is able to serve as a criterion for affine Fano varieties of a certain type. In this section, we study affine Fano varieties that are quotients of $\mathbb{A}^{n}$ by actions of finite subgroups of $\text{GL}_{n}(\mathbb{C})$ . We verify that the $\unicode[STIX]{x1D6FC}$ -invariant completely determines super-rigidity of such affine Fano varieties of dimensions up to $4$ .

Let $G$ be a finite subgroup in $\text{GL}_{n}(\mathbb{C})$ , where $n\geqslant 2$ . Put $\mathbb{V}=\mathbb{A}^{n}$ and consider $\mathbb{V}$ as a linear representation of the group $G$ . In addition, let us identify $\text{GL}_{n}(\mathbb{C})$ with a subgroup of $\text{PGL}_{n+1}(\mathbb{C})=\text{Aut}(\mathbb{P}^{n})$ by means of the natural embedding $\text{GL}_{n}(\mathbb{C}){\hookrightarrow}\text{GL}_{n+1}(\mathbb{C})$ and the quotient homomorphism $\text{GL}_{n+1}(\mathbb{C})\rightarrow \text{PGL}_{n+1}(\mathbb{C})$ . In this way, we thus consider $G$ as a subgroup of $\text{Aut}(\mathbb{P}^{n})$ .

Let $H$ be the $\text{GL}_{n}(\mathbb{C})$ -invariant hyperplane $\mathbb{P}^{n}\setminus \mathbb{V}$ . The action of $G$ on $H$ is not necessarily faithful. Denote by $\overline{G}$ the image of $G$ in $\text{Aut}(H)=\text{PGL}_{n}(\mathbb{C})$ , and denote by $Z$ the kernel of the group homomorphism $G\rightarrow \overline{G}$ . Then $Z$ is a cyclic group of order $m\geqslant 1$ , and $G$ is a central extension of the group $\overline{G}$ by $Z$ . Letting

we can identify the quotient $S_{X}=H/\overline{G}$ with a prime divisor in $X$ , so that by construction

Recall that an element $g\in G\subset \text{GL}_{n}(\mathbb{C})$ is called a quasi-reflection if there is a hyperplane in $H\cong \mathbb{P}^{n-1}$ that is pointwise fixed by the image of $g$ in $\overline{G}$ . If $G$ is generated by quasi-reflections, then by virtue of the Chevalley–Shephard–Todd theorem $\mathbb{A}^{n}/G$ is isomorphic to $\mathbb{A}^{n}$ , hence is in particular an affine Fano variety. More generally, we have the following.

In view of the above discussion, to address the question whether the quotient (3.1) is a super-rigid affine Fano variety, we may and will assume in the sequel that $G$ is small, i.e., does not contain any non-trivial quasi-reflection. Let $\unicode[STIX]{x1D6FC}_{\overline{G}}(H)$ be the number defined as

Then $\unicode[STIX]{x1D6FC}_{\overline{G}}(H)$ is the $\overline{G}$ -equivariant $\unicode[STIX]{x1D6FC}$ -invariant of the projective space $H$ . Moreover, it follows from the proof of [Reference Cheltsov and ShramovCS11, Theorem 3.16] that

Furthermore, one has $\unicode[STIX]{x1D6FC}_{\overline{G}}(H)\geqslant 1$ if and only if the quotient singularity $\mathbb{A}^{n}/G$ is weakly-exceptional in the notation of [Reference Cheltsov and ShramovCS11, Reference Cheltsov and ShramovCS14, Reference SakovicsSak12, Reference SakovicsSak14]. In particular, if $\unicode[STIX]{x1D6FC}_{\overline{G}}(H)\geqslant 1$ , then $\mathbb{V}$ must be an irreducible representation of the group $G$ by [Reference Cheltsov and ShramovCS11, Theorem 3.18]. In this case, the subgroup $\overline{G}\subset \text{Aut}(H)$ is usually called transitive.

One can show that super-rigid affine Fano quotients (3.1) exist in all dimensions.

We now give a complete classification of super-rigid affine Fano varieties (3.1) of dimensions at most $4$ .

Using Theorem 3.7 together with the results of Miyanishi and Sugie on $\mathbb{Q}$ -homology planes with quotient singularities [Reference Miyanishi and SugieMS91], we obtain the following classification of all two-dimensional super-rigid affine Fano varieties.

