Cylinders in Fano varieties

This paper is a survey about cylinders in Fano varieties and related problems.


Introduction
A cylinder in a projective variety X is a Zariski open subset U ⊂ X such that for an affine variety Z. If X contains a cylinder, we say that X is cylindrical. Since cylindrical varieties have negative Kodaira dimension, we will focus our attention on cylindrical Fano varieties, because they are building blocks of projective varieties with negative Kodaira dimension. Example 1.1. For positive integers m, n with m < n, let X be the Grassmannian Gr(m, n) of m-dimensional subspaces of an n-dimensional vector space over k. Then X is a smooth projective variety of dimension m(n − m), and −K X ∼ nH, where H is an ample generator of the group Pic(X). Since X contains an open Schubert cell isomorphic to A m(n−m) , it is a cylindrical Fano variety.
α(X) = sup λ ∈ Q the log pair (X, λD) is log canonical for any effective Q-divisor D ∼ Q −K X .
This number plays an important role in K-stability of Fano varieties, since X is K-stable if α(X) > dim(X) dim(X) + 1 .
On the other hand, we have the following result.
Theorem 1. 26. Let X be a Fano variety that has at most Kawamata log terminal singularities. If α(X) 1, then X does not contain (−K X )-polar cylinders.
Proof. Suppose X contains a (−K X )-polar cylinder. Then U ∼ = Z × A 1 for an affine variety Z, and for some effective Q-divisor D on X such that D ∼ Q −K X . Arguing as in the proof Corollary 2.7, we see that the log pair (X, D) is not log canonical in this case, so that α(X) < 1.
Let us show how to use this obstruction.
Example 1. 27. Let X be a del Pezzo surface with Du Val singularities of degree K 2 X = 1 such that one of the following two conditions holds: (1) either X has 2 singular points of type A 3 and 2 singular points of type A 1 ; (2) or the surface X has 4 singular points of type A 2 . By [217,Theorem 1.2], the surface X exists, and it is uniquely determined by its singularities. Moreover, it follows from [217, Table 4.1] that the pencil | − K X | contains exactly 4 singular fibers. They are singular fibers of types I 4 and I 2 (in the first case) or of types I 2 (in the second case). This gives α(X) = 1 by [34,Theorem 1.25], so that X contains no anticanonical polar cylinders. Since the group Cl(X) is of rank 1, the surface X contains no cylinders at all. Remark 1.28. Implicitly, Theorem 1.26 has been already used by many people for quite some time. For instance, Miyanishi conjectured in [94] that the smooth locus of a del Pezzo surface with quotient singularities and Picard rank 1 admits a finite unramified covering that contains a cylinder. It turned out to be wrong. Namely, in [122,Example 21.3.3], Keel and McKernan have constructed a singular del Pezzo surface X with quotient singularities such that ρ(X) = 1 and α(X) 1, but its smooth locus has trivial algebraic fundamental group. Thus, its smooth locus does not admit non-trivial unramified coverings, and X does not contain cylinders by Theorem 1.26.
Using Theorem 1.26, we can create many rational Fano varieties without anticanonical polar cylinders. Indeed, if X and Y are Fano varieties that have Kawamata log terminal singularities, then it follows from [44,Lemma 2.29] and [136,Proposition 8.11] that α X × Y = min α(X), α(Y ) .
Thus, if S is a general smooth del Pezzo surface with K 2 S = 1, then α(S) = 1 by [30,Theorem 1.7], which implies that we also have α(X) = 1 for the 2n-dimensional smooth Fano variety X = S × S × · · · × S n times , so that X does not contain (−K X )-polar cylinders, but X is cylindrical, because S is cylindrical. We can construct many similar examples using [34,36,37,46,197].
Surprisingly, we do not know a single example of a cylindrical smooth Fano threefold that contains no anticanonical polar cylinder (cf. Examples 3.14, 3.15, 3.16 and 3.17). Problem 1.30. Find a cylindrical smooth Fano threefold without anticanonical polar cylinder.
Note that there are Fano varieties without cylinders whose α-invariant of Tian is smaller than 1. For instance, if X is the del Pezzo surface from Example 1.2, then α(X) = 1 2 by [34,Theorem 1.25]. On the other hand, this surface does not contain cylinders [39]. Note that it is K-polystable [165]. Surprisingly, all known K-unstable Fano varieties are also cylindrical. Example 1.31 ([70-72, 114]). Let X be a smooth Fano variety of dimension n 2 such that where H is an ample divisor such that H n = 5. Then n ∈ {2, 3, 4, 5, 6}, and X is unique for each n. The divisor H is very ample, and the linear system |H| gives an embedding X ֒→ P n+3 such that the image is a section of the Grassmannian Gr(2, 5) ⊂ P 9 by a linear subspace of dimension 3 + n. Moreover, if n = 2, then Pic(X) = Z [H]. Furthermore, the following assertions hold.
• The variety X contains a Zariski open subset isomorphic to A n , so that it is cylindrical. If n = 5, this follows from Example 1.1 and Theorems 3.6 and 3.20 (see also [71,189]). If n = 5, then X contains a plane Π such that there exists the following Sarkisov link: where α is the blowup of the plane Π, and β is the blowup of a smooth cubic scroll in P 5 . This easily implies that X contains a Zariski open subset isomorphic to A 5 . • If n ∈ {2, 3, 6}, then X is known to be K-polystable (see, for example, [30,45,173,209,219]).
Keeping in mind Theorem 1. 26 and examples of K-stable Fano varieties without anticanonical polar cylinders (for example, smooth del Pezzo surfaces of degree 1, 2 and 3), we pose On the other hand, if K 2 X 3, then −K X ∈ Amp cyl (X). This gives an evidence for Conjecture 1. 34. If X is a Fano variety, then −K X ∈ Amp cyl (X) ⇐⇒ Amp cyl (X) = Amp(X).
Let us describe the structure of this survey. In Section 2 we review results about polar cylinders in rational surfaces. In Section 3, we describe results about cylinders in smooth Fano threefolds, smooth Fano fourfolds, and del Pezzo fibrations. In Section 4, we survey results on three topics that are closely related to the main topic of this survey: flexibility of affine varieties with a special accent on the flexibility of affine cones over Fano varieties, cylinders in the complements to hypersurfaces in weighted projective spaces, and compactifications of C n . Finally, in Appendix A, we present some results about singularities of two-dimensional log pairs, which are used in Section 2 to prove the absence of polar cylinders in some del Pezzo surfaces.
Notations. Throughout this paper, we will use the following notation: • µ n is a cyclic subgroup of order n; • G a is a one-dimensional unipotent additive group; • G m is a one-dimensional algebraic torus; • F n is the Hirzebruch surface; • P n is the n-dimensional projective space over k; • A n is the n-dimensional affine space over k; • P(a 1 , . . . , a n ) is the weighted projective space; • for a variety X, we denote by ρ(X) the rank of its Picard group.
A Gorenstein del Pezzo surface is a del Pezzo surface whose anticanonical divisor is Cartier, equivalently a del Pezzo surface with only Du Val singularities. Such surface is either a quadric, or its minimal resolution of singularities can be obtained by blowing up P 2 in at most 8 points such that at most 3 of them are contained in a line, and at most 6 of them are contained in a conic. First, let us go over basic facts about cylinders in rational surfaces.
2.1. Cylinders in rational surfaces. Observe that every smooth rational surface is cylindrical. This immediately follows from the fact that P 2 contains a cylinder and the following Lemma 2.1. Let C be an irreducible curve in F n that is a section of the natural projection F n → P 1 , and let F 1 , . . . , F r be fibers of this projection, where r 1. Then F n \ (C ∪ F 1 ∪ · · · F r ) is a cylinder.
Proof. Performing appropriate elementary birational transformations, we may assume that C 2 = 0, so that n = 0. In this case, the required assertion is obvious.
However, as we have seen already in Example 1.2, there are singular rational surfaces that contain no cylinders. Let us explain how to find many such rational surfaces and provide an obstruction for the existence of cylinders (see Remark 2.3 below), which will be used in Section 2.2 to show the absence of anticanonical polar cylinders in smooth del Pezzo surfaces of degree 1, 2 and 3.
Let S be a rational surface with quotient singularities and suppose that S contains a cylinder U. Then U is a Zariski open subset in S such that U ∼ = A 1 × Z for some affine curve Z. We then have the following commutative diagram x r r r r r r r r r r r r r r r r r r r r r r r r r r r r r where p Z , p 2 and p 2 are the natural projections to the second factors, ψ is the rational map induced by p Z , π is a birational morphism resolving the indeterminacy of ψ and ϕ is a morphism. By construction, a general fiber of ϕ is P 1 . Let C 1 , . . . , C n be the irreducible curves in S such that The curves C 1 , · · · , C n generate the divisor class group Cl(S) of the surface S, because Cl(U) = 0. In particular, one has Let E 1 , . . . , E r be all exceptional curves of the morphism π (if any), and let Γ = P 1 ×P 1 \A 1 ×P 1 . Denote by C 1 , . . . , C n and Γ the proper transforms S of the curves C 1 , . . . , C n and Γ, respectively. Then Γ is a section of the conic bundle ϕ, and Γ is one of the curves C 1 , . . . , C n and E 1 , . . . , E r . Moreover, all other curves among C 1 , . . . , C n and E 1 , . . . , E r are components of some fibers of ϕ. Thus, we may assume that either Γ = C 1 or Γ = E r . Then ψ is a morphism ⇐⇒ Γ = C 1 .
We are mostly interested in cylinders in del Pezzo surfaces. Applying our Remark 2.3 to them, we obtain the following special case of Theorem 1.26, which we already applied in Example 1.27.
Proof. There exists an effective Q-divisor D ′ on the surface S such that D ′ ∼ Q −K S and for some positive rational numbers a 1 , . . . , a n . Let D = D ′ . Then K S + D ∼ Q 0 is pseudo-effective, so that (S, D) is not log canonical by Remark 2.3, which implies that α(S) < 1. Now, we state main result of this section, which implies negative answer to Question 1.12.
Theorem 2.8 ( [38,39,123,126]). Let S be a del Pezzo surface that has at most Du Val singularities. Then S does not contain (−K S )-polar cylinders exactly when • K 2 S = 1 and S allows at most singular points of types A 1 , A 2 , A 3 , D 4 if any; • K 2 S = 2 and S allows at most singular points of type A 1 if any; • K 2 S = 3 and S is smooth. Corollary 2.9. A smooth del Pezzo surface S contains a (−K S )-polar cylinder ⇐⇒ K 2 S 4.
