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 $\unicode[STIX]{x1D716}$ .

Lemma 2.1 (Log Noether–Fano inequality I).

Suppose that the pair $(Y,S_{Y})$ is $\unicode[STIX]{x1D716}$ -purely log terminal for some positive rational number $\unicode[STIX]{x1D716}$ . Let $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D706}$ be non-negative rational numbers such that $1-\unicode[STIX]{x1D716}\leqslant \unicode[STIX]{x1D707}\leqslant 1$ and

$$\begin{eqnarray}K_{X}+\unicode[STIX]{x1D707}S_{X}+\unicode[STIX]{x1D706}{\mathcal{M}}_{X}{\sim}_{\mathbb{Q}}0.\end{eqnarray}$$

Then the log discrepancy of $\widetilde{S}_{Y}$ with respect to $(X,\unicode[STIX]{x1D707}S_{X}+\unicode[STIX]{x1D706}{\mathcal{M}}_{X})$ is less than $1-\unicode[STIX]{x1D707}$ .

Proof. We have

$$\begin{eqnarray}K_{W}+\unicode[STIX]{x1D707}\widetilde{S}_{X}+\unicode[STIX]{x1D706}{\mathcal{M}}_{W}{\sim}_{\mathbb{Q}}f^{\ast }(K_{X}+\unicode[STIX]{x1D707}S_{X}+\unicode[STIX]{x1D706}{\mathcal{M}}_{X})+a\widetilde{S}_{Y}+\sum a_{i}E_{i},\end{eqnarray}$$

and

$$\begin{eqnarray}K_{W}+\unicode[STIX]{x1D707}\widetilde{S}_{Y}+\unicode[STIX]{x1D706}{\mathcal{M}}_{W}{\sim}_{\mathbb{Q}}g^{\ast }(K_{Y}+\unicode[STIX]{x1D707}S_{Y}+\unicode[STIX]{x1D706}{\mathcal{M}}_{Y})+b\widetilde{S}_{X}+\sum b_{i}E_{i},\end{eqnarray}$$

where each $E_{i}$ is simultaneously $f$ -exceptional and $g$ -exceptional. The hypothesis that $(Y,S_{Y})$ is $\unicode[STIX]{x1D716}$ -purely log terminal immediately implies that $\unicode[STIX]{x1D707}+b$ is positive.

Suppose $a+\unicode[STIX]{x1D707}\geqslant 0$ . Since $\unicode[STIX]{x1D707}+b$ is positive, applying the negativity lemma [Reference ShokurovSho93, 1.1] to

$$\begin{eqnarray}(\unicode[STIX]{x1D707}+a)\widetilde{S}_{Y}-g^{\ast }(K_{Y}+\unicode[STIX]{x1D707}S_{Y}+\unicode[STIX]{x1D706}{\mathcal{M}}_{Y}){\sim}_{\mathbb{Q}}(\unicode[STIX]{x1D707}+b)\widetilde{S}_{X}+\sum (b_{i}-a_{i})E_{i}\end{eqnarray}$$

we see that $K_{Y}+\unicode[STIX]{x1D707}S_{Y}+\unicode[STIX]{x1D706}{\mathcal{M}}_{Y}$ is positive.

Let $\unicode[STIX]{x1D706}^{\prime }$ be the number such that $K_{Y}+\unicode[STIX]{x1D707}S_{Y}+\unicode[STIX]{x1D706}^{\prime }{\mathcal{M}}_{Y}{\sim}_{\mathbb{Q}}0$ . Note that $\unicode[STIX]{x1D706}^{\prime }<\unicode[STIX]{x1D706}$ . We have

$$\begin{eqnarray}K_{W}+\unicode[STIX]{x1D707}\widetilde{S}_{X}+\unicode[STIX]{x1D706}^{\prime }{\mathcal{M}}_{W}{\sim}_{\mathbb{Q}}f^{\ast }(K_{X}+\unicode[STIX]{x1D707}S_{X}+\unicode[STIX]{x1D706}^{\prime }{\mathcal{M}}_{X})+a^{\prime }\widetilde{S}_{Y}+\sum \mathop{a}_{i}^{\prime }E_{i},\end{eqnarray}$$

and

$$\begin{eqnarray}K_{W}+\unicode[STIX]{x1D707}\widetilde{S}_{Y}+\unicode[STIX]{x1D706}^{\prime }{\mathcal{M}}_{W}{\sim}_{\mathbb{Q}}g^{\ast }(K_{Y}+\unicode[STIX]{x1D707}S_{Y}+\unicode[STIX]{x1D706}^{\prime }{\mathcal{M}}_{Y})+b\widetilde{S}_{X}+\sum b_{i}E_{i},\end{eqnarray}$$

where $a^{\prime }\geqslant a$ and $a_{i}^{\prime }\geqslant a_{i}$ . We then get

$$\begin{eqnarray}(\unicode[STIX]{x1D707}+b)\widetilde{S}_{X}-f^{\ast }(K_{X}+\unicode[STIX]{x1D707}S_{X}+\unicode[STIX]{x1D706}^{\prime }{\mathcal{M}}_{X}){\sim}_{\mathbb{Q}}(\unicode[STIX]{x1D707}+a^{\prime })\widetilde{S}_{Y}+\sum (\mathop{a}_{i}^{\prime }-b_{i})E_{i}.\end{eqnarray}$$