A consequence of Theorem 3.7 is that for $n=2$ all three conditions of Lemma 3.4 are actually equivalent. This is no longer true for $n\geqslant 3$ as illustrated by the following example.

In fact, for $n=3$ and $4$ , we can obtain criteria for $X\setminus S_{X}$ to be super-rigid which are similar to Theorem 3.7. For example, we have the following.

Similarly, for $n=4$ we have the following.

4 Complements to del Pezzo surfaces

Let $S$ be a del Pezzo surface of anticanonical degree at most $3$ , i.e., $K_{S}^{2}\leqslant 3$ , with at worst Du Val singularities. If the anticanonical degree is $3$ , then $S$ is a cubic surface in $\mathbb{P}^{3}$ . Similarly, if the anticanonical degree is $2$ , then $S$ is a quartic surface in $\mathbb{P}(1,1,1,2)$ . Finally, if the anticanonical degree is $1$ , then $S$ is a sextic surface in $\mathbb{P}(1,1,2,3)$ .

In this section, we study automorphism groups of affine Fano varieties whose boundary is $S$ and whose completion is the corresponding ambient weighted projective space. Using the result obtained in this section we are able to yield many examples of non-trivial non-super-rigid affine Fano threefolds.

Let $\mathbb{P}$ be the ambient spaces $\mathbb{P}^{3}$ , $\mathbb{P}(1,1,1,2)$ and $\mathbb{P}(1,1,2,3)$ in the cases of anticanonical degrees $3$ , $2$ and $1$ , respectively. We have

where the variables $x$ , $y$ , $z$ and $w$ are of weights $(1,1,1,1)$ , $(1,1,1,2)$ and $(1,1,2,3)$ according to anticanonical degrees $3$ , $2$ and $1$ . Denote by $d$ the degree of the surface $S$ as a hypersurface in $\mathbb{P}$ . Then $d$ is equal to $3$ , $4$ and $6$ according to anticanonical degrees $3$ ,  $2$ and $1$ , respectively.

The following theorem summarizes the current knowledge towards the structure of the automorphism groups of the complements of smooth del Pezzo surfaces in $\mathbb{P}$ .

A consequence of Theorem 4.1 is the following generalization of [Reference Dubouloz and KishimotoDK15, Proposition 10].

It is known that a smooth del Pezzo surface of anticanonical degree at most $3$ never contains any $(-K_{S})$ -polar cylinder. For the singular case, we obtain a complete description for $(-K_{S})$ -polar cylinders from [Reference Cheltsov, Park and WonCPW16a, Theorem 1.5].

Existence of open $\mathbb{A}^{1}$ -cylinders in the complements of singular normal cubic surfaces with singularities strictly worse than $\text{A}_{2}$ was first established in [Reference Kishimoto, Prokhorov and ZaidenbergKPZ11, Proposition 3.7]. The first example of a single nodal cubic surface whose complement contains an open $\mathbb{A}^{1}$ -cylinder was constructed later on by Lamy (unpublished). Combined with the above results on $(-K_{S})$ -polar cylinders of singular del Pezzo surfaces and Example 1.7, this leads to anticipate that the complement of a del Pezzo surface with a $(-K_{S})$ -polar cylinder contains an open $\mathbb{A}^{1}$ -cylinder. Ideas evolving from the proof of [Reference Cheltsov, Park and WonCPW16a, Theorem 4.1] and the study of open ‘vertical’ $\mathbb{A}^{1}$ -cylinders in del Pezzo fibrations [Reference Dubouloz and KishimotoDK18] turn out to confirm this expectation, as follows.

Before we proceed the proof, let us first prepare setups for the proof. Due to Theorem 4.3 above the surface $S$ contains a singular point $P$ . Furthermore, we may assume that the singular point $P$ is not of type $\text{A}_{1}$ if the anticanonical degree of $S$ is $2$ ; it is not of types $\text{A}_{1}$ , $\text{A}_{2}$ , $\text{A}_{3}$ , $\text{D}_{4}$ if the anticanonical degree is $1$ . By suitable coordinate changes, we may assume that the point $P$ is located at $[1:0:0:0]$ . Under such conditions, we immediately observe the following.