In the next two subsections, we will explain how to prove Theorem 2.8. Now let us use this result to find all del Pezzo surfaces with Du Val singularities that contain no cylinder. We may assume that ρ(S) 2. Let us seek for a contradiction. Since every smooth rational surface contains a cylinder, we see that S is singular. Then K 2 S 2 by Theorem 2.8. Let π : S → Y be the contraction of an extremal ray of the Mori cone NE(S) of the surface S. Then it follows from [161] that one of the following cases hold: • either π is a conic bundle, Y = P 1 and ρ(S) = 2; • or π is birational, Y is a del Pezzo surface with Du Val singularities, ρ(Y ) = ρ(S) + 1; the morphism π is a weighted blowup of a smooth point in Y with weights (1, k) for k 1, and K 2 Y = K 2 S + k. Suppose that π is a conic bundle. Then we have the following commutative diagram: where α is a minimal resolution of singularities, β is a birational map, and F n → P 1 is a natural projection. On the other hand, it follows from Tsen's theorem that S contains a smooth irreducible curve Z that is a section of the conic bundle π. Let C be its proper transform on F n . Then where T 1 , . . . , T r are fibers of π that contain singular points of the surface S, and F 1 , . . . , F r are fibers of the projection F n → P 1 over the points π(T 1 ), . . . , π(T r ), respectively. Then S contains a cylinder by Lemma 2.1, which is a contradiction. We see that π is birational. Let E be the π-exceptional curve. If Y contains a cylinder U, then it also contains a cylinder U ′ ⊂ U such that π(E) ∈ U ′ , so that its preimage in S is a cylinder as well. Thus, the surface Y does not contain cylinders. Then Y is singular and K 2 Y 2 by Theorem 2.8. We see that K 2 Y = 2 and π is a blowup of a smooth point in Y . If ρ(Y ) 2, then we can apply the same arguments to Y to show that it contains a cylinder. Hence, we conclude that ρ(Y ) = 1. On the other hand, all singularities of the surface Y are ordinary double points by Theorem 2.8. We see that K 2 Y = 2 and Y has 7 singular points of type A 1 . But such a surface does not exist. Let us conclude this subsection by presenting few results about polar cylinders in arbitrary rational surfaces. To do this, fix an ample Q-divisor H on the surface S. If S contains an H-polar cylinder, we say that H is cylindrical. The cylindrical ample Q-divisors on S form a cone, which we denoted earlier by Amp cyl (S). To investigate this cone, consider the following number: Remark 2.11. The number µ H is known as the Fujita invariant of the divisor H, because it was implicitly used by Fujita in [73][74][75][76]. It plays an essential role in Manin's conjecture (see [14,98] 3, and the self-intersection of every smooth rational curve in S is at least −1. Then S does not contain H-polar cylinders. Note that if S is smooth del Pezzo surface, then the self-intersection of every smooth rational curve in S is at least −1. Moreover, it follows from [54,Proposition 2.4] that this condition also holds if S is obtained by blowing up P 2 at any number of points in general position. On the other hand, we have the following complimentary result: Theorem 2.14 ( [40,150]). Suppose that S is a smooth rational surface. If K 2
If K 2 S = 3 and −K S is not ample, then Amp cyl (S) = Amp(S). If K 2 S = 3 and −K S is ample, then If S is a smooth rational surface and K 2 S 2, then Amp cyl (S) is poorly understood (see [40]).

2.2.
Absence of polar cylinders. Now, we show that smooth del Pezzo surfaces of degree 3 does not contain any anticanonical polar cylinders, which is one way implication of Corollary 2.9. For singular del Pezzo surfaces of degree 2 with types of singular points listed in Theorem 2.8, the same implication can be verified in a similar way (see [39] for the details).
Let S be a smooth del Pezzo surface of degree K 2 S = d 3, and let D be an effective Q-divisor on the surface S, i.e., we have where every C i is an irreducible curve on S, and every a i is a non-negative rational number. Suppose that D ∼ Q −K S . If d ∈ {2, 3}, then each a i does not exceed 1 by Lemmas A.9 and A.10. Similarly, if d = 1, we have which immediately implies that a i 1 for each i.
Theorem 2.15. Let P be a point in S. Suppose that the log pair (S, D) is not log canonical at P . Then there exists a curve T ∈ | − K S | such that • the curve T is singular at P ; • the log pair (S, T ) is not log canonical at P ; Proof. We consider the cases d = 1, d = 2 and d = 3 separately. See the proof of [38,Theorem 1.12] for an alternative proof in the case d = 3. Suppose that K 2 S = 1. Let C be a curve in | − K S | that passes through P . Then C is irreducible. If C is not contained in the support of D, then it follows from Lemma A.3 that This shows that C ⊂ Supp(D). If (S, C) is not log canonical at P , then we can put T = C and we are done. Thus, we may assume that (S, C) is log canonical at P . Then Remark A.2 implies the existence of an effective Q-divisor D ′ such that D ′ ∼ Q −K S , the curve C is not contained in the support of D ′ , and (S, D ′ ) is not log canonical at P . Now Lemma A.3 implies that Now, we suppose that K 2 S = 2. In this case there exists a double cover τ : S → P 2 branched over a smooth quartic curve C. Moreover, we have where L is a line in P 2 . By Lemma A.10, we have τ (P ) ∈ C. Now let us choose L to be the tangent line to C at the point τ (P ), and let R be the curve in | − K S | such that τ (R) = L. Then mult P (R) = 2. If R is irreducible and is not contained in the support of D, then Lemma A.3 gives Note that either R is irreducible or R consists of two (−1)-curves that both pass through P . Therefore, if one component of the curve R is not contained in the support of the divisor D, then we obtain a contradiction in a similar way by intersecting D with this irreducible component. Thus, we may assume that all irreducible component of the curve R are contained in Supp(D). Now we can use Remark A.2 as in the case d = 1 to conclude that (S, R) is not log canonical at P . Hence, we can let T = R.
Finally, we suppose that K 2 S = 3. Then S is a smooth cubic surface in P 3 , and −K S is rationally equivalent to its hyperplane section. Let T P be the intersection of the surface S with the hyperplane that is tangent to S at the point P . Then T P is a reduced cubic curve that is singular at P . If (S, T P ) is not log canonical at P and Supp(T P ) ⊆ Supp(D), we can let T = T P and we are done. Therefore, we may assume that at least one of the following two conditions hold: (1) the log pair (S, T P ) is log canonical at P ; (2) Supp(D) does not contain at least one irreducible components of the curve T P .
To obtain a contradiction, we may assume by Remark A.2 that at least one irreducible component of the curve T P is not contained in Supp(D). If L P is a line that passes through P , then L P ⊆ Supp(D), since otherwise we would get 1 D · L P mult P (D)mult P (L P ) mult P (D) > 1 by Lemma A.3. Thus, we see that mult P (T P ) = 2. Let f : S → S be the blowup of the point P , let E be the exceptional curve of the blowup f , and let D be the proper transform on S of the Q-divisor D. Then mult P (D) > 1 by Lemma A.3. Moreover, if follows from Lemma A.5 that the log pair where ψ is a projection from P , the morphism g is a contraction of the proper transforms of all lines in S that pass through P , and h is a double cover branched over a quartic curve. This quartic curve has at most two ordinary double points, because mult P (T P ) = 3.
Let T P be the proper transform on S of the curve T P . Then Q ∈ E ∩ T P by Lemma A.10. Note that T P is one of the following curves: an irreducible cubic curve, a union of a conic and a line, a union of three lines. Let us consider these cases separately.
Suppose that T P is a union of a conic and a line, so that T P = L P + C P , where L P is a line, and C P is an irreducible conic. Then L P ⊂ Supp(D), so that C P is not contained in Supp(D). Thus, we write D = aL P + Ω, where a ∈ Q >0 , and Ω is an effective Q-divisor on S whose support contains none of the curves L P and C P . Put m = mult P (Ω). Then mult P (D) = m + a and 2 − 2a = Ω · C P m, which gives m + 2a 2. Similarly, we obtain 1 + a m by using 1 + a = L P · D = Ω · L P m.
Denote by C P the proper transform of the conic C P on the surface S, denote by L P the proper transform of the line L P on the surface S, and denote by Ω the proper transform of the divisor Ω on the surface S. Put m = mult Q ( Ω). Then the log pair (2.16) S, a L P + Ω + (m + a − 1) E is not log canonical at P . Now, applying Lemma A.3 to this log pair, we obtain 2a + m + m > 2.
On the other hand, if Q ∈ C P , then so that Q ∈ C P . Since Q ∈ T P , we see that Q ∈ L P . Then we have 1 + a − m = Ω · L P m, so that 2 1 + a m + m 2 m, which gives m 1. Thus, we can apply Theorem A.8 to the log pair (2.16) at the point Q. This gives so that we get 3a + m > 3 or 2a + m > 2, which is impossible since a 1 and m + 2a 2. Therefore, we conclude that the curve T P a union of three lines. Hence, we have T P = L 1 +L 2 +L 3 , where L 1 , L 2 , L 3 are lines in S such that P = L 1 ∩ L 2 and P ∈ L 3 . Then L 1 ⊂ Supp(D) ⊃ L 2 . Therefore, we can write D = a 1 L 1 + a 2 L 2 + ∆, where a 1 and a 2 are some positive rational numbers, and ∆ is an effective Q-divisor whose support does not contain L 1 and L 2 . Put m = mult P (∆). Then because L 1 · L 2 = 1 and L 2 1 = −1 on the surface S. Similarly, we see that This gives m 1. Thus, we can apply Theorem A.8 to the log pair (S, D) at the point P . Then which implies that a 1 + a 2 > 1. On the other hand, we have which implies that a 1 + a 2 1. The obtained contradiction completes the solution.
We now claim that a smooth del Pezzo surface of degree d 3 cannot contain a (−K S )-cylinder. If d 2, the claim is [ Suppose that S contains a (−K S )-polar cylinder U. Then S \ U = C 1 ∪ · · · ∪ C n for some irreducible curves C 1 , . . . , C n in S, and there are positive rational numbers λ 1 , . . . , λ n such that Put D = λ 1 C 1 + · · · + λ n C n . Then (S, D) is not log canonical at some point P ∈ S by Remark 2.3. Hence, by Theorem 2.15, there exists a curve T ∈ | − K S | such that Then D = T , because n > 3 by (2.2), and T does not have more than d 3 irreducible components.
Thus, there exists a rational number µ > 0 such that (1 + µ)D − µT is effective, and its support does not contain at least one irreducible component of the curve T . Then (S, (1 + µ)D − µT ) is not log canonical at P by Remark 2.3, which contradicts to Theorem 2.15, since

4.
Then the surface S contains a (−K S )-polar cylinder.
