AFFINE CONES OVER SMOOTH CUBIC SURFACES

. We show that aﬃne cones over smooth cubic surfaces do not admit non-trivial G a -actions.


Introduction
One of the motivations for the present article originates from the articles of H. A. Schwartz ( [34]) and G. H. Halphen ( [17]) in the middle of 19th century, where they studied polynomial solutions of Brieskorn-Pham polynomial equations in three variables after L. Euler (1756), J. Liouville (1879) and so fourth ( [12]). Meanwhile, since the middle of 20th century the study of rational singularities has witnessed great development ( [2], [5], [26]). These two topics, one classic and the other modern, encounter each other in contemporary mathematics. For instance, there is a strong connection between the existence of a rational curve on a normal affine surface, i.e., a polynomial solution to algebraic equations, and rational singularities ( [15]).
As an additive analogue of toric geometry, unipotent group actions, specially G a -actions, on varieties are very attractive objects to study. Indeed, G a -actions have been investigated for their own sake ( [3], [18], [29], [35], [40]). We also observe that G a -actions appear in the study of rational singularities. In particular, the article [15] shows that a Brieskorn-Pham surface singularity is a cyclic quotient singularity if and only if the surface admits a non-trivial regular G a -action. Considering its 3-dimensional analogue, H. Flenner  Does the affine Fermat cubic threefold x 3 + y 3 + z 3 + w 3 = 0 in A 4 admit a non-trivial regular G a -action?
Even though it is simple-looking, this problem stands open for 10 years. It turns out that this problem is purely geometric and can be considered in a much wider setting ( [19], [20], [21], [22], [31]).
To see the problem from a wider view point, we let X be a smooth projective variety with a polarisation H, where H is an ample divisor on X. The generalized cone over (X, H) is the affine variety defined byX = Spec n 0 H 0 (X, O X (nH)) .
Remark 1.1. The affine varietyX is the usual cone over X embedded in a projective space by the linear system |H| provided that H is very ample and the image of the variety X is projectively normal.
Let S d be a smooth del Pezzo surface of degree d and letŜ d be the generalized cone over (S d , −K S d ). For 3 d 9, the anticanonical divisor −K S d is very ample and the generalized The second author has been supported by the Research Center Program (Grant No. CA1205-02) of Institute for Basic Science in Korea. coneŜ d is the affine cone in A d+1 over the smooth variety anticanonically embedded in P d . In particular, for d = 3, the cubic surface S 3 is defined by an cubic homogenous polynomial equation F (x, y, z, w) = 0 in P 3 , and hence the coneŜ 3 is the affine hypersurface in A 4 defined by the equation F (x, y, z, w) = 0. For d = 2, the generalized coneŜ 2 is the affine cone in A 4 over the smooth hypersurface in the weighted projective space P(1, 1, 1, 2) defined by a quasihomogeneous polynomial of degree 4. For d = 1, the coneŜ 1 is the affine cone in A 4 over the smooth hypersurface in the weighted projective space P(1, 1, 2, 3) defined by a quasihomogeneous polynomial of degree 6 ( [16,Theorem 4.4]).
It is natural to ask whether the affine varietyŜ d admits a non-trivial G a -action. The problem at the beginning is just a special case of this.
T. Kishimoto, Yu. Prokhorov and M. Zaidenberg have been studying this generalised problem and proved the following: Theorem 1.2. If 4 d 9, then the generalized coneŜ d admits an effective G a -action.
Proof. See [19,Theorem 3.19]. Their proofs make good use of a geometric property called cylindricity, which is worthwhile to be studied for its own sake. Indeed, what T. Kishimoto, Yu. Prokhorov and M. Zaidenberg proved for their two theorems is that the del Pezzo surface S d has a (−K S d )-polar cylinder if 4 d 9 but no (−K S d )-polar cylinder if d 2.
The main result of the present article is Theorem 1.7. A smooth cubic surface S 3 in P 3 does not contain any (−K S 3 )-polar cylinders.
Together with Theorems 1.2 and 1.3, this makes us reach the following conclusion. In particular, we here present a long-expected answer to the question raised by H. Flenner and M. Zaidenberg. Corollary 1.9. The affine Fermat cubic threefold x 3 + y 3 + z 3 + w 3 = 0 in A 4 does not admit a non-trivial regular G a -action.