Let $\unicode[STIX]{x1D70B}:S{\dashrightarrow}\unicode[STIX]{x1D6F1}$ be the projection from the point $P$ to the hyperplane $\unicode[STIX]{x1D6F1}$ defined by  $x=0$ in $\mathbb{P}$ , i.e., $\unicode[STIX]{x1D70B}([x:y:z:w])=[y:z:w]$ . The hyperplane $\unicode[STIX]{x1D6F1}$ is isomorphic to $\mathbb{P}^{2}$ , $\mathbb{P}(1,1,2)$ , $\mathbb{P}(1,2,3)$ according to the anticanonical degrees, $3$ , $2$ , $1$ , respectively. We denote by $g(y,z,w)$ the coefficient quasi-homogenous polynomial of $x$ in each of the quasi-homogenous equations of (4.6), (4.7), (4.8) and (4.9), i.e., for the case $K_{S}^{2}=1$

for the case $K_{S}^{2}=2$

and for the case $K_{S}^{2}=3$

Let $D$ be the divisor on $S$ cut by the equation $g(y,z,w)=0$ . In the case of $K_{S}^{2}=3$ , the divisor $D$ consists of the lines in $S$ that pass through $P$ . There are at most six such lines and they are defined by the system of homogenous equations

in $\mathbb{P}^{3}$ . In the case of $K_{S}^{2}=2$ , $D$ consists of at most six curves passing through the point  $P$ . They are defined by the system of quasi-homogeneous equations

in $\mathbb{P}(1,1,1,2)$ . Finally, in the case of $K_{S}^{2}=1$ , it consists of at most five curves passing through the point  $P$ , which are defined by the system of quasi-homogeneous equations

in $\mathbb{P}(1,1,2,3)$ . In each case, the number of curves in $D$ is completely determined by the number of points determined by the system of quasi-homogeneous equations on $\unicode[STIX]{x1D6F1}$ . Denote these curves by $L_{1},\ldots ,L_{r}$ in each case. The map $\unicode[STIX]{x1D70B}$ contracts each curve $L_{i}$ to a point on $\unicode[STIX]{x1D6F1}$ .

Now we are ready to prove Theorem C.

Let $\unicode[STIX]{x1D711}:\overline{S}\rightarrow S$ be the blow up at the point $P$ . Then there exists a commutative diagram

where $\unicode[STIX]{x1D719}$ is the birational morphism that contracts exactly the proper transforms of the lines $L_{1},\ldots ,L_{r}$ . Let $E$ be the exceptional divisor of the blow up $\unicode[STIX]{x1D711}$ . Then the image of the curve $E$ by $\unicode[STIX]{x1D719}$ in $\unicode[STIX]{x1D6F1}$ is the conic curve ${\mathcal{C}}$ . Then ${\mathcal{C}}$ contains all the points $\unicode[STIX]{x1D70B}(L_{i})$ . If $P$ is an ordinary double point of the surface $S$ , then ${\mathcal{C}}$ is a smooth conic. Moreover, if $P$ is a singular point of the surface $S$ of type $\text{A}_{n}$ for $n\geqslant 2$ , then ${\mathcal{C}}$ consists of two distinct lines. Finally, if $P$ is either of type $\text{D}_{n}$ or of type $\text{E}_{6}$ , then ${\mathcal{C}}$ is a double line.

If ${\mathcal{C}}$ is smooth, let $\ell$ be a general line in $\unicode[STIX]{x1D6F1}$ that is tangent to ${\mathcal{C}}$ . If ${\mathcal{C}}$ is singular, let $\ell$ be a general line in $\unicode[STIX]{x1D6F1}$ that passes through a singular point of the conic ${\mathcal{C}}$ . By a suitable coordinate change, we may assume that $\ell$ is defined by the equation $y=0$ on $\unicode[STIX]{x1D6F1}$ . Let $H$ be the divisor on $S$ cut by the equation $y=0$ in $\mathbb{P}^{3}$ .

Then

so that $S\setminus (H\cup L_{1}\cup \cdots \cup L_{r})$ is an $\mathbb{A}^{1}$ -cylinder.

For the rest of the cases we also denote by $\ell$ the curve defined by $y=0$ on $\unicode[STIX]{x1D6F1}$ . In addition let $H$ be the divisor on $S$ cut by $y=0$ in $\mathbb{P}$ .