Proof. We may assume that S = P 1 × P 1 . Then there exists a birational map σ : S → P 2 that blows up k 5 distinct points. Let E 1 , . . . , E k be the σ-exceptional curves, let C be an irreducible conic in P 2 that contains all points σ(E 1 ), . . . , σ(E k ), and let L be a line in P 2 that is tangent to the conic C at some point that is different from σ(E 1 ), . . . , σ(E k ). Denote by C and L the proper transforms on S of the curves C and L, respectively. Then On the other hand, we have so that the surface S contains a (−K S )-polar cylinder. Now let us present an example of a singular del Pezzo surface of degree 2 that has one singular point of type A 2 and contains an anticanonical polar cylinder.
Example 2.18. Let h : S → P 2 be a composition of 10 blowups, let E 1 , . . . , E 10 be the exceptional curves of the birational morphism h, let L 1 and L 2 be two distinct lines in P 2 , and let L 1 and L 2 be their proper transforms on S, respectively. Now, let us choose h such that the intersections of these twelve curves are depicted as follows: Note that the 10 blowups are arranged in the order indicated by the indices of their exceptional curves E i . To describe the intersection form of the curves L 1 , L 2 , E 1 , . . . , E 10 , observe that . . = E 2 10 = −1. Let g : S → S be the contraction of the curves L 1 , E 2 , E 3 , and let L 2 , E 1 , E 4 , . . . , E 10 be the proper transforms on S of the curves L 2 , E 1 , E 4 , . . . , E 10 , respectively. Then S is smooth and K 2 S = 2.
Moreover, the divisor −K S is nef. To show this, fix an arbitrary positive rational number ǫ < 1 3 , let D S be the following Q-divisor: and denote by D S its proper transform on S. Then D S is effective, This shows that −K S is nef. Moreover, we also see that L 2 and E 1 are the only (−2)-curves in S. Let f : S → S be the birational contraction of these two (−2)-curves. Then S is a del Pezzo surface with one singular point of type A 2 such that K 2 One can use the construction in Example 2.18 to construct an anticanonical polar cylinder in every del Pezzo surface of degree 2 that has a single singular point of type A 2 (see [39, §4.3]). Similarly, we can prove the existence part of Theorem 2.8. However, there is an alternative proof, which is more algebraic. Let us describe it following [33].
Let S be a singular del Pezzo surface of degree K 2 S 3 that has at most Du Val singularities, and let P be its singular point. Suppose, in addition, that the following conditions hold: • the singular point P is not of type . Now, let us prove that S contains a (−K S )-polar cylinder (cf. Theorem 2.8).
Let D be the divisor on S that is cut out by g(y, z, w) = 0. If K 2 S = 3, then D consists of the lines that contains P . There are at most six such lines and they are defined in P 3 by g(y, z, w) = 0, Similarly, if K 2 S = 2, then the divisor D consists of at most six curves passing through the point P . They are defined in P(1, 1, 1, 2) by g(y, z, w) = 0, Finally, if K 2 S = 1, then the divisor D consists of at most five curves passing through the point P , which are defined in P(1, 1, 2, 3) by g(y, z, w) = 0, In each case, the number of curves in D is the same as the number of points determined by the corresponding system of equations in Π. We denote these curves by L 1 , . . . , L r in each case. The map π contracts each curve L i to a point on Π.
Let C be the curve on Π defined by g(y, z, w) = 0. Then C can be reducible or non-reduced.
Proof. Observe that Im( π) = Π \ C. Let ϕ : S → S be the blowup of the point P . Then there exists a commutative diagram where ψ is the birational morphism that contracts the proper transforms of the lines L 1 , . . . , L r . Let E be the exceptional curve of the blowup ϕ. Then ψ(E) = C, and C contains each point π(L i ).
If P is an ordinary double point of the cubic surface S, then the curve C is a smooth conic. Similarly, if P is a singular point of type A n for n 2, then C splits as a union of two distinct lines. Finally, if P is either of type D n or of type E 6 , then C is a double line.
If C is smooth, let ℓ be a general line in Π that is tangent to C. If C is singular, let ℓ be a general line in Π that passes through a singular point of the conic C. By a suitable coordinate change, we may assume that ℓ is defined by x = y = 0. Let H be the curve in S cut out by y = 0. Then To deal with the cases K 2 S = 1 and K 2 S = 2, let ℓ y be the curve in P that is given by x = y = 0, and let H y be the curve in the surface S that is cut out by y = 0.
Therefore, in both cases, the complement S \ (H y ∪ L 1 ∪ · · · ∪ L r ) is a cylinder. Now, arguing as in the proof of Lemma 2.23, we see that S \ (H y ∪ L 1 ∪ · · · ∪ L r ) is a (−K S )-polar cylinder.
Finally, to deal with the remaining case, let ℓ z be the curve in P that is given by x = z = 0, and let H z be the hyperplane section of S that is cut by z = 0. Lemma 2.25. Suppose that K 2 S = 1 and the del Pezzo surface S is given by equation (2.22).
Since f 5 (y, 1) is an odd polynomial in y, the isomorphism A 2 → A 2 defined by y, w → y, w + f 5 (y, 1) is µ 2 -equivariant and gives an isomorphism between the complement Π \ (C ∪ ℓ z ) and the complement in A 2 /µ 2 of the image of the curve defined by w = 0, which is isomorphic to We see that S \ (H z ∪ L 1 ∪ · · · ∪ L r ) is a cylinder. Now, arguing as in the proof of Lemma 2.23,

Cylinders in higher-dimensional varieties
In this section, we describe known results about cylinders in smooth Fano threefolds and fourfolds, and varieties fibred into del Pezzo surfaces. Let us say few words about Fano varieties [108,114,163].
Let V be a smooth Fano variety of dimension n 3. The number (−K V ) n is known as the degree of the Fano variety V . Put Then ι(V ) is known as the (Fano) index of the variety V . It is well known that 1 ι(V ) n + 1. Moreover, one has Similarly, we have ι(V ) = n if and only if V is a quadric (see [114,128]).
for some ample divisor H ∈ Pic(V ). In this case, the variety V is usually called a del Pezzo variety. If ρ(V ) = 1, then there are just the following possibilities: Let us prove cylindricity of any higher-dimensional smooth intersection of two quadrics. 123]). Let V be a smooth complete intersection of two quadric hypersurfaces in P n+2 . Then V is cylindrical.
Proof. Let ℓ be a line in V , let D be an irreducible divisor in X swept out by lines meeting ℓ, let σ : V → V be the blowup of the line ℓ, let E be its exceptional divisor, and let D be the proper transform on V of the divisor D. There exists the following commutative diagram: where ϕ is a birational morphism that contracts D, and ψ is the projection from ℓ. Thus, we have But ϕ(E) is a quadric that contains a one-parameter family of linear subspaces of dimension n − 2. Hence, this quadric is singular, so that P n \ ϕ(E) contains a cylinder. Moreover, the following result has been recently proved in [142].
In Subsection 3.1, we will outline several known results about cylindrical smooth Fano threefolds. Then, in Subsection 3.2, we will present constructions of cylinders in some smooth Fano fourfolds. In particular, we will explain how to prove the following result: Finally, in Subsection 3.3, we will present results about cylinders in Mori fibrations.
3.1. Cylindrical Fano threefolds. Let X be a smooth Fano variety that has dimension three. Then X belongs to one of 105 families, which have been explicitly described in [106-108, 110, 157-160]. Their automorphism groups have been studied in [42,140,143,164,182]. In particular, we have Theorem 3.5. Let X be a smooth Fano threefold such that ρ(X) = 1 and Aut(X) is infinite. Then X and Aut(X) can be described as follows: (1) X = P 3 and Aut(X) ∼ = PGL 4 (k); (2) X is a smooth quadric in P 4 and Aut(X) ∼ = PSO 5 (k); (3) X is the quintic del Pezzo threefold described in Example 1.31 and Aut(X) ∼ = PGL 2 (k); (4) X is one of the following Fano threefolds in P 13 of degree 22 and genus 12: Before we describe some cylindrical smooth Fano threefolds, observe that we have the following implications: X contains (−K X )-polar cylinder =⇒ X is cylindrical =⇒ X is rational. Moreover, the rationality problem for smooth Fano threefolds is almost completely solved (see [114]). In particular, for general member of every family, we know whether it is rational or irrational. It is expected that the same answer holds for every smooth member in each family.
On the other hand, if H 3 4, then X contains a (−K X )-polar cylinder. Indeed, if H 3 = 4, this follows from Lemma 3.2. If H 3 6, this is obvious. Finally, if H 3 = 5, this follows from Theorem 3.6. Let V 5 be the quintic del Pezzo threefold in P 6 that is described in Example 1.31.
Proof. Let us give two constructions of the required hyperplane section. First, let L be a line in X.
Let α : V 5 → V 5 be the blowup of the line L. Then we have the following commutative diagram: where Q is a smooth quadric in P 4 , and β is the blowup of a twisted cubic curve C contained in Q.
Let H C be the unique hyperplane section of Q that contains C, and let H L be the unique hyperplane section of V 5 that is singular along L. Then H L is the proper transform of the β-exceptional surface, and H C is the proper transform of the α-exceptional surface. Note that H L is swept out by the lines that intersects the line L. Moreover, it follows from [114,143] that The lines in V 5 are parameterized by P 2 , and the lines of the type (1, −1) are parameterized by a smooth conic in this plane (see [83,108,143]). Furthermore, the surface H C is smooth if and only if L is a line of type (1, −1). Thus, if we choose L to be a line of type (1, −1) and put H = H L , To present the second construction, let P be a point in V 5 . Recall that Aut(V 5 ) ∼ = PGL 2 (k). Moreover, it follows from [47,83,108,143,164] that Aut(V 5 ) has exactly three orbits on V 5 : (1) a closed one-dimensional orbit C, which is a twisted rational sextic curve in P 6 ; (2) a two-dimensional orbitS whose closure is a surface S ∼ −K V which is singular along C; Furthermore, let k P be the number of lines in V 5 passing through P . Then Observe also that S is swept out by the lines of type (1, −1).
Let σ : V 5 → V 5 be the blowup of the point P . Then it follows from [82] that there exists the following Sarkisov link: where χ is a composition of flops of the proper transforms of lines in V 5 that pass through P , the morphism ϕ is a P 1 -bundle, and ψ is given by the linear system of hyperplane sections that are singular at the point P . Now we suppose that P ∈ C.
Let E be the σ-exceptional surface, and let E be its proper transform on the threefold V 5 . Then E is a del Pezzo surface of degree 6 with at most Du Val singularities, and its singular locus consists of one singular point of type A 2 . Moreover, the P 1 -bundle ϕ : V 5 → P 2 induces a birational map E → P 2 that contracts a single curve Γ ⊂ E to a point in P 2 .
Let L be a line in P 2 that passes through the point ϕ(Γ), let H be its preimage in V 5 via ϕ, let H be its proper transform on V 5 , and let H = σ( H). Then and H is a hyperplane section of the threefold V 5 that is singular at P . Furthermore, one can show that the surface H is smooth away from P , and H has Du Val singularity of type A 4 at this point.