Since $\unicode[STIX]{x1D707}+b>0$ and $K_{X}+\unicode[STIX]{x1D707}S_{X}+\unicode[STIX]{x1D706}^{\prime }{\mathcal{M}}_{X}$ is negative, the negativity lemma implies $\unicode[STIX]{x1D707}+a^{\prime }\leqslant 0$ . This is absurd. Therefore, $a+1<1-\unicode[STIX]{x1D707}$ .◻

Corollary 2.2. Let $\unicode[STIX]{x1D706}$ be a non-negative rational number such that

$$\begin{eqnarray}K_{X}+S_{X}+\unicode[STIX]{x1D706}{\mathcal{M}}_{X}{\sim}_{\mathbb{Q}}0.\end{eqnarray}$$

Then the log discrepancy of $\widetilde{S}_{Y}$ with respect to the pair $(X,S_{X}+\unicode[STIX]{x1D706}{\mathcal{M}}_{X})$ is negative.

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 $\unicode[STIX]{x1D713}$ .

Lemma 2.3 (Log Noether–Fano inequality II).

Let $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D706}$ be non-negative rational numbers such that $\unicode[STIX]{x1D707}\leqslant 1$ and

$$\begin{eqnarray}K_{X}+\unicode[STIX]{x1D707}S_{X}+\unicode[STIX]{x1D706}{\mathcal{H}}_{X}{\sim}_{\mathbb{Q}}0.\end{eqnarray}$$

Then the log discrepancy of $\widetilde{S}_{Y}$ with respect to $(X,\unicode[STIX]{x1D707}S_{X}+\unicode[STIX]{x1D706}{\mathcal{H}}_{X})$ is less than $1-\unicode[STIX]{x1D707}$ .

Proof. As in the proof of Lemma 2.1, we have

$$\begin{eqnarray}K_{V}+\unicode[STIX]{x1D707}\widetilde{S}_{X}+\unicode[STIX]{x1D706}{\mathcal{H}}_{V}{\sim}_{\mathbb{Q}}p^{\ast }(K_{X}+\unicode[STIX]{x1D707}S_{X}+\unicode[STIX]{x1D706}{\mathcal{H}}_{X})+a\widetilde{S}_{Y}+\sum a_{i}E_{i},\end{eqnarray}$$

and

$$\begin{eqnarray}K_{V}+\unicode[STIX]{x1D707}\widetilde{S}_{Y}+\unicode[STIX]{x1D706}{\mathcal{H}}_{V}{\sim}_{\mathbb{Q}}q^{\ast }(K_{\overline{Y}}+\unicode[STIX]{x1D707}\overline{S}_{Y}+\unicode[STIX]{x1D706}{\mathcal{H}}_{\overline{Y}})+b\widetilde{S}_{X}+\sum b_{j}F_{j},\end{eqnarray}$$

where each $E_{i}$ is $p$ -exceptional, and each $F_{j}$ is $q$ -exceptional. Then

$$\begin{eqnarray}(a+\unicode[STIX]{x1D707})\widetilde{S}_{Y}{\sim}_{\mathbb{Q}}q^{\ast }(K_{\overline{Y}}+\unicode[STIX]{x1D707}\overline{S}_{Y}+\unicode[STIX]{x1D706}{\mathcal{H}}_{\overline{Y}})+(b+\unicode[STIX]{x1D707})\widetilde{S}_{X}+\sum b_{j}F_{j}-\sum a_{i}E_{i}.\end{eqnarray}$$

Thus, we have

$$\begin{eqnarray}(a+\unicode[STIX]{x1D707})\overline{S}_{Y}{\sim}_{\mathbb{Q}}K_{\overline{Y}}+\unicode[STIX]{x1D707}\overline{S}_{Y}+\unicode[STIX]{x1D706}{\mathcal{H}}_{\overline{Y}}+(b+\unicode[STIX]{x1D707})q(\widetilde{S}_{X})-\sum a_{i}q(E_{i}).\end{eqnarray}$$

On the other hand, each $p$ -exceptional divisor $E_{i}$ is either $q$ -exceptional or mapped by  $\bar{\unicode[STIX]{x1D70C}}\circ q$ to a proper subvariety of $\overline{B}$ . This shows that

$$\begin{eqnarray}-(a+\unicode[STIX]{x1D707})S_{Y_{\unicode[STIX]{x1D702}}}{\sim}_{\mathbb{Q}}-(K_{Y_{\unicode[STIX]{x1D702}}}+\unicode[STIX]{x1D707}S_{Y_{\unicode[STIX]{x1D702}}})=-(K_{Y_{\unicode[STIX]{x1D702}}}+S_{Y_{\unicode[STIX]{x1D702}}})+(1-\unicode[STIX]{x1D707})S_{Y_{\unicode[STIX]{x1D702}}}.\end{eqnarray}$$