As in the case of $K_{S}^{2}=3$ , the isomorphism $\tilde{\unicode[STIX]{x1D70B}}$ maps $S\setminus (H\cup L_{1}\cup \cdots \cup L_{r})$ onto $\unicode[STIX]{x1D6F1}\setminus ({\mathcal{C}}\cup \ell )$ . Meanwhile, the projection $\unicode[STIX]{x1D70B}$ maps $S\setminus H$ onto $\unicode[STIX]{x1D6F1}\setminus \ell \cong \mathbb{A}^{2}$ . The affine variety $S\setminus (H\cup L_{1}\cup \cdots \cup L_{r})$ is therefore isomorphic to the complement of the curve defined by

for type (4.7) (respectively $aw+f_{5}(1,z)=0$ for type (4.8)) in $\unicode[STIX]{x1D6F1}\setminus \ell \cong \mathbb{A}^{2}$ . This immediately implies the claim.

In this case, the isomorphism $\tilde{\unicode[STIX]{x1D70B}}$ maps $S\setminus (H_{z}\cup L_{1}\cup \cdots \cup L_{r})$ onto $\unicode[STIX]{x1D6F1}\setminus ({\mathcal{C}}\cup \ell _{z})$ , where  $\ell _{z}$ is the curve on $\unicode[STIX]{x1D6F1}$ defined by $z=0$ . The projection $\unicode[STIX]{x1D70B}$ maps $S\setminus H_{z}$ onto $\unicode[STIX]{x1D6F1}\setminus \ell _{z}$ . The affine variety $\unicode[STIX]{x1D6F1}\setminus \ell _{z}$ is isomorphic to the quotient space $\mathbb{A}^{2}/\boldsymbol{\unicode[STIX]{x1D707}}_{2}$ , where the $\boldsymbol{\unicode[STIX]{x1D707}}_{2}$ -action is given by $(y,w)\mapsto (-y,-w)$ . Notice that we abuse our notations $y$ and $w$ . The affine open subset $\unicode[STIX]{x1D6F1}\setminus ({\mathcal{C}}\cup \ell _{z})$ is an $\mathbb{A}^{1}$ -cylinder. To see this, notice that $\unicode[STIX]{x1D6F1}\setminus ({\mathcal{C}}\cup \ell _{z})$ is the complement of the image of the curve defined by

in $\unicode[STIX]{x1D6F1}\setminus \ell _{z}\cong \mathbb{A}^{2}/\boldsymbol{\unicode[STIX]{x1D707}}_{2}$ . Since $f_{5}(y,1)$ is an odd polynomial in $y$ , the isomorphism $\unicode[STIX]{x1D713}:\mathbb{A}^{2}\rightarrow \mathbb{A}^{2}$ defined by $(y_{1},w_{1})=\unicode[STIX]{x1D713}(y,w)=(y,w+f_{5}(y,1))$ is $\boldsymbol{\unicode[STIX]{x1D707}}_{2}$ -equivariant for the action $(y_{1},w_{1})\mapsto (-y_{1},-w_{1})$ so that we have a $\boldsymbol{\unicode[STIX]{x1D707}}_{2}$ -equivariant diagram

where the morphisms from $\mathbb{A}^{2}$ to $\mathbb{A}^{2}/\boldsymbol{\unicode[STIX]{x1D707}}_{2}$ are the respective quotient maps. This shows that $\unicode[STIX]{x1D6F1}\setminus ({\mathcal{C}}\cup \ell _{z})$ is isomorphic to the complement in $\mathbb{A}^{2}/\boldsymbol{\unicode[STIX]{x1D707}}_{2}$ of the image of the curve defined by $w_{1}=0$ . Letting $s=w_{1}^{2}$ , $t=y_{1}^{2}$ and $u=w_{1}y_{1}$ , $\mathbb{A}^{2}/\boldsymbol{\unicode[STIX]{x1D707}}_{2}$ is isomorphic the affine variety defined by the equation $st=u^{2}$ in $\mathbb{A}^{3}$ . The image in $\mathbb{A}^{2}/\boldsymbol{\unicode[STIX]{x1D707}}_{2}$ of the curve defined by $w_{1}=0$ coincides with the curve defined by $s=u=0$ . Its complement is thus isomorphic to $\mathbb{A}^{1}\setminus \{0\}\times \mathbb{A}^{1}$ , which is an $\mathbb{A}^{1}$ -cylinder as desired.