Now, we assume that ι(X) = 1. This leaves us 95 families of smooth Fano threefolds [114,157]. If ρ(X) = 1, ι(X) = 1 and g(X) 6, then we have the following possibilities: (1) g(X) = 2 and X is a sextic hypersurface in P(1 4 , 3); (2) g(X) = 3 and X is an intersection of a quadric and a quartic in P(1 5 , 2); (3) g(X) = 4 and X is a complete intersection of a quadric and a cubic in P 5 ; (4) g(X) = 5 and X is a complete intersection of three quadrics in P 6 ; (5) g(X) = 6 and X is a section of the cone in P 8 over the smooth quintic del Pezzo fourfold described in Example 1.31 by a quadric and a hyperplane.
All these deformation families are irreducible. General members of the family (2) are smooth quartic hypersurfaces in P 4 , and special members are double covers of the quadric threefold branched over octic surfaces. Similarly, general members of the family (5) are sections of the smooth quintic del Pezzo fourfold in P 7 by quadrics, and special members are double covers of the smooth quintic del Pezzo threefold branched over anticanonical surfaces.
In the first two cases, the Fano threefold X is known to be irrational even if we allow mild isolated singularities [35,105,109,113,144,152,193,206]. In the case (4), the threefold X is also irrational [15]. General threefolds of the families (3) and (5) are irrational [15,99,112,115,198], and every smooth member is also expected to be irrational. Therefore, in all these cases, the threefold X is either non-cylindrical or it is expected to be irrational and, thus, non-cylindrical.
Remark 3.7. Let V 5 be the smooth quintic del Pezzo threefold, see Example 1.31, and let π : X → V 5 be a double cover branched over a surface S ∈ | − K V 5 |. If S has an isolated ordinary double point, then X is rationally connected [218], it is Q-factorial [52], and it follows from [187] that there exists the following Sarkisov link: where α is the blow up of the singular point of X, and β is a standard conic bundle, whose discriminant curve has degree 6. Hence, in this case, the threefold X is irrational by [205,Theorem 10.2]. Now, using [131, Theorem IV.1.8.3], we conclude that X is also irrational if S is a very general surface in the linear system | − K V 5 |.
In these cases, the divisor −K X is very ample, and | − K X | gives an embedding X ֒→ P g(X)+1 . Moreover, all the known constructions of cylinders in X use the double projection from a line in X (see [110]). Recall from [114,125,183] that X can contain two types of lines depending on their normal bundles. Namely, for a line ℓ ⊂ X, we have the following two possibilities: If X is a sufficiently general member of one of these three families of smooth Fano threefolds, then X does not contain lines of type (1, −2). Moreover, one can show that the threefolds containing lines of type (1, −2) form a codimension one subset in the corresponding moduli spaces. On the other hand, we have the following result: Proof. Let ℓ be a line in the Fano threefold X, and let σ : X → X be the blowup of the line ℓ. Then it follows from [108,110,114,184] that there is the Sarkisov link: where Y is a smooth Fano threefold described below, the morphism ϕ is the blowup of a smooth rational curve Γ, and χ is a composition of flops of the proper transforms of the lines that meet ℓ. Moreover, we have the following options: • if g(X) = 9, then Y = P 3 , and Γ is a curve of degree 7 and genus 3; • if g(X) = 10, then Y is a smooth quadric in P 4 , and Γ is a curve of degree 7 and genus 2.
Let E be the σ-exceptional surface, let E be its proper transform on X, and let S = ϕ( E). Then S is a (maybe singular or non-normal) del Pezzo surface of degree g(X) − 3 that contains Γ. Similarly, let S be the proper transform of the ϕ-exceptional surface on the Fano threefold X. Then S is a hyperplane section of X such that mult ℓ (S) = 3. Using this, we conclude that Moreover, if ℓ is a line of type (1, −2), then the surface S is not normal. This implies that the complement Y \ S contains a cylinder, so that X is cylindrical.
In fact, we believe that the following is true: Conjecture 3.9. Let X be a very general smooth Fano threefold such that ρ(X) = 1, ι(X) = 1, and g(X) = 9 or g(X) = 10. Then X is not cylindrical.
Proof. Let ℓ be a line in X. Then there exists a unique surface S ∈ |−K X | such that mult ℓ (S) = 3. Moreover, it follows from [108,110,114,184] that there exists the following Sarkisov link: where σ is the blowup of the line ℓ, the variety V 5 is a smooth quintic del Pezzo threefold in P 6 , the morphism ϕ is the blowup of a rational quintic curve Γ, and χ is a composition of flops. Let E be the σ-exceptional surface, let E be its proper transform on X, and let S = ϕ( E). Then S is a hyperplane section of the threefold V 5 that contains the curve Γ, and S is the proper transform of the ϕ-exceptional surface. Moreover, we have Let us show that V 5 \ S contains a cylinder. In fact, this follows from the proof of Theorem 3.6. We will use the notation and assumptions introduced in this proof.
Let L be a line in V 5 that is contained in S (it does exists). If S = H L , let S be the proper transform on Q of the surface S . Otherwise, we let S = H C . Then the surface S is a hyperplane section of the quadric Q. Thus, we see that Now taking the linear projection Q P 3 from a sufficiently general point in S ∩ H C , one can easily show that the complement Q \ (S ∪ H C ) contains a cylinder, so that X is cylindrical.
Remark 3.12 ([184]). In the notation and assumptions of the proof of Theorem 3.10, let ℓ be a line of type (−1, 2). Then S is a non-normal surface whose singular locus is a line in V 5 . Letting L to be this line gives S = H L , so that Thus, if we also have N L/V 5 ∼ = O L (1) ⊕O L (−1), then H C is singular (see the proof of Theorem 3.6), so that X \ S ∼ = A 3 . We can always find such ℓ and L if Aut(X) is infinite (see Theorem 3.5).
We do not know examples of cylindrical smooth Fano threefolds of Picard rank 1 and genus 7. In fact, we believe that any such threefold is not cylindrical. Conjecture 3.13. Let X be a smooth Fano threefold such that ρ(X) = 1, ι(X) = 1, and g(X) = 7. Then X is not cylindrical.
Before we close this section, let us mention that most of smooth Fano threefolds with ρ(X) 2 are rational [112,114,188], and many of them are known to be cylindrical. However, we do not know the existence of anticanonical polar cylinders in majority of cylindrical smooth Fano threefolds. Let us list few examples.
Example 3.14. Let Y be a smooth Fano threefold such that Y is a del Pezzo threefold or Y = P 3 . Take H ∈ Pic(Y ) on Y such that −K Y ∼ 2H. Choose a smooth curve C ⊂ Y that is a complete intersection of two surfaces from |H|. Suppose that X is a blowup of the threefold Y along C . Then X is a smooth Fano threefold. Moreover, if H 3 4, then X is cylindrical.
Example 3.15. Suppose that X is a blowup of P 3 along a smooth curve that is a complete intersection of two cubic surfaces. Then X is a cylindrical smooth Fano threefold.
Example 3.16. Suppose that X is a blowup of P 3 along a smooth curve of degree 6 and genus 3, which is an intersection of cubic hypersurfaces. Then X is a cylindrical smooth Fano threefold.
Example 3.17. Let Q be a smooth quadric threefold in P 4 , and let H be its hyperplane section. Suppose that X is a blowup of Q along a smooth curve that is a complete intersection of two surfaces from |2H|. Then X is a cylindrical smooth Fano threefold.
Each smooth Fano threefold described in Examples 3.14, 3.15, 3.16 and 3.17 is cylindrical, but we do not know whether any of these threefolds contains anticanonical polar cylinders or not.
3.2. Cylindrical Fano fourfolds. Now, let X be a smooth Fano fourfold such that ρ(X) = 1. By Corollary 1.6 we have the following implications: X is cylindrical =⇒ X is rational.
If ι(X) = 5 or ι(X) = 4, then X = P 4 or X is a smooth quadric fourfold, so that X is cylindrical. Similarly, if ι(X) = 3, then it follows from Remark 3.1 that X is one of the following fourfolds: (1) a smooth sextic hypersurface in P(1, 1, 1, 1, 2, 3); (2) a smooth quartic hypersurface in P(1, 1, 1, 1, 1, 2); (3) a smooth cubic fourfold in P 5 ; (4) a smooth complete intersection of two quadrics in P 6 ; (5) the quintic del Pezzo fourfold described in Example 1.31. In the first two cases, we expect that X is always irrational. In fact, we know that a very general quartic hypersurface in P(1, 1, 1, 1, 1, 2) is irrational [97], so that it is definitely not cylindrical. Similarly, general cubic fourfold in P 5 is expected to be irrational. But there are rational smooth cubic fourfolds (see [95,96,199,210]), so that it is very natural to ask the following question: Question 3.18. Are there smooth rational cylindrical cubic fourfolds? Remark 3.19. Every smooth cubic fourfold in P 5 containing two skew planes is rational (see [96]). In particular, the Fermat cubic fourfold is rational. If it is cylindrical, then the affine cone over it admits an effective action of the group G a by Theorem 1.15, which contradicts Conjecture 1.22.
By Lemma 3.2, we know that a smooth complete intersection of two quadrics in P 6 is cylindrical. Let us prove that the quintic del Pezzo fourfold described in Example 1.31 is cylindrical as well. To do this, let us present a detailed description of this fourfold given in [186].
Let V 5 be the quintic del Pezzo fourfold in P 7 . By [179, Theorem 6.6], we have the following exact sequence of groups: so that the group Aut(V 5 ) is not reductive. In particular, the fourfold V 5 is not K-polystable [1]. The planes on V 5 belong to one of the following two classes: (i) a unique plane Ξ which is a Schubert variety of type σ 2,2 ; (ii) a one-parameter family of planes Π t that are Schubert varieties of type σ 3,1 . We say that Ξ is the plane of type σ 2,2 , and Π t are planes of type σ 3,1 . They are distinguished by the types of the normal bundles: c 2 (N Ξ/X ) = 2 and c 2 (N Πt/X ) = 1. Moreover, there is a hyperplane section H of the fourfold V 5 that contains all planes in V 5 . Furthermore, one has Sing(H) = Ξ, the threefold H is the union of all the σ 3,1 -planes in V 5 , and Ξ contains a special conic C such that • the intersection Π t ∩ Ξ is a tangent line to the conic C; • two distinct σ 3,1 -planes Π t 1 and Π t 2 meet in a point in Ξ \ C. The automorphism group Aut(V 5 ) has the following orbits in V 5 : (1) the open orbit X \ H; (2) the three-dimensional orbit H \ Ξ; (3) the two-dimensional orbit Ξ \ C; (4) the one-dimensional closed orbit C.