Since $-(K_{Y_{\unicode[STIX]{x1D702}}}+S_{Y_{\unicode[STIX]{x1D702}}})$ and $S_{Y_{\unicode[STIX]{x1D702}}}$ are ample, we obtain $a+\unicode[STIX]{x1D707}<0$ . Therefore, the log discrepancy of  $\widetilde{S}_{Y}$ with respect to $(X,\unicode[STIX]{x1D707}S_{X}+\unicode[STIX]{x1D706}{\mathcal{H}}_{X})$ is less than $1-\unicode[STIX]{x1D707}$ .◻

Corollary 2.4. Let $\unicode[STIX]{x1D706}$ be a non-negative rational number such that

$$\begin{eqnarray}K_{X}+S_{X}+\unicode[STIX]{x1D706}{\mathcal{H}}_{X}{\sim}_{\mathbb{Q}}0.\end{eqnarray}$$

Then the log discrepancy of $\widetilde{S}_{Y}$ with respect to the pair $(X,S_{X}+\unicode[STIX]{x1D706}{\mathcal{H}}_{X})$ is negative.

Theorem 2.5 (Theorem A).

Let $X\setminus S_{X}$ be an affine Fano variety with completion $X$ and boundary $S_{X}$ . If for any mobile linear system ${\mathcal{M}}_{X}$ on $X$ and a positive rational number $\unicode[STIX]{x1D706}$ such that

$$\begin{eqnarray}K_{X}+S_{X}+\unicode[STIX]{x1D706}{\mathcal{M}}_{X}{\sim}_{\mathbb{Q}}0,\end{eqnarray}$$

the pair $(X,S_{X}+\unicode[STIX]{x1D706}{\mathcal{M}}_{X})$ is log canonical along $S_{X}$ , then $X\setminus S_{X}$ is super-rigid.

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

Proof. Indeed, otherwise Theorem A implies that there exists a mobile linear system ${\mathcal{M}}$ on $\mathbb{P}_{\Bbbk }^{3}$ of degree $m$ such that the singularities of the pair

$$\begin{eqnarray}\biggl(\mathbb{P}^{3},S+\frac{1}{m}{\mathcal{M}}\biggr)\end{eqnarray}$$

are not log canonical along $S$ . Write $D=(1/m){\mathcal{M}}|_{S}$ . Then the pair $(S,D)$ is not log canonical by Inversion of adjunction (see [Reference KollárKol97, Theorem 7.5]), and $D$ is an effective $\mathbb{Q}$ -divisor on $S$ such that $D{\sim}_{\mathbb{Q}}-K_{S}$ .

Let $S_{\overline{\Bbbk }}$ be the surface obtained from $S$ by base change to an algebraic closure $\overline{\Bbbk }$ of $\Bbbk$ . Similarly, let $D_{\overline{\Bbbk }}$ be the $\mathbb{Q}$ -divisor obtained from $D$ by the same base change. Then $D_{\overline{\Bbbk }}$ is an effective $\mathbb{Q}$ -divisor on $S_{\overline{\Bbbk }}$ such that

$$\begin{eqnarray}D_{\overline{\Bbbk }}{\sim}_{\mathbb{Q}}-K_{S_{\overline{\Bbbk }}}.\end{eqnarray}$$

Moreover, the pair $(S_{\overline{\Bbbk }},D_{\overline{\Bbbk }})$ is not log canonical at some point $P\in S_{\overline{\Bbbk }}$ . Furthermore, it follows from [Reference CheltsovChe08a, Lemma 3.7] that this pair is log canonical outside of the point $P$ . Thus, the point $P$ must be invariant under the action of the Galois group $\text{Gal}(\overline{\Bbbk }/\Bbbk )$ on $S_{\overline{\Bbbk }}$ , and hence corresponds to a $\Bbbk$ -rational point of $S$ . This is a contradiction.◻

Example 2.7. Let $X$ be the cubic hypersurface in $\mathbb{P}_{\mathbb{Q}}^{4}$ that is given by

$$\begin{eqnarray}5x^{3}+9y^{3}+10z^{3}+12w^{3}+uf_{2}(x,y,z,w,u)=0,\end{eqnarray}$$

where $f_{2}$ is a general homogeneous polynomial of degree $2$ . Let $S_{X}$ be its hyperplane section cut by $u=0$ . Then $X$ has at most canonical singularities, the surface $S_{X}$ is smooth, and  $X\setminus S_{X}$ is an affine cubic hypersurface in $\mathbb{A}_{\mathbb{Q}}^{4}$ . Moreover, Cassels and Guy proved in [Reference Cassels and GuyCG66] that the surface $S_{X}$ violates the Hasse principle. In particular, it does not contain any $\mathbb{Q}$ -rational points. Then the proof of Corollary 2.6 shows verbatim that  $X\setminus S_{X}$ is super-rigid.

Proof. Suppose that $X\setminus S_{X}$ is not super-rigid. Then Theorem A above implies that the variety  $X$ must carry a mobile linear system ${\mathcal{M}}_{X}$ such that the singularities of the pair $(X,S_{X}+\unicode[STIX]{x1D706}{\mathcal{M}}_{X})$ are not log canonical along $S_{X}$ for a positive rational number $\unicode[STIX]{x1D706}$ satisfying

$$\begin{eqnarray}K_{X}+S_{X}+\unicode[STIX]{x1D706}{\mathcal{M}}_{X}{\sim}_{\mathbb{Q}}0.\end{eqnarray}$$

However, the condition $\unicode[STIX]{x1D6FC}(S_{X},\text{Diff}_{S_{X}}(0))\geqslant 1$ and Inversion of adjunction (see [Reference KollárKol97, Theorem 7.5]) immediately show that this is impossible.◻

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.