In order to show the non-existence of a (−K S 3 )-polar cylinder on a cubic del Pezzo surface S 3 , we apply the following statement. Lemma 1. 10. Let S d be a smooth del Pezzo surface of degree d 4. Suppose that S d contains a (−K S d )-polar cylinder, i.e., there is an open affine subset U ⊂ S d and an effective anticanonical Q-divisor 1 D such that U = S d \ Supp(D) and U ∼ = Z × A 1 for some smooth rational affine curve Z. Then there exists a point P on S d such that • the log pair (S d , D) is not log canonical at the point P ; • if there exists a unique divisor T in the anticanonical linear system |− K S d | such that the log pair (S d , T ) is not log canonical at the point P , then there is an effective anticanonical Q-divisor D ′ on the surface S d such that the log pair (S d , D ′ ) is not log canonical at the point P ; -the support of D ′ does not contain at least one irreducible component of the support of the divisor T .
Proof. This follows from [19,Lemma 4.11] and the proof of [19,Lemma 4.14] (cf. the proof of [22,Lemma 5.3]). Since the proof is presented implicitly and dispersedly in [19], for the convenience of the reader, we give a detailed proof in Appendix A.
Applying Lemma 2.2, we easily obtain Corollary 1.11. Let S 3 be a smooth del Pezzo surface of degree 3. Suppose that S 3 contains a (−K S 3 )-polar cylinder. Then there is an effective anticanonical Q-divisor D on S 3 such that • the pair (S 3 , D) is not log canonical at some point P on S 3 ; • the support of D does not contain at least one irreducible component of the tangent hyperplane section T P of S 3 at the point P .
The lemma above may be one example that shows how important it is to study singularities of effective anticanonical Q-divisors on Fano manifolds. In addition, it shows that the problem proposed at the beginning is strongly related to the log canonical thresholds of effective anticanonical Q-divisors on del Pezzo surfaces.
In this article, we prove the following Theorem 1.12. Let S d be a smooth del Pezzo surface of degree d 3 and let D be an effective anticanonical Q-divisor on S d . Suppose that the log pair (S d , D) is not log canonical at a point P . Then there exists a unique divisor T in the anticanonical linear system | − K S d | such that the log pair (S d , T ) is not log canonical at the point P . Moreover, the support of D contains all the irreducible components of Supp(T ).
Corollary 1.13. Let S 3 be a smooth cubic surface in P 3 and let D be an effective anticanonical Q-divisor on S 3 . Suppose that the log pair (S 3 , D) is not log canonical at a point P . Then for the tangent hyperplane section T P at the point P , the log pair (S 3 , T P ) is not log canonical at P and Supp(D) contains all the irreducible components of Supp(T P ).
Note that Corollary 1.13 contradicts the conclusion of Corollary 1.11. It simply means that the hypothesis of Corollary 1.11 fails to be true. This shows that Theorem 1.12 implies Theorem 1.7. Moreover, we see that Theorem 1.12 recovers Theorem 1.3 through Lemma 1.10 as well. 1 An anticanonical Q-divisor on a variety X is a Q-divisor Q-linearly equivalent to an anticanonical divisor of X, meanwhile, an effective anticanonical divisor on X is a member of the anticanonical linear system | − KX |. Remark 1.14. The condition d 3 is crucial in Theorem 1.12. Indeed, if d 4, then the assertion of Theorem 1.12 is no longer true (cf. the proof of [19,Theorem 3.19]). For example, consider the case when d = 4. There exists a birational morphism f : S 4 → P 2 such that f is the blow up of P 2 at five points that lie on a unique irreducible conic. Denote this conic by C. LetC be the proper transform of the conic C on the surface S 4 and let E 1 , . . . , E 5 be the exceptional divisors of the morphism f . Put It is an effective anticanonical Q-divisor on S 4 and the log pair (S 4 , D) is not log canonical at any point P onC. Moreover, for any T ∈ | − K S 4 |, its support cannot be contained in the support of the divisor D.
To our surprise, Theorem 1.12 has other applications that are interesting for their own sake. From here to the end of this section, let X be a projective variety with at worst Kawamata log terminal singularities and let H be an ample divisor on X.