The Hilbert scheme of lines on the del Pezzo fourfold V 5 is smooth, irreducible, and four-dimensional. Moreover, if ℓ is a line in V 5 , then ℓ belongs to one the following five classes: (a) ℓ ⊂ H, ℓ ∩ Ξ = ∅, and l ∩ H is a point; (b) ℓ ⊂ H, l ∩ Ξ is a point, and ℓ ∩ C = ∅; (c) ℓ ⊂ H, and l ∩ Ξ = l ∩ C is a point; (d) ℓ ⊂ Ξ, and the intersection ℓ ∩ C consists of two points; (e) ℓ ⊂ Ξ and ℓ is tangent to C. The group Aut(V 5 ) acts transitively on the lines in each of these classes. For a line ℓ ⊂ V 5 , the lines meeting ℓ sweep out a hyperplane section H ℓ of the fourfold V 5 that is singular along the line ℓ. Vice versa, if H is a hyperplane section of the quintic del Pezzo fourfold V 5 that has non-isolated singularities, then H = H ℓ for some line ℓ ⊂ V 5 . Proof. If ℓ is a line of type (d) or (e), then H ℓ = H. On the other hand, there exists the following Aut(V 5 )-equivariant Sarkisov link: where σ is the blowup of the plane Ξ, ϕ is the blowup of a twisted cubic curve C, and ψ is the linear projection from Ξ. Then the ϕ-exceptional divisor is the proper transform of the threefold H. Moreover, if E is the σ-exceptional divisor, then ϕ(E) is the hyperplane in P 4 that contains C. Thus, if ℓ is a line of type (d) or (e), then Let π : V 5 be the blowup of the line ℓ. Then there exists the following Sarkisov link: where Q is an irreducible quadric in P 5 , the map ζ is the projection from ℓ, and η is a birational morphism that contracts the proper transform of the hyperplane section H ℓ to a surface of degree 3. Let H ℓ be the proper transform on V 5 of the threefold H ℓ , and let F be the π-exceptional divisor. Then V 5 \ H ℓ ∼ = Q \ η(F ), and η(F ) is a singular hyperplane section of the quadric Q.
If ℓ is a line of type (a), then all fibers of η are one-dimensional, so that Q is smooth (see [2]). Thus, in this case, we have To complete the proof, we may assume that ℓ is of type (c). Then ℓ is contained in a plane in V 5 , so that η has a two-dimensional fiber. Hence, in this case, the quadric Q can be singular (cf. [3]). Analyzing the situation more carefully, we see that In the remaining part of this subsection, we present known constructions of cylinders in some smooth Fano-Mukai fourfolds. Basically, our main goal is to explain how to prove Theorem 3.4. Thus, we suppose that X is a smooth Fano-Mukai fourfold, ρ(X) = 1 and g(X) ∈ {7, 8, 9, 10}.
Let H be an ample Cartier divisor on X such that −K X ∼ 2H.
Then H 4 = 2 g(X) − 2 ∈ {12, 14, 16, 18}. Moreover, the divisor H is very ample, and the linear system |H| gives an embedding X ֒→ P g(X)+2 . Let us deal with four cases separately. If g(X) = 10, then X = X 18 is a hyperplane section of the homogeneous fivefold G 2 /P ⊂ P 13 , where G 2 is the simple algebraic group of exceptional type G 2 , and P is its parabolic subgroup that corresponds to a short root (see [162,163]). The family X of all such fourfolds is one-dimensional. Moreover, if X = X 18 is a general member of X, then Aut(X) ∼ = G 2 m ⋊ µ 2 . Besides, there are three distinguished fourfolds in this family: (0) X r 18 such that Aut(X r 18 ) ∼ = G 2 m ⋊ µ 6 ; (1) X s 18 such that Aut(X s 18 ) ∼ = GL 2 (C) ⋊ µ 2 ; (2) X a 18 such that Aut(X a 18 ) ∼ = (G a × G m ) ⋊ µ 2 . See [191] for details, where the following result has been proved: 191]). Let X be a smooth Fano-Mukai fourfold in P 12 of genus 10 with ρ(X) = 1. Then there exists an Aut 0 (X)-invariant hyperplane section H of X such that the complement X \H is Aut 0 (X)-equivariantly isomorphic to A 4 .
This theorem implies, in particular, that any smooth Fano-Mukai fourfolds of genus 10 is cylindrical. See also Example 4.16 for another application of Theorem 3.23.
If g(X) = 8, then X = X 14 is a section of the Grassmannian Gr(2, 6) ⊂ P 14 by a linear subspace of dimension 10 (see [162,163]). Some of these fourfolds are cylindrical.

Example 3.24 ([189]
). Suppose that g(X) = 8 and X contains a plane Π which is a Schubert variety of type σ 4,2 , and X does not contain planes meeting Π along a line. Such fourfolds do exist and form a subspace of codimension one in the moduli space of all Fano-Mukai fourfolds of genus 8. Then it follows from [185] that there exists the following Sarkisov link: where V 5 is the del Pezzo quintic fourfold in P 7 (see Theorem 3.20), σ is the blowup of the plane Π, and ϕ is the blowup of a smooth rational surface S of degree 7 such that K 2 S = 3. Then where H V 5 is the proper transform on V 5 of the σ-exceptional divisor, and H X is the proper transform on X of the ϕ-exceptional divisor. On the other hand, the divisor H V 5 is a hyperplane section of the fourfold V 5 that contains S, and H X is a hyperplane section of X containing Π. Thus, the set V 5 \ H V 5 contains a cylinder by [189,Theorem 4.1], so that X is cylindrical.
If g(X) = 7, then X = X 12 is a section of the orthogonal Grassmannian OGr(4, 9) ⊂ P 15 by a linear subspace of dimension 9 (see [162,163]). In this case, we also have cylindrical fourfolds.  189]). Suppose that g(X) = 7 and X contains a plane Π. Such fourfolds do exist. Suppose that X is a sufficiently general Fano-Mukai fourfold of genus 7 that contains the plane Π. Then by [185] there exists the Sarkisov link where V 4 is a smooth complete intersection of two quadrics in P 6 , σ is the blowup of the plane Π, and ϕ is the blowup of a smooth del Pezzo surface S such that K 2 S = 5. Arguing as in Example 3.24, we conclude that X is cylindrical.
If g(X) = 9, then X = X 16 is a section of the Lagrangian Grassmannian LGr(3, 6) ⊂ P 13 by a linear subspace of dimension 11 (see [162,163]). There are cylindrical fourfolds in this family.
Example 3.26 ([190]). Suppose that g(X) = 9. Then X 16 contains an irreducible two-dimensional quadric surface S. Suppose, for simplicity, that X 16 is a general Fano-Mukai fourfold of genus 9 that contains S. Then there exists the following Sarkisov link: where V 5 is the del Pezzo quintic fourfold, σ is the blowup of the surface S, and ϕ is the blowup along a smooth del Pezzo surface of degree 6. Arguing as in Example 3.24, we see that X is cylindrical.
The interested reader can consult also the recent preprint [103] for further examples of cylindrical Fano fourfolds.

Cylinders in Mori fibrations. This subsection is inspired by the following
Question 3.27. Given a family of cylindrical varieties, when its total space is cylindrical?
For example, irrational three-dimensional conic bundles are not cylindrical, though their general fibers are. In general, this question is very subtle and has birational nature, so that it is natural to consider it for Mori fibred spaces first.
Let V be a projective variety with terminal Q-factorial singularities, let π : V → B be a dominant projective non-birational morphism such that −K V is π-ample, π * O V = O B and ρ(V ) = ρ(B) + 1. Let X η be the fiber of the morphism π over the (scheme-theoretic) generic point η of the base B. Then X η is a Fano variety that has at most terminal singularities, which is defined over K = k(B), i.e. the field of rational functions on B. Over the (algebraically non-closed) field K, the divisor class group of the Fano variety X η is of rank 1, because we assume that ρ(V ) = ρ(B) + 1.

Definition 3.28 ([61]
). If the variety V contains a (Zariski open) cylinder U = A 1 × Z, we say that the cylinder U is vertical (with respect to π) if there is a morphism h : Z → B such that the restriction π| U : U → B is a composition h • pr Z , where pr Z : U → Z is the natural projection. In this case, we have commutative diagram: If V contains a vertical cylinder U = A 1 × Z, then the Fano variety X η contains a cylinder where U η and Z η are generic (scheme) fibers of the morphisms h • pr Z and h in (3.29), respectively. Vice versa, if the Fano variety X η contains a cylinder defined over the field K, then V does contain a vertical cylinder by [61,Lemma 3]. This gives a motivation to study cylinders in Fano varieties defined over arbitrary fields (cf. [17, 100, 141, 142]) The first step in this direction is Theorem 3.30 ([61]). Let S be a geometrically irreducible smooth del Pezzo surface defined over a field F of characteristic 0. Suppose that ρ(S) = 1. Then the following conditions are equivalent: (i) the surface S contains a cylinder defined over F; (ii) the surface S is rational over F; (iii) K 2 S 5 and S has an F-point.
Proof. It is commonly known that the conditions (ii) and (iii) are equivalent (see, for example, [111]). Moreover, the implication (iii)⇒(i) can be shown using well-known Sarkisov links that start at S, which are described in [111]. For details, see the proof of [61,Proposition 12]. Thus, we just have to show that (i) implies (iii). This can also be shown using Sarkisov links, but we present another proof. Suppose that S contains a cylinder U which is defined over F. Then U ∼ = A 1 × Z for some affine curve Z defined over F. Let Z be the completion of the curve Z. Then Z is a geometrically irreducible curve. Moreover, we have the following commutative diagram y y r r r r r r r r r r r r r r r r r r r r r r r r r r r r r Z _ Z where p Z , p 2 and p 2 are the natural projections to the second factors, ψ is the rational map induced by p Z , π is a birational morphism resolving the indeterminacy of ψ and ϕ is a morphism. By construction, a general fiber of ϕ is isomorphic to P 1 . Let Γ be the section of p 2 that is the complement of A 1 × Z in P 1 × Z, and let Γ be the proper transform on S of the curve Γ. Then Γ ∼ = Γ ∼ = Z, the curve Γ is a section of ϕ, and the curve Γ is π-exceptional, because ρ(S) = 1. Let P = π( Γ). Then P is an F-point. Now, we can proceed in two (slightly different) ways. First, as in the proof of [61, Theorem 1], we can let M to be the linear system on S that gives the map ψ. Then, arguing as in Section 2.2, we conclude that (S, λM) is not log canonical at P for a some λ ∈ Q >0 such that λM ∼ Q −K S . Such number exists, since ρ(S) = 1. Let M 1 and M 2 be two general curves in M. Then Theorem 3.1]. This gives K 2 S 5, so that (i) implies (iii). Alternatively, we can use Corollary 2.9. Let C 1 , . . . , C n be the irreducible curves in S that lie in the complement S \ U. Then we put D = λ(C 1 + · · · + C n ) for λ ∈ Q >0 such that D ∼ Q −K S . Therefore, we conclude that S contains a (−K S )-polar cylinder, so that K 2 S 4 by Corollary 2.9. Thus, we may assume that K 2 S = 4. Then our point P is not contained in any (−1)-curve in S ⊗ F F, where F is an algebraic closure of the field F. Indeed, otherwise the Gal(F/F)-orbit of this curve would consist of at least four (−1)-curves that all pass through the point P , which is impossible. Let ξ : S → S be the blowup of the point P , and let E be the exceptional curve of the blowup ξ. Then S is a smooth del Pezzo surface of degree K 2 S = 3 and where mult P (D) > 1 by Remark 2.3 and Lemma A.3. Then S contains a (−K S )-polar cylinder, which is impossible by Corollary 2.9. This again shows that (i) implies (iii). Note that if k is uncountable and the general fiber of π contains a cylinder, then it follows from [63,116] that the total space of the family V × B B ′ → B contains a vertical cylinder for an appropriate finite base change B ′ → B. This basically means that X η ⊗ K K ′ contains a cylinder defined over K ′ for an appropriate finite extension of fields K ⊂ K ′ .