Example 2.9. Suppose that $X=\mathbb{P}^{n}$ and $d=n\geqslant 4$ . Then

$$\begin{eqnarray}\text{Aut}(X\setminus S_{X})=\text{Aut}(X,S_{X}),\end{eqnarray}$$

because $S_{X}$ is not ruled (see [Reference Iskovskih and ManinIM71, Reference PukhlikovPuk98, Reference de FernexdeF13]). We do not know whether the affine Fano variety  $X\setminus S_{X}$ is super-rigid or not. However, if $n\geqslant 6$ and $S_{X}$ is a general hypersurface of degree  $n$ , then $\unicode[STIX]{x1D6FC}(S_{X})=1$ by [Reference PukhlikovPuk05], so that $X\setminus S_{X}$ should be super-rigid by Theorem B.

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.

Example 2.10. Suppose that $n=3$ , so that $X\setminus S_{X}$ should be an affine Fano threefold. Then it follows from [Reference Cheltsov, Park and ShramovCPS10, Reference Cheltsov and ShramovCS13] that $\unicode[STIX]{x1D6FC}(S_{X})>1$ if and only if $(a_{0},a_{1},a_{2},a_{3},d)$ is one of the following the quintuples:

$$\begin{eqnarray}\begin{array}{@{}llll@{}}(2,3,5,9,18), & (3,3,5,5,15), & (3,5,7,11,25), & (3,5,7,14,28),\\ (3,5,11,18,36), & (5,14,17,21,56), & (5,19,27,31,81), & (5,19,27,50,100),\\ (7,11,27,37,81), & (7,11,27,44,88), & (9,15,17,20,60), & (9,15,23,23,69),\\ (11,29,39,49,127), & (11,49,69,128,256), & (13,23,35,57,127), & (13,35,81,128,256),\\ (3,4,5,10,20), & (3,4,10,15,30), & (5,13,19,22,57), & (5,13,19,35,70),\\ (6,9,10,13,36), & (7,8,19,25,57), & (7,8,19,32,64), & (9,12,13,16,48),\\ (9,12,19,19,57), & (9,19,24,31,81), & (10,19,35,43,105), & (11,21,28,47,105),\\ (11,25,32,41,107), & (11,25,34,43,111), & (11,43,61,113,226), & (13,18,45,61,135),\\ (13,20,29,47,107), & (13,20,31,49,111), & (13,31,71,113,226), & (14,17,29,41,99),\\ (5,7,11,13,33), & (5,7,11,20,40), & (11,21,29,37,95), & (11,37,53,98,196),\\ (13,17,27,41,95), & (13,27,61,98,196), & (15,19,43,74,148), & (9,11,12,17,45),\\ (10,13,25,31,75), & (11,17,20,27,71), & (11,17,24,31,79), & (11,31,45,83,166),\\ (13,14,19,29,71), & (13,14,23,33,79), & (13,23,51,83,166), & (11,13,19,25,63),\\ (11,25,37,68,136), & (13,19,41,68,136), & (11,19,29,53,106), & (13,15,31,53,106),\\ (11,13,21,38,76). & & & \end{array}\end{eqnarray}$$

In all these cases, the affine Fano variety $X\setminus S_{X}$ is super-rigid by Theorem B.

Example 2.11. With the notation and assumptions of Example 2.10, one has $\unicode[STIX]{x1D6FC}(S_{X})=1$ if $(a_{0},a_{1},a_{2},a_{3},d)$ is one of the following the quintuples:

$$\begin{eqnarray}\begin{array}{@{}ll@{}}(1,3,5,8,16),\quad (2,3,4,7,14), & (5,6,8,9,24),\quad (5,6,8,15,30),\\ (2,2n+1,2n+1,4n+1,8n+4), & (3,3n,3n+1,3n+1,9n+3),\\ (3,3n+1,3n+2,3n+2,9n+6), & (3,3n+1,3n+2,6n+1,12n+5),\\ (3,3n+1,6n+1,9n,18n+3), & (3,3n+1,6n+1,9n+3,18n+6),\\ (4,2n+1,4n+2,6n+1,12n+6), & (4,2n+3,2n+3,4n+4,8n+12),\\ (6,6n+3,6n+5,6n+5,18n+15), & (6,6n+5,12n+8,18n+9,36n+24),\\ (6,6n+5,12n+8,18n+15,36n+30), & (8,4n+5,4n+7,4n+9,12n+23),\\ (9,3n+8,3n+11,6n+13,12n+35), & \end{array}\end{eqnarray}$$

where $n$ is any positive integer. Moreover, we also have $\unicode[STIX]{x1D6FC}(S_{X})=1$ if $S_{X}$ is a general hypersurface of degree $d$ in $\mathbb{P}(a_{0},a_{1},a_{2},a_{3})$ and $(a_{0},a_{1},a_{2},a_{3},d)$ is one of the following quintuples: $(1,1,2,3,6)$ , $(1,2,3,5,10)$ , $(1,3,5,7,15)$ , $(2,3,4,5,12)$ . In all these cases, the affine Fano variety $X\setminus S_{X}$ is super-rigid by Theorem B.