Definition 1.15. The α-invariant of the log pair (X, H) is the number defined by The invariant α(X, H) has been studied intensively by many people who used different notations for α(X, H) ( [1], [6], [14], [ [11,Theorem A.3]. In the case when X is a Fano variety, the invariant α(X, −K X ) is known as the famous α-invariant of Tian and it is denoted simply by α(X). The α-invariant of Tian plays a very important role in Kähler geometry due to the following. Theorem 1.16 ( [13], [30], [36]). Let X be a Fano variety of dimension n with at worst quotient singularities. If α(X) > n n+1 , then X admits an orbifold Kähler-Einstein metric. The exact values of the α-invariants of smooth del Pezzo surfaces, as below, have been obtained in [7,Theorem 1.7]. Those of del Pezzo surfaces defined over a field of positive characteristic are presented in [28,Theorem 1.6] and those of del Pezzo surface with du Val singularities in [8] and [33]. Suppose that α(S d ) < µ. Then there are an effective anticanonical Q-divisor D and a positive rational number λ < µ such that (S d , λD) is not log canonical at some point P on S d . Since λ < 1, the log pair (S d , D) is not log canonical at the point P either. By Theorem 1.12, there exists a divisor T ∈ | − K S d | such that (S d , T ) is not log canonical at P . Since (S d , λT ) is log canonical at P and (S d , λD) is not log canonical at P , the log pair (S d , λD ǫ ) is not log canonical at P either (see Lemma 2.2). In particular, the log pair (S d , D ǫ ) is not log canonical at P . However, this contradicts Theorem 1.12 since D ǫ is an effective anticanonical Q-divisor. Therefore, α(S d ) = µ. The problem on the existence of Kähler-Einstein metrics on smooth del Pezzo surfaces is completely solved by G. Tian and S.-T. Yau in [37] and [39]. In particular, Corollary 1.19 follows from [37,Main Theorem].
The invariant α(X, H) has a global nature. It measures the singularities of effective Q-divisors on X in a fixed Q-linear equivalence class. F. Ambro suggested in [1] a function that encodes the local behavior of α(X, H).

Definition 1.20 ( [1]
). The α-function α H X of the log pair (X, H) is a function on X into real numbers defined as follows: for a given point P ∈ X, α H X (P ) = sup λ ∈ Q the log pair (X, λD) is log canonical at the point P ∈ X for every effective Q-divisor D on X with D ∼ Q H. . Lemma 1.21. The identity α(X, H) = inf P ∈X α H X (P ) holds. Proof. It is easy to check.
In the case when X is a Fano variety, we denote the α-function of the log pair (X, −K X ) simply by α X . Example 1.22. One can easily see that α P n (P ) 1 n+1 for every point P on P n . This implies that the α-function α P n is the constant function with the value 1 n+1 since α(P n ) = 1 n+1 . Example 1.23. It is easy to see α P 1 ×P 1 (P ) 1 2 for every point P on P 1 × P 1 . Since α(P 1 × P 1 ) = 1 2 by Theorem 1.17, the α-function α P 1 ×P 1 is the constant function with the value 1 2 by Lemma 1.21. Moreover, if X is a Fano variety with at most Kawamata log terminal singularities, then the proof of [11,Lemma 2.21] shows that α X×P 1 (P ) = min 1 2 , α X (pr 1 (P )) for every point P on X × P 1 , where pr 1 : X × P 1 → X is the projection on the first factor. Using the similar argument in the proof of [11,Lemma 2.29], one can show that the α-function of a product of Fano varieties with at most Gorenstein canonical singularities is the point-wise minimum of the pull-backs of the α-functions on the factors.
As shown in Remark 1.18, the following can be obtained from Theorem 1.12 in a similar manner.
Corollary 1.24. Let S d be a smooth del Pezzo surface of degree d 3. Then the α-function of S d is as follows: if there is an effective anticanonical divisor that consists of two 0-curves intersecting tangentially at the point P ; Let us describe the structure of this article. In Section 2, we describe the results that will be used in the proofs of Theorems 1.12 and 1.25. We also prove Theorem 1.12 for a smooth del Pezzo surface of degree 1 (see Lemma 2.3). In Section 3, we prove two results about singular del Pezzo surfaces of degree 2 that will be used in the proofs of Theorems 1.12 and 1.25. In addition, we verify Theorem 1.12 for a smooth del Pezzo surface of degree 2 (see Lemma 3.5). In Section 4, we prove Theorem 1.12 omitting the proof of Lemma 4.8 that plays a crucial role in the proof of Theorem 1.12. In Section 5, we prove Lemma 4.8. In Section 6, Theorem 1.25 is shown. In Appendix A, we prove Lemma 1.10.

Preliminaries
This section presents simple but essential tools for the article. Most of the described results here are well-known and valid in much more general settings (cf. [23], [24] and [25]).
Let S be a projective surface with at most du Val singularities, let P be a smooth point of the surface S and let D be an effective Q-divisor on S.
Lemma 2.1. If the log pair (S, D) is not log canonical at the point P , then mult P (D) > 1.
Proof. This is a well-known fact. See [25, Proposition 9.5.13], for instance.
s are distinct prime divisors on the surface S and a i 's are positive rational numbers.
(3) the support of the divisor D µ does not contain at least one component of Supp(T ); (4) if (S, T ) is log canonical at P but (S, D) is not log canonical at P , then (S, D µ ) is not log canonical at P .
Proof. The first assertion is obvious. For the rest we put For some index k we have c = b k a k . Suppose that c 1. Then a i b i for every i. It means that