Let us conclude this section with the following generalization of Theorem 3.3.

Theorem 3.34 ([142])
. Let X be a smooth Fano threefold defined over a field F of characteristic 0.

Beyond cylindricity
4.1. Flexible affine varieties. Let X be an affine variety. Given a G a -action on X, it induces a representation of the group G a on the structure k-algebra O(X) of the form for t ∈ G a and f ∈ O(X), where the infinitesimal generator ∂ of the G a -subgroup is a locally nilpotent derivation of O(X), which means that every element f ∈ O(X) is annihilated by ∂ (m) for some sufficiently large m that depends on the element f . Conversely, any locally nilpotent derivation of the k-algebra O(X) generates a G a -action on X (see [67]).
Recall that the derivations of O(X) correspond to the regular vector fields on X. We say that a vector field on X is locally nilpotent if the corresponding derivation is.
If an open variety X admits a G a -action, then the log-Kodaira dimension of X is negative. However, the converse does not hold, in general. Indeed, there are smooth affine surfaces of negative log-Kodaira dimension which admit no effective G a -action [92]. Let us stay on this in more detail.
As we mentioned already, any smooth affine surface X of negative log-Kodaira dimension contains a cylinder [154, Ch. 2, Theorem 2.1.1]. Moreover, X is affine-ruled, that is, there is a morphism X → C onto a smooth curve C with general fiber A 1 . The base curve C could be affine or projective. However, a smooth affine surface X admits an effective G a -action if and only if it admits an A 1 -ruling X → C over an affine curve C, or, which is equivalent, a principal cylinder, see Theorem 1.13. In [92] there are examples of smooth rational affine surfaces A 1 -ruled over P 1 and with no A 1 -ruling over an affine curve. Hence, such a surface admits no effective G a -action.
To construct such a surface X we start with the quadric P 1 ×P 1 endowed with the first projection to P 1 . We blow up three distinct points on the section S = P 1 × {∞} and infinitesimally near points in such a way that each of the 3 resulting reducible fibers has a unique (−1)-component of multiplicity 2 and the union of the section S and the remaining components of the reducible fibers forms a connected divisor D. The complement of D in the resulting projective surface is a smooth affine surface X. It comes equipped with an A 1 -fibration X → P 1 . Each fiber of this fibration is irreducible, and three of them are multiple of multiplicity 2. According to [93,Theorem 4.1], such a surface X does not carry any A 1 -fibration over an affine curve. Hence, X admits no G a -action.
Definition 4.1. A point P ∈ X is said to be flexible if locally nilpotent vector fields on X span the tangent space T P X. The variety X is said to be flexible if every smooth point of X is flexible. We also say that X is generically flexible if every point in a non-empty Zariski open subset of X is flexible.
Let SAut(X) be the subgroup of Aut(X) generated by all the G a -subgroups. The flexibility of X is ultimately related to the transitivity of the action of the group SAut(X). Indeed, we have the following criteria of flexibility. (1) the variety X is flexible; (2) the group SAut(X) acts transitively on the smooth locus of X; (3) the group SAut(X) acts highly transitively on the smooth locus of X.
One says that a group acts highly transitively 1 . on an infinite set if it acts m-transitively for any natural number m. Remark 4.3. A dimension count shows that an algebraic group cannot act highly transitively on an affine variety. Moreover, it cannot act even 3-transitively on an affine variety [22,127].
Example 4.4. Let X = A n , where n 2. Then the subgroup of translations in SAut(A n ) acts transitively on the variety X, so that X is flexible by Theorem 4.2 (cf. [120]).
Example 4.5. Let X be the nth Calogero-Moser space defined as follows: where PGL n (k) acts via g.(A, B) = (gAg −1 , gBg −1 ). Then X is a smooth rational irreducible affine algebraic variety of dimension 2n [181,215], and it follows from [18,138] that Aut(X) acts highly transitively on X for every n 1. Moreover, the variety X is flexible by [4,Proposition 2.9].
There are several constructions producing new flexible varieties from given ones (see [8,12,65,120]). For instance, the product of flexible varieties is flexible. Some further examples of flexible varieties are as follows.
Example 4.6. Suppose that X is an affine G-variety of dimension 2, where G is a connected linear algebraic group that acts on X with an open orbit. Then X is flexible in the following cases: • X is a normal toric variety with no torus factor [ See also [21,56,57,87,117,139]. If we replace the smooth locus of X in Theorem 4.2 by the open orbit of the group SAut(X), we obtain a criterion for the generic flexibility [8]. If X contains A n as a principal Zariski open set, then X is generically flexible. Generically flexible varieties are unirational, but they are not always stably rational (see [148,Proposition 4.9] and [180,Example 1.22]).
Example 4.7. Suppose that X is a normal affine surface such that X can be completed by a simple normal crossing chain of rational curves. Then X is often called a Gizatullin surface. If , then it is generically flexible [86], but it is not necessarily flexible [137].  To explain why this is the case, let us present two explicit criteria of flexibility of affine cones. To do this, fix a smooth projective variety V . Let H be a very ample divisor on the variety X. Then the linear system |H| gives an embedding V ֒→ P n . Let V be the affine cone in A n+1 over V . We are interested in the case when V is a smooth cylindrical Fano variety.
If the variety V is uniformly cylindrical, then each point of V is contained in a cylinder, so that the variety V admits a covering where each U i is a Zariski open subset in V such that U i ∼ = A 1 × Z i for some affine variety Z i . In this case, a subset Y ⊂ V is said to be invariant with respect to a cylinder U i if where π i : U i → Z i is the natural projection.
Definition 4.10. If V is uniformly cylindrical, then we say that the covering (4.9) is transversal if no proper subset Y ⊂ X is invariant with respect to every cylinder U i in the covering (4.9). Now, we are ready to state the first flexibility criterion for affine cones.
Theorem 4.11 ([174]). Suppose that V is uniformly cylindrical and has a covering (4.9) such that (i) the covering (4.9) is transversal; (ii) each cylinder in the covering (4.9) is H-polar.
Then the affine cone V is flexible.
The second useful criterion is given by the following Using these criteria and the proof of Lemma 2.17, one can prove the following result: 172,174]). Suppose that V is a smooth del Pezzo surface such that K 2 V 4. Then the affine cone V is flexible for every very ample divisor H on the surface V .
Unfortunately, we cannot apply Theorems 4.11 and 4.12 to the affine cone in A 4 over a smooth cubic surface in P 3 , simply because its anticanonical divisor is not cylindrical by Corollary 2.9. On the other hand, in this case, we know from Theorem 2.14 that every ample Q-divisor that is not a multiple of the anticanonical divisor is cylindrical. Using this and the construction of cylinders given in the proof of Theorem 2.14, Perepechko very recently proved the following result:  Proof. If ρ(V ) = 1, then it follows from Theorem 3.5 that one of the following four cases occurs: (i) V = P 3 ; (ii) V is the smooth quadric threefold in P 4 ; (iii) V is the smooth quintic del Pezzo threefold V 5 ⊂ P 6 described in Example 1.31; (iv) V is the Mukai-Umemura threefold X = X mu 22 ⊂ P 13 . We may assume that we are in the case (iii) or (iv), because the required assertion is clear in the remaining cases. Then it follows from the proofs of Theorems 3.6 and 3.10 that V contains a one-parameter family of hyperplane sections H ℓ such that each H ℓ is singular along a line ℓ and The group PSL 2 (k) acts transitively on this family. So, to apply Theorem 4.12, we need to check that the intersection of all these hyperplane sections is empty. Suppose that this is not the case. Then this intersection is PSL 2 (k)-invariant, so that it contains a closed PSL 2 (k)-orbit of minimal dimension. But the variety V does not contain PSL 2 (k)-fixed points, and the only one-dimensional closed PSL 2 (k)-orbit in V is not contained in any hyperplane section singular along a line.   191]). It follows from Theorems 3.20 and 3.23 that the following smooth cylindrical Fano fourfolds admit coverings by affine charts isomorphic to A 4 : (1) the quintic del Pezzo fourfold V 5 described in Example 1.31 (see Theorem 3.20); (2) the Fano-Mukai fourfold X s 18 of genus 10 with Aut(X s 18 ) ∼ = GL 2 (C) ⋊ µ 2 ; (3) the Fano-Mukai fourfolds X 18 of genus 10 with Aut 0 (X 18 ) ∼ = G 2 m (there is a one-parameter family of these, up to isomorphism). Hence, all of them have flexible affine cones.
By Theorem 3.23, every smooth Fano-Mukai fourfold in P 12 of genus 10 contains a Zariski open subset isomorphic to A 4 . Moreover, the following result has been recently proved in [192].

4.2.
Cylinders in complements to hypersurfaces. This section is motivated by the following folklore conjecture that first appeared in 2005 [68].
Conjecture 4.18. Let S be a smooth cubic surface in P 3 . Then any automorphism of the affine variety P 3 \ S is induced by an automorphism of P 3 , i.e., we have Aut P 3 \ S = Aut P 3 , S .
If S is smooth surface in P 3 of degree 4, then it is easy to see that Aut(P 3 \ S) = Aut(P 3 , S). Vice versa, if S is either a smooth quadric surface or a plane in P 3 , then Aut(P 3 \ S) = Aut(P 3 , S). Moreover, it is not hard to see that Conjecture 4.18 fails for some singular cubic surfaces.
Example 4.19. Let S be one of the three cubic surfaces with Du Val singularities in P 3 that admits an effective G a -action (see [41,151,200]). Then Aut(P 3 , S) contains a subgroup isomorphic to G a , so that Aut(P 3 \ S) also contains a subgroup isomorphic to G a . Then Aut(P 3 \ S) must be infinite dimensional (see [67]), so that Aut(P \ S) = Aut(P, S), because Aut(P, S) is algebraic.