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.

Example 2.12. For the case where $n=4$ , $d=\sum _{i=0}^{4}a_{i}-1$ , Iano-Fletcher verified that there are exactly $95$ quintuples $(a_{0},a_{1},a_{2},a_{3},a_{4})$ that define the hypersurface $S_{X}$ with only terminal singularities and listed such quintuples in [Reference Iano-FletcherIan00]. Moreover, it follows from [Reference Corti, Pukhlikov and ReidCPR00, Reference Cheltsov and ParkCP17] that

$$\begin{eqnarray}\text{Aut}(X\setminus S_{X})=\text{Aut}(X,S_{X}).\end{eqnarray}$$

Furthermore, if $-K_{S_{X}}^{3}\leqslant 1$ and the hypersurface $S_{X}$ is general, then $\unicode[STIX]{x1D6FC}(S_{X})=1$ by [Reference CheltsovChe08b, Reference CheltsovChe09], so that the corresponding affine Fano fourfold $X\setminus S_{X}$ should be super-rigid by 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

$$\begin{eqnarray}X=\mathbb{P}^{n}/G,\end{eqnarray}$$

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

(3.1) $$\begin{eqnarray}X\setminus S_{X}\cong \mathbb{A}^{n}/G.\end{eqnarray}$$

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.

Proof. Since the subgroup of $G$ generated by quasi-reflections is normal, we may assume that  $G$ does not contain non-trivial quasi-reflection. Note that $G$ when considered as a subgroup in  $\text{GL}_{n+1}(\mathbb{C})$ may contain quasi-reflections. These are just elements of $Z$ that are different from identity. To take these quasi-reflexions into account, we consider the commutative diagram

where $\unicode[STIX]{x1D70B}$ is the quotient map by the group $G$ , the morphism $q$ is the quotient map by the group  $Z$ , and $\overline{\unicode[STIX]{x1D70B}}$ is the quotient map by the group $\overline{G}$ . By construction, the finite morphism $q$ is branched over $q(H)$ , which is a smooth hypersurface in $\mathbb{P}(1^{n},m)$ of degree $n$ , that does not contain the singular point of the weighted projective space $\mathbb{P}(1^{n},m)$ . Moreover, the finite morphism $\overline{\unicode[STIX]{x1D70B}}$ is unramified in codimension one. This implies that the divisor

$$\begin{eqnarray}-(K_{X}+S_{X})\end{eqnarray}$$

is ample, and $(X,S_{X})$ has purely log terminal singularities. Thus, the quotient (3.1) is an affine Fano variety as desired.◻

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

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}(S_{X},\text{Diff}_{S_{X}}(0))=\unicode[STIX]{x1D6FC}_{\overline{G}}(H).\end{eqnarray}$$

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.

Proof. The first implication follows from Theorem B and (3.3). For the second one, suppose on the contrary that $\mathbb{V}$ splits as the direct sum of two non-trivial representations

$$\begin{eqnarray}\mathbb{V}=\mathbb{V}_{1}\oplus \mathbb{V}_{2}.\end{eqnarray}$$

Then the projection $\text{pr}_{1}:\mathbb{V}\rightarrow \mathbb{V}_{1}$ descends to a fibration $\unicode[STIX]{x1D70C}:X\setminus S_{X}\rightarrow \mathbb{V}_{1}/G$ , whose general fibers are isomorphic to $\mathbb{V}_{2}/G^{\prime }$ , where $G^{\prime }\subset G$ denotes the stabilizer of a general fiber of $\text{pr}_{1}$ . Since $\mathbb{V}_{2}/G^{\prime }$ is an affine Fano variety by Lemma 3.2, we see that the generic fiber of $\unicode[STIX]{x1D70C}$ is an affine Fano variety. In other words, $\unicode[STIX]{x1D70C}:X\setminus S_{X}\rightarrow \mathbb{V}_{1}/G$ is a relative affine Fano variety in contradiction to the super-rigidity hypothesis.◻

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 $4$ .

Proof. We have $S_{X}\cong H\cong \mathbb{P}^{1}$ . For every point $P\in S_{X}$ , denote by $n_{P}$ the order of the stabilizer in $\overline{G}$ of any point on $H$ that is mapped to $P$ by the quotient map $\unicode[STIX]{x1D70B}:\mathbb{P}^{2}\rightarrow X$ . Then

$$\begin{eqnarray}\text{Diff}_{S_{X}}(0)=\mathop{\sum }_{P\in S_{X}}\frac{n_{P}-1}{n_{P}}P.\end{eqnarray}$$

Note that $\text{Diff}_{X}(0)\neq 0$ provided that $\overline{G}$ is not trivial. Let $Q$ be the point in $S_{X}$ with the largest  $n_{Q}$ . Then it follows from [Reference Cheltsov and ShramovCS11, Example 3.3] that

Furthermore, since we assumed that $G$ is small, it is abelian if and only if it is cyclic. Finally, observe that $\unicode[STIX]{x1D6FC}_{\overline{G}}(H)$ is the length of the smallest $\overline{G}$ -orbit in $H$ divided by 2, so that

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}(S_{X},\text{Diff}_{S_{X}}(0))=\unicode[STIX]{x1D6FC}_{\overline{G}}(H)=\frac{|\overline{G}|}{2n_{Q}}\end{eqnarray}$$

by virtue of (3.3). Now the conclusion follows from Lemma 3.4 together with the fact that the representation $\mathbb{V}$ splits when $G$ is cyclic.◻

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.