From now on, let ${\mathcal{H}}$ (respectively ${\mathcal{G}}$ ) be the hyperplane defined by $y=0$ (respectively $z=0$ ) in $\mathbb{P}$ and let ${\mathcal{Q}}$ be the (weighted) hypersurface (possibly reducible or non-reduced) defined by $g(y,z,w)=0$ in $\mathbb{P}$ .

We first consider the surface $S$ dealt with in Claims 1 and 2.

From Claims 1 and 2 we obtain the property that

is an $\mathbb{A}^{1}$ -cylinder. In addition we suppose that the weighted surface $S$ is defined by $f(x,y,z,w)=0$ in $\mathbb{P}$ . The equation must be one of the types in Lemma 4.5.

The pencil on $\mathbb{P}$ generated by the surfaces $S$ and ${\mathcal{H}}+{\mathcal{Q}}$ consists of (weighted) surfaces with equations of the form

where $[\unicode[STIX]{x1D6FC}:\unicode[STIX]{x1D6FD}]\in \mathbb{P}^{1}$ . This pencil gives a rational map $\unicode[STIX]{x1D70C}:\mathbb{P}{\dashrightarrow}\mathbb{P}^{1}$ . Its generic fiber $S_{\unicode[STIX]{x1D702}}$ is the weighted surface defined by equation

in the corresponding weighted projective space $\mathbb{P}_{\Bbbk (\unicode[STIX]{x1D706})}$ over the field $\Bbbk (\unicode[STIX]{x1D706})$ , where $\unicode[STIX]{x1D706}$ is a parameter. It is singular at the point $[1:0:0:0]$ , and the intersection of $S_{\unicode[STIX]{x1D702}}$ with the surface in $\mathbb{P}_{\Bbbk (\unicode[STIX]{x1D706})}$ given by $f(x,y,z,w)=0$ consists of the $\Bbbk (\unicode[STIX]{x1D706})$ -rational hyperplane section

and the $\Bbbk (\unicode[STIX]{x1D706})$ -rational (weighted) hypersurface section

As in the case of $S\setminus ({\mathcal{H}}\cup {\mathcal{Q}})$ , the complement $S_{\unicode[STIX]{x1D702}}\setminus ({\mathcal{H}}_{\unicode[STIX]{x1D702}}\cup {\mathcal{Q}}_{\unicode[STIX]{x1D702}})$ is an $\mathbb{A}^{1}$ -cylinder in $S_{\unicode[STIX]{x1D702}}$ , defined over the field $\Bbbk (\unicode[STIX]{x1D706})$ , and hence $\mathbb{P}\setminus S$ contains an open $\mathbb{A}^{1}$ -cylinder (see [Reference Dubouloz and KishimotoDK18, Lemma 3]).

Now we consider the surface $S$ defined by a quasi-homogenous equation of type (4.9). Letting $S_{\unicode[STIX]{x1D702}}$ be the generic member of the pencil of hypersurfaces of degree $6$ generated by $S$ and $3{\mathcal{G}}$ , we deduce from Claim 3 in the same way as in the previous cases that the complement of $S_{\unicode[STIX]{x1D702}}\setminus {\mathcal{G}}$ contains an open $\mathbb{A}^{1}$ -cylinder defined over the field $\Bbbk (\unicode[STIX]{x1D706})$ , and hence that $\mathbb{P}\setminus S$ contains an open $\mathbb{A}^{1}$ -cylinder.◻

Combined with Corollary 4.2, Theorem C fully settles the question of existence of open $\mathbb{A}^{1}$ -cylinders in the complements to cubic surfaces with at worst Du Val singularities. Since the cubic surface $S$ has a $(-K_{S})$ -polar cylinder if and only if $S$ is singular, it follows from Corollary 4.2 and Theorem C that the following two conditions are equivalent:

• ∙ the affine Fano variety $\mathbb{P}\setminus S$ contains an open $\mathbb{A}^{1}$ -cylinder;

• ∙ the cubic surface $S$ contains a $(-K_{S})$ -polar cylinder.

It is natural to expect that the same holds in the cases of lower anticanonical degrees.