Based on the results in [33,38,39], we may generalize the problem to del Pezzo surfaces that are hypersurfaces in weighted projective spaces. To be precise, let S be a del Pezzo surface that has at most Du Val singularities such that K 2 S 3. Then we have one of the following three cases: (1) K 2 S = 1, and S is a hypersurface of degree 6 in P(1, 1, 2, 3); (2) K 2 S = 2, and S is a hypersurface of degree 4 in P(1, 1, 1, 2); (3) K 2 S = 3, and S is a hypersurface of degree 3 in P 3 . Denote by P the weighted projective space in these three cases: P(1, 1, 2, 3), P(1, 1, 1, 2) or P 3 . Then, very surprisingly, we have the following result: Theorem 4.20 ([33, 171]). The following three conditions are equivalent: • the surface S contains a (−K S )-polar cylinder; • the complement P \ S is cylindrical; • the group Aut(P \ S) contains a unipotent subgroup.
Combining this result with Theorem 2.8, we obtain  Proof. By Theorem 4.20, the group Aut(P \ S) contains a unipotent subgroup, so that it is infinite dimensional, which implies that Aut(P \ S) = Aut(P, S), because Aut(P, S) is algebraic.  If K 2 S = 2 or K 2 S = 3, then Aut(P \ S) does not contain nontrivial connected algebraic groups. The proof of this result depends on irrationality of some del Pezzo threefolds (see [50,88,89,213]). Taking into account Theorem 4.23, Corollary 2.9 and Corollary 4.22, we propose the following If S is a smooth cubic surface, then it does not contain any (−K S )-polar cylinder by Theorem 2.8. In this case, Conjecture 4.24 claims that Aut (P \ S) = Aut (P, S), which is Conjecture 4.18.
In [171], Theorem 4.20 has been generalized as follows. Let X be a normal projective variety, and let D be an ample Cartier divisor on X. Suppose that the following conditions are satisfied: (1) the section ring of (X, D) is a hypersurface, i.e., one has where k[x 0 , . . . , x n ] is a polynomial ring in variables x 0 , . . . , x n with weights a 0 = wt(x 0 ) a 1 = wt(x 1 ) . . . a n = wt(x n ), and F is a quasi-homogeneous polynomial of degree d, so that X is a hypersurface in the weighted projective space P(a 0 , a 1 , . . . , a n ) = Proj(k[x 0 , x 1 , . . . , x n ]); (2) the Veronese map v d : P(a 0 , a 1 , . . . , a n ) P N given by O P(a 0 ,a 1 ,...,an) (d) is an embedding. Recall from [123,Proposition 3.5] that the complement P(a 0 , a 1 , . . . , a n ) \ X admits a nontrivial G a -action if and only if it is cylindrical. On the other hand, we have the following result: . Suppose that P(a 0 , a 1 , . . . , a n )\X has a nontrivial G a -action. Then X contains a D-polar cylinder.
Based on the results on non-ruledness of smooth hypersurfaces of low degrees in the projective spaces such as [29,50,55,113,130,194,195,203] one can extend Conjecture 4.18 as follows: Conjecture 4.26. Let X be a smooth hypersurface in P n of degree d 3. Then Aut P n \ X = Aut P n , X .
The conjecture holds when d > n since the hypersurface X has non-negative Kodaira dimension. It remains true if d = n 4 and (n, d) = (4, 3) due to the results by [29,50,55,113,194,195].

4.3.
Compactifications of C n . In this subsection, we assume that varieties are defined over C. In this case, the problem of existence of (Zariski open) cylinders in smooth Fano varieties is closely related to the following famous problem posed by Hirzebruch 65 years ago in [102]. This problems asks to describe all compact complex manifolds X with b 2 (X) = 1 that contain an open subset U which is biholomorphic to C n and whose complement A = X \ U is a closed complex analytic subspace. Thus, we call a compactification of C n a pair (X, A) consisting of • a compact complex manifold X with b 2 (X) = 1; • and a closed complex analytic subset A ⊂ X such that X \ A ∼ = bihol C n .
A compactification (X, A) of C n is said to be algebraic if X is a smooth projective variety, and the biholomorphism X \ A ∼ = bihol C n is an algebraic isomorphism. Thus, we see that (X, A) is an algebraic compactification of C n =⇒ X is a cylindrical Fano variety. The following deep result is due to Kodaira [129,Theorem 3]: for every m > 0, where ω X is the sheaf of holomorphic n-forms on X.
Thus, if (X, A) is a compactification of C n and X is projective, then X is a smooth Fano variety, and A is an ample divisor on X that generates Pic(X).
Example 4.30. Let (X, A) be one of the following polarized smooth Fano varieties: (1) X = P n and A is a hyperplane; (2) X is a smooth quadric in P n+1 and A is its singular hyperplane section; (3) X = Gr(m, k) and A is its Schubert subvariety of codimension 1, where n = m(k − m); (4) X = G/P and A is its open cell isomorphic to C n (such a cell does exist by [23,131]), where G is a semisimple connected complex linear algebraic group, and P is its maximal parabolic subgroup. Then (X, A) is a compactification of C n .
In two-dimensional case, Problem 4.27 has an easy solution: if (X, A) is a compactification of C 2 , then X = P 2 and A is a line in X. In the three-dimensional case, Problem 4.27 has been solved in the series of papers [77-82, 176, 177, 184]. In particular, we have the following result: Theorem 4.31. Let (X, A) be a compactification of C 3 . Suppose that X is a projective threefold. Then this compactification is algebraic and (X, A) can be described as follows: (1) X = P 3 and A is a plane; (2) X is a smooth quadric in P 4 and A is its singular hyperplane section; (3) X is the quintic del Pezzo threefold in P 5 described in Example 1.31 and A is its singular hyperplane section that can be described as follows: (a) a surface whose singular locus is a line L with normal bundle N L/X = O L (1)⊕O L (−1); (b) a normal del Pezzo surface that has a unique singular point of type A 4 ; (4) X is a smooth Fano threefold of index 1 and genus 12 in P 13 and A is its certain hyperplane section whose singular locus is a line ℓ with normal bundle Proof. We know that X is a smooth Fano threefold, and the surface A generates Pic(X), so that where ι(X) is the Fano index of the threefold X. If ι(X) = 4, then X = P 3 and A is a plane. Similarly, if ι(X) = 3, then X is a smooth quadric threefold in P 4 , and A is its hyperplane section. In this case, the surface A must be singular, since H 2 (A, Z) = Z by Proposition 4.28.
If ι(X) = 1, then the surface A must be a non-normal K3 surface, and the proof uses a delicate analysis of its singularities. As a result, one can show that X is a Fano threefold of genus 12 in P 13 , and A is its hyperplane section that is singular along a line of type (1, −2). One construction of such compactification is described in Remark 3.12. We will not dwell into further details in this case.
Suppose that ι(X) = 2. Let us show that X is the quintic del Pezzo threefold in P 5 , and A is its singular hyperplane section described above. Note that in this case (X, A) is indeed a compactification of C 3 , which follows from the proof of Theorem 3.6.
First, we suppose that the surface A is normal. Then −K A is ample by the adjunction formula, so that A is a del Pezzo surface with isolated Gorenstein singularities. If its singularities are worse than Du Val, then A must be a (generalized) cone over an elliptic curve [101], so that χ top (A) = 1. The latter contradicts (4.32). Thus, we see that A is a del Pezzo surface with Du Val singularities. Then ρ(A) = 1, because H 2 (A, Z) = Z. Then χ top (A) = 3, so that we have h 1,2 (X) = 0 by (4.32). Now, using Remark 3.1, we conclude that X is the quintic del Pezzo threefold in P 5 as required. Moreover, we have K 2 A = 5, so that A is a quintic del Pezzo surface that has Du Val singularities. Since ρ(A) = 1, it follows from [77,156] that A has a unique singular point of type A 4 . Now, we suppose that A is non-normal, so that it has a singular locus of positive dimension. It is easy to show that any hyperplane section of a smooth complete intersection has only isolated singularities, and the same result holds for hyperplane sections of weighed smooth hypersurfaces. Therefore, using Remark 3.1, we conclude again that X is the quintic del Pezzo threefold in P 5 , and A is its hyperplane section. Using the adjunction formula, we see that a general hyperplane section of the surface A is an irreducible singular curve of arithmetic genus 1, so that it has one singular point. Thus, the non-normal locus of the surface A is some line L. Hence, it follows from the proof of Theorem 3.6 that Sing(A) = L and where Q is a smooth quadric threefold in P 4 , and H is its hyperplane section. Since X \ A ∼ = C 3 , we conclude that the surface H is singular. As we already mentioned in the proof of Theorem 3.6, this implies that N L/X = O L (1) ⊕ O L (−1) as required.
Corollary 4.33. Let (X, A) be a compactification of C 3 . Suppose that X is a projective threefold. Then H k (X, Z) ∼ = H k (P 3 , Z) for all k.
It would be interesting to find an alternative proof of Theorem 4.31 that does not heavily rely on the classification of smooth Fano threefolds.
Remark 4.34. Let X be a smooth Fano threefold such that ρ(X) = 1, ι(X) = 1 and g(X) = 12. If X is a compactification of C 4 , then X contains a line ℓ such that N ℓ/X = O ℓ (1) ⊕ O ℓ (−2). However, this condition does not always guarantee that X is a compactification of C 4 (see [184]). In higher dimensions, we know very few results on Problem 4.27. Let us present one of them, which follows from Theorem 3.20 and its proof. We use here the notation introduced in Section 3.2.
Theorem 4.36 ([186]). Let (X, A) be a compactification of C 4 , where X is a smooth Fano fourfold. Suppose that ι(X) = 3. Then X is the quintic del Pezzo fourfold in P 7 and (1) either A = H ℓ , where ℓ is a line in X that is not a line of type (b); (2) or A is a singular hyperplane section of the del Pezzo fourfold X such that its singular locus consists of a single ordinary double point that is not contained in the divisor H. Each of these compactifications is algebraic and unique up to isomorphism.
Proof. We prove the existence part only. In the first case, the existence follows from Theorem 3.20. To deal with the second case, let us use the notation introduced in the proof of Theorem 3.20. Consider the Sarkisov link (3.21) with ℓ being a line of type (a). We already know that Q is smooth, and so we may assume that it is given in P 4 by Similarly, we may assume that η(F ) is cut out by x 0 = 0. Moreover, the surface η( H ℓ ) is a smooth cubic scroll in this case. Hence, we may assume that it is cut out on Q by      x 0 = 0, Let D be the hyperplane section of the quadric Q that is cut out by x 3 = 0, and let D be its proper transform on V 5 . Then D is singular. We claim that x 4 , so that V 5 \ D is given by y 0 z 0 = (y 5 − y 2 4 )z 1 in A 4 × P 1 , where z 0 and z 1 are coordinates on P 1 . Then V 5 \ ( D ∪ H ℓ ) is given in A 4 × A 1 by is a hyperplane section of V 5 whose singular locus consists of a single ordinary double point not contained in H.