Theorem 3.8. Let $S\setminus C$ be an affine Fano variety of dimension $2$ with completion $S$ and boundary $C$ . Then $S\setminus C$ is super-rigid if and only if the pair $(S,C)$ is isomorphic to a pair

$$\begin{eqnarray}(\mathbb{P}^{2}/G,H/\overline{G})\end{eqnarray}$$

for some non-cyclic small finite subgroup $G\subset \text{GL}_{2}(\mathbb{C})$ .

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.

Example 3.9. Suppose that $n=3$ . Let $G=\mathfrak{A}_{5}$ and let $\mathbb{V}$ be an irreducible three-dimensional representation of $G$ . Then $\overline{G}\cong G$ , and the center $Z$ is trivial. Moreover, there exists a $\overline{G}$ -invariant smooth conic $C$ in $H$ . Let $\unicode[STIX]{x1D70B}:W\rightarrow \mathbb{P}^{3}$ be the blow up of the conic $C$ with exceptional divisor $E$ , and let $\widetilde{H}$ be the proper transform of the plane $H$ on the threefold $W$ . Then there exists a $G$ -equivariant commutative diagram

where $Q$ is a smooth quadric threefold in $\mathbb{P}^{4}$ , the morphism $\unicode[STIX]{x1D702}$ is the contraction of the surface $\widetilde{H}$ to a smooth point of $Q$ , and $\unicode[STIX]{x1D713}$ is a linear projection from this point. Then $(Q,\unicode[STIX]{x1D702}(E))$ is purely log terminal and $-(K_{Q}+\unicode[STIX]{x1D702}(E))$ is ample, so that $(Q,\unicode[STIX]{x1D702}(E))$ is an affine Fano variety. By construction, we have $Q\setminus \unicode[STIX]{x1D702}(E)\cong \mathbb{P}^{3}\setminus H$ . Let $Y=Q/G$ and $S_{Y}=\unicode[STIX]{x1D702}(E)/G$ . Then $(Y,S_{Y})$ is an affine Fano variety and $Y\setminus S_{Y}\cong X\setminus S_{X}$ , so that $(X,S_{X})$ could not be super-rigid.

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.

Proof. By [Reference Cheltsov and ShramovCS11, Theorem 4.3], Theorem B and (3.3), it is enough to show that (1) implies (3). If the group $\overline{G}$ is not transitive, then $X\setminus S_{X}$ is not super-rigid by Lemma 3.4. Thus, we may assume that $\overline{G}$ is transitive. Let us show that $X\setminus S_{X}$ is not super-rigid in the case where the hyperplane $H$ contains one of the following $\overline{G}$ -invariant subvarieties: a quadric surface, a cubic surface, or a twisted cubic curve.

Suppose first that the hyperplane $H$ contains a $\overline{G}$ -invariant surface $S_{d}$ of degree $d=2$ or $3$ . Because $\overline{G}$ is transitive, this surface must be smooth. This is obvious in the case when $d=2$ while in the case when $d=3$ , the conclusion follows from the classification of singular cubic surfaces.

We may assume that $H$ is given by $u=0$ , and $S_{d}$ is defined by

where $f_{d}(x,y,z,w)$ is a homogeneous polynomial of degree $d$ , and $x$ , $y$ , $z$ , $w$ , $u$ are homogeneous coordinates on $\mathbb{P}^{4}$ . Let $V$ be a hypersurface in $\mathbb{P}(1^{5},d-1)$ given by

$$\begin{eqnarray}ut=f_{d}(x,y,z,w),\end{eqnarray}$$

where $x$ , $y$ , $z$ , $w$ , $u$ , $t$ are quasi-homogeneous coordinates on $\mathbb{P}(1^{5},d-1)$ such that $t$ is a coordinate of weight $d-1$ . Then there exists a $G$ -equivariant commutative diagram

where $\unicode[STIX]{x1D70B}$ is the blow up of the surface $S_{d}$ with exceptional divisor $E$ , the morphism $\unicode[STIX]{x1D702}$ is the contraction of the proper transform of the hyperplane $H$ to the point $[0:0:0:0:0:1]$ , and $\unicode[STIX]{x1D713}$ is the linear projection from this point. Then $\unicode[STIX]{x1D702}(E)$ is cut out on $V$ by $u=0$ . Let $Y=V/G$ and $S_{Y}=\unicode[STIX]{x1D702}(E)/G$ . Then $(Y,S_{Y})$ is an affine Fano variety and $Y\setminus S_{Y}\cong X\setminus S_{X}$ . In particular, we see that $X\setminus S_{X}$ is not super-rigid.