In dimension 4, we know very few compactifications (X, A) of C 4 . They can be listed as follows: • X = P 4 and A is a hyperplane; • X is a smooth quadric and A is its singular hyperplane section; • X is the del Pezzo quintic fourfold and A is described in Theorem 4.36; • X is a smooth Fano-Mukai fourfold of genus 10 and A is described in Theorem 3.23. In particular, in every known example of a compactification (X, A) of C 4 with X ∼ = P 4 , one has for all k, where Q is a smooth quadric in P 5 . We wonder whether this is just a coincidence. Before we conclude this survey, let us set the following question: Question 4.38. Is it true that any compactification of C n is rational?
Note that the answer to this question is not obvious, since the isomorphism X \ A ∼ = C n in the definition of a compactification of C n is a biholomorphism, which is not necessarily algebraic.

Appendix A. Singularities of pairs
Let S be a surface with at most quotient singularities, let D be an effective non-zero Q-divisor on S, let P be a point of S, and let where C 1 , . . . , C r are distinct irreducible curves on S, and each a i is a non-negative rational number. We call (S, D) a log pair.
Let π : S → S be a birational morphism such that S is smooth. For each C i , denote by C i its proper transform on the surface S. Let F 1 , . . . , F n be π-exceptional curves. Then for some rational numbers b 1 , . . . , b n . Suppose that C 1 + · · · + C 2 + F 1 + · · · + F n is a divisor with simple normal crossings. Then we say that π : S → S is a log resolution of the log pair (S, D).
Definition A.1. The log pair (S, D) is said to be log canonical at the point P if the following two conditions are satisfied: • a i 1 for every C i such that P ∈ C i ; • b j 1 for every F j such that π(F j ) = P . The log pair (S, D) is called log canonical if it is log canonical at every point of S. 38 This definition does not depend on the choice of the log resolution π : S → S.
Remark A.2. Let R be an effective Q-divisor on S such that R ∼ Q D. For a rational number ǫ, let Then D ǫ ∼ Q D. Suppose that R = D. Then there exists the greatest rational number ǫ 0 0 such that the divisor D ǫ 0 is effective. By construction, the support of the divisor D ǫ 0 does not contain at least one curve contained in the support of the divisor R. Moreover, if (S, D) is not log canonical at P , but (S, R) is log canonical at P , then (S, D ǫ 0 ) is not log canonical at P . Now, we suppose that the surface S is smooth at P . Proof. The required assertion follows from Definition A.1 and Lemma A.3.
The following handy statement is a very special case of a much more general result, which is known as Inversion of Adjunction (see, for example, [134,Theorem 6.29]).
Therefore, we may assume that m 1. This implies that the log pair (S, D) is log canonical in a punctured neighborhood of the point P ∈ S. Since the log pair (S, D) is not log canonical at P , there exists a birational morphism h : S → S that is a composition of s 1 blowups of points dominating P such that e s > 1, where e s is a rational number determined by where each e i is a rational number, each E i is an h-exceptional divisor, ∆ is a proper transform on the surface S of the divisor ∆, and C 1 is a proper transform on S of the curve C 1 .
Let ∆ and C 1 be the proper transforms on S of the divisor ∆ and the curve C 1 , respectively. Then (S, a 1 C 1 + (a 1 + m − 1)E + ∆) is not log canonical at some point Q ∈ E by Lemma A.5.
Let us prove the inequality (C 1 · ∆) P > 1 by induction on s. If s = 1, then a 1 + m − 1 > 1, which implies that m > 2 − a 1 1, so that (C 1 · ∆) P m > 1 as required. Thus, we may assume that s 2 and a 1 + m − 1 2. Since it is enough to show that m + (C 1 · ∆) Q > 1. We may also assume that m 1, since (C 1 · ∆) P m.
Corollary A.7. In the notation and assumptions of Lemma A.5, suppose that mult P (D) 2. Then there exists a unique point Q ∈ E such that (A.4) is not log canonical at Q.
Proof. If (A.4) is not log canonical at two distinct points P 1 and P 2 , then 2 mult P (D) = D · E D · E P 1 + D · E P 2 > 2 by Lemma A.6. Now use Lemma A.5.
The following result plays an essential role in the proof of Theorem 2.15 given in Section 2.2. In fact, this theorem has been discovered [31] in an attempt to give a simple proof of Theorem 2.15, since its original proof in [38] is very technical. For other applications of Theorem 2.15, see [32,212].
Proof. We may assume that a 1 1 and a 2 1. There is a morphism h : S → S that is a composition of s 1 blowups of points dominating P such that e s > 1 for e s ∈ Q that is determined by where each e i is a rational number, each E i is an h-exceptional divisor, C 1 and C 2 , are proper transforms on S of the curves C 1 and C 2 , respectively, and ∆ is a proper transform of the divisor ∆. Let ∆, C 1 , C 2 be the proper transforms on S of the divisors ∆, C 1 and C 2 , respectively. Then S, a 1 C 1 + a 2 C 2 + a 1 + a 2 + m − 1 E + ∆ is not log canonical at some point Q ∈ E by Lemma A.5. If s = 1, then a 1 + a 2 + m − 1 > 1. If m > 2 − a 1 − a 2 , then m > 2(1 − a 1 ) or m > 2(1 − a 2 ), because otherwise we would have 2m 4 − 2(a 1 + a 2 ), which contradicts to m > 2 − a 1 − a 2 . Then (∆ · C 1 ) P > 2(1 − a 2 ) or (∆ · C 2 ) P > 2(1 − a 1 ) if s = 1.
Let us prove the required assertion by induction on s. The case s = 1 is already done, so that we may assume that s 2 and a 1 + a 2 + m 2. If Q = E ∩ C 1 and Q = E ∩ C 2 , then m = ∆ · E > 1 40 by Lemma A.6, which is impossible by assumption. Thus, either Q = E ∩ C 1 or Q = E ∩ C 2 . Without loss of generality, we may assume that Q = E ∩ C 1 .
Almost all results we have considered so far in this subsection are local (except for Remark A.2). Let us conclude this subsection by two global statements. The first of them is due to Puhklikov: Lemma A.9 ([134, Lemma 5.36]). Suppose that S is a smooth surface in P 3 , and D is Q-linearly equivalent to its hyperplane section. Then each a i does not exceed 1.
Proof. Let X be a cone over the curve C i whose vertex is a general enough point in P 3 . Then where C i is an irreducible curve of degree (deg(S) − 1)deg(C i ). Moreover, C i is not contained in the support of the divisor D, and the intersection C i ∩ C i consists of exactly deg( C i ) points. Then deg C i = D · C i a i C i · C i a i deg C i , which implies that a i 1.
The second global result we want to mention is the following lemma about del Pezzo surfaces of degree 2 that have at most two ordinary double points.
Lemma A.10. Suppose that there is a double cover τ : S → P 2 branched over an irreducible quartic curve B that has at most two ordinary double points, and Then each a i does not exceed 1. Moreover, if (S, D) is not log canonical at P , then τ (P ) ∈ B.
Proof. Write D = a 1 C 1 + ∆, where ∆ = a 2 C 2 + · · · + a r C r . Suppose that a 1 > 1. Let us seek for a contradiction. Since 2 = −K S · D = −K S · (a 1 C 1 + ∆) = −a 1 K S · C 1 − K S · ∆ −a 1 K S · C 1 > −K S · C 1 , we have −K S · C 1 = 1. Then τ (C 1 ) is a line. Hence, the surface S contains an irreducible curve Z 1 such that C 1 + Z 1 ∼ −K S and τ (C 1 ) = τ (Z 1 ). Note that the curves C 1 and Z 1 are interchanged by the biregular involution of the surface S induced by the double cover τ . Then which implies that C 1 · Z 1 = 1 − C 2 1 . Since C 1 and Z 1 are smooth rational curves, we have where k is the number of singular points of S that lie on C 1 . Now we write D = a 1 C 1 + b 1 Z 1 + Θ, where b 1 is a non-negative rational number, and Θ is an effective Q-divisor whose support does not contains the curves C 1 and Z 1 . Then 1 = C 1 · a 1 C 1 + b 1 Z 1 + Θ a 1 C 2 1 + b 1 C 1 · Z 1 = a 1 C 2 1 + b 1 1 − C 2 1 , and hence 1 a 1 C 2 1 + b 1 (1 − C 2 1 ). Similarly, from Z 1 · D = 1, we obtain 1 b 1 C 2 1 + a 1 1 − C 2 1 . The obtained two inequalities imply that a 1 1 and b 1 1, because C 2 1 = −1 + k 2 and k 2. Since a 1 > 1 by our assumption, this is a contradiction.
We see that a 1 1. Similarly, we see that a i 1 for every i. Now we suppose that the log pair (S, D) is not log canonical at P . Let us show that τ (P ) ∈ B. Suppose that τ (P ) ∈ B. Then S is smooth at P . Let us seek for a contradiction.
Let H be a general curve in | − K S | that passes through the point P . Then Since τ (P ) ∈ B, there exists a unique (possibly reducible) curve R ∈ | −K S | such that its proper transform on S passes through the point Q. Note that R is smooth at P . This enables us to assume that the support of D does not contain at least one irreducible component of R by Remark A.2. Denote by R the proper transform of R on the surface R. If the curve R is irreducible, then 2 − mult P (D) = 2 − mult P (C)mult P (D) = R · D mult Q (R)mult Q (D) = mult Q (D), which is impossible, since mult P (D) + mult Q (D) > 2. Thus, the curve R must be reducible.
Write R = R 1 + R 2 , where R 1 and R 2 are irreducible smooth curves. Without loss of generality we may assume that the curve R 1 is not contained in Supp(D). Then P ∈ R 2 , because otherwise we would have 1 = D · R 1 mult P (D) > 1, since mult P (D) > 1 by Lemma A.3. Thus, we put D = aR 2 + Ω, where a is a non-negative rational number and Ω is an effective Q-divisor whose support does not contain the curve R 2 . Then where l is the number of singular points of the surface S contained in R 1 . Denote by R 2 the proper transform on S of the curve R 2 , and denote by Ω the proper transform on S of the divisor Ω. Then the log pair S, aR 2 + Ω + mult P (D) − 1 E is not log canonical at Q. Note that we already proved that a 1. Thus, using Lemma A.6, we get 2 − 1 2 l a = R 2 · Ω + mult P (D) − 1 E > 1.
This is a contradiction. 42