Next suppose that $H$ contains a $\overline{G}$ -invariant twisted cubic curve $C$ . Then the group $\overline{G}$ acts faithfully on the curve $C$ . Using the classification of finite subgroups in $\text{PGL}_{2}(\mathbb{C})$ , we see that $\overline{G}$ is isomorphic to one of the three groups $\mathfrak{A}_{4}$ , $\mathfrak{S}_{4}$ , $\mathfrak{A}_{5}$ , because we assumed that $\overline{G}$ is transitive. Let $\unicode[STIX]{x1D70E}:U\rightarrow \mathbb{P}^{4}$ be the blow up of the curve $C$ with exceptional divisor $F$ , and denote by $\widetilde{H}$ the proper transform of the hyperplane $H$ on the fourfold $U$ . Then there exists a $G$ -equivariant morphism $\unicode[STIX]{x1D710}:\widetilde{H}\rightarrow \mathbb{P}^{2}$ that is a $\mathbb{P}^{1}$ -bundle (see [Reference Szurek and WiśniewskiSW90, Application 1]). Its fibers are the proper transforms of the secants of the curve $C$ . It follows from [Reference Prokhorov and ZaidenbergPZ16, Proposition 3.9] that there exists a $G$ -equivariant commutative diagram

where $W_{5}$ is a smooth del Pezzo fourfold in $\mathbb{P}^{7}$ of degree $5$ , the morphism $\unicode[STIX]{x1D708}$ is a contraction of the proper transform of $\widetilde{H}$ to a plane in $W_{5}$ such that the restriction $\unicode[STIX]{x1D708}|_{\widetilde{H}}:\widetilde{H}\rightarrow \unicode[STIX]{x1D708}(\widetilde{H})$ is the $\mathbb{P}^{1}$ -bundle $\unicode[STIX]{x1D710}$ , and $\unicode[STIX]{x1D719}$ is the linear projection from the plane $\unicode[STIX]{x1D708}(\widetilde{H})$ . Then $\unicode[STIX]{x1D708}(F)$ is a singular hyperplane section of the fourfold $W_{5}$ such that the pair $(W_{5},\unicode[STIX]{x1D708}(F))$ has purely log terminal singularities. As above, we let $M=W_{5}/G$ and $S_{M}=\unicode[STIX]{x1D708}(F)/G$ . Then $(M,S_{M})$ is an affine Fano variety and $M\setminus S_{M}\cong X\setminus S_{X}$ , so that $X\setminus S_{X}$ could not be super-rigid.◻

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

$$\begin{eqnarray}\mathbb{P}=\text{Proj}(\mathbb{C}[x,y,z,w]),\end{eqnarray}$$

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 $\mathbb{P}$ .

Theorem 4.1. Suppose that $S$ is smooth. If its anticanonical degree is $1$ , then

$$\begin{eqnarray}\text{Aut}(\mathbb{P}\setminus S)=\text{Aut}(\mathbb{P},S).\end{eqnarray}$$

In particular, $\text{Aut}(\mathbb{P}\setminus S)$ is a finite group. If its anticanonical degree is either $2$ or $3$ , then $\text{Aut}(\mathbb{P}\setminus S)$ does not contain non-trivial connected algebraic groups.

Proof. We follow the same strategy as [Reference Dubouloz and KishimotoDK15, §2.2], which consists in shifting the question to a suitable finite étale cover of $\mathbb{P}\setminus S$ . Namely, let $\unicode[STIX]{x1D70B}:V\rightarrow \mathbb{P}$ be a cyclic Galois cover of degree $d$ branched along $S$ and étale elsewhere. Then $V$ is a smooth Fano threefold of index $2$ , isomorphic to a hypersurfaces of degree $d$ in $\mathbb{P}^{4}$ , $\mathbb{P}(1,1,1,1,2)$ and $\mathbb{P}(1,1,1,2,3)$ in the cases $K_{S}^{2}=3$ , $2$ and $1$ , respectively. Furthermore, the ramification divisor of $\unicode[STIX]{x1D70B}$ coincides with a hyperplane section $H$ of $V$ . We then have a split exact sequence of groups

$$\begin{eqnarray}0\rightarrow \text{Aut}(V\setminus H,\unicode[STIX]{x1D70B})\rightarrow \text{Aut}(V\setminus H)\rightarrow \text{Aut}(\mathbb{P}\setminus S)\rightarrow 0,\end{eqnarray}$$

where $\text{Aut}(V\setminus H,\unicode[STIX]{x1D70B})\cong \mathbb{Z}/d\mathbb{Z}$ denotes the group of the deck transformations of the induced étale Galois cover $\unicode[STIX]{x1D70B}|_{V\setminus H}:V\setminus H\rightarrow \mathbb{P}\setminus S$ . The surjectivity of the right hand side homomorphism follows from the fact that

$$\begin{eqnarray}\unicode[STIX]{x1D70B}_{\ast }{\mathcal{O}}_{V\setminus H}\cong \bigoplus _{i=0}^{d-1}(\unicode[STIX]{x1D714}_{\mathbb{ P}\setminus S}^{-1})^{\otimes i},\end{eqnarray}$$

where $\unicode[STIX]{x1D714}_{\mathbb{P}\setminus S}^{-1}=\unicode[STIX]{x1D714}_{\mathbb{P}}^{-1}|_{\mathbb{P}\setminus S}\cong {\mathcal{O}}_{\mathbb{P}}(d+1)|_{\mathbb{P}\setminus S}\cong {\mathcal{O}}_{\mathbb{P}}(1)|_{\mathbb{P}\setminus S}$ denotes the anticanonical sheaf of the variety $\mathbb{P}\setminus S$ .

If $K_{S}^{2}=1$ , then $\text{Bir}(V)=\text{Aut}(V)$ by [Reference GrinenkoGri03, Reference GrinenkoGri04], so that $\text{Aut}(V\setminus H)=\text{Aut}(V,H)$ . This implies in turn that $\text{Aut}(\mathbb{P}\setminus S)=\text{Aut}(\mathbb{P},S)$ , which is a finite group.

If $K_{S}^{2}=2$ or $K_{S}^{2}=3$ , then $V$ is not rational by [Reference VoisinVoi88, Corollaire 4.8] and [Reference Clemens and GriffithsCG72], respectively. Now suppose that $\text{Aut}(\mathbb{P}\setminus S)$ contains a non-trivial connected algebraic subgroup $G$ . First note that $G$ is necessarily an affine algebraic group. Indeed, otherwise by Chevalley’s theorem, there would exist a maximal proper normal linear algebraic subgroup $G^{\prime }$ of $G$ such that $G/G^{\prime }$ is abelian variety, and so the affine variety $\mathbb{P}\setminus S$ would inherit a non-trivial action of a proper connected algebraic group, which is absurd. Since it is affine and connected, $G$ thus contains either $\mathbb{G}_{m}$ or $\mathbb{G}_{a}$ , which implies in turn that $V\setminus H$ admits a non-trivial action of either $\mathbb{G}_{m}$ or $\mathbb{G}_{a}$ . By virtue of Rosenlicht’s theorem [Reference RosenlichtRos56], it would follow that $V$ is birational to $Z\times \mathbb{P}^{1}$ for some rational surface $Z$ . But then, the threefold $V$ itself would be rational, which is a contradiction.◻

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 $K_{S}^{2}=1$

$$\begin{eqnarray}g(y,z,w)=ay^{2}w~(\text{or }zw)+f_{5}(y,z),\end{eqnarray}$$

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

$$\begin{eqnarray}g(y,z,w)=ayw+f_{3}(y,z),\end{eqnarray}$$

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

$$\begin{eqnarray}g(y,z,w)=f_{2}(y,z,w).\end{eqnarray}$$

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.

Proof of Theorem C.

Lemma 4.5 immediately implies that the projection $\unicode[STIX]{x1D70B}$ is a birational map. Moreover, it induces an isomorphism

$$\begin{eqnarray}\tilde{\unicode[STIX]{x1D70B}}:S\setminus (L_{1}\cup \cdots \cup L_{r})\rightarrow \text{Im}(\tilde{\unicode[STIX]{x1D70B}})\subset \unicode[STIX]{x1D6F1}.\end{eqnarray}$$

Let ${\mathcal{C}}$ be the curve on $\unicode[STIX]{x1D6F1}$ defined by

$$\begin{eqnarray}g(y,z,w)=0.\end{eqnarray}$$

Note that this can be reducible or non-reduced.

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

$$\begin{eqnarray}S\setminus (H\cup L_{1}\cup \cdots \cup L_{r})\cong \unicode[STIX]{x1D6F1}\setminus ({\mathcal{C}}\cup \ell )\cong (\mathbb{A}^{1}\setminus \{0\})\times \mathbb{A}^{1},\end{eqnarray}$$

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

$$\begin{eqnarray}aw+f_{3}(1,z)=0\end{eqnarray}$$

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

$$\begin{eqnarray}w+f_{5}(y,1)=0\end{eqnarray}$$

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

$$\begin{eqnarray}S\setminus ({\mathcal{H}}\cup {\mathcal{Q}})\cong \unicode[STIX]{x1D6F1}\setminus ({\mathcal{C}}\cup \ell )\end{eqnarray}$$

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

$$\begin{eqnarray}\unicode[STIX]{x1D6FC}f(x,y,z,w)+\unicode[STIX]{x1D6FD}yg(y,z,w)=0,\end{eqnarray}$$

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

$$\begin{eqnarray}f(x,y,z,w)+\unicode[STIX]{x1D706}yg(y,z,w)=0\end{eqnarray}$$

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

$$\begin{eqnarray}{\mathcal{H}}_{\unicode[STIX]{x1D702}}=S_{\unicode[STIX]{x1D702}}\cap \{y=0\}\end{eqnarray}$$

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

$$\begin{eqnarray}{\mathcal{Q}}_{\unicode[STIX]{x1D702}}=S_{\unicode[STIX]{x1D702}}\cap \{g(y,z,w)=0\}.\end{eqnarray}$$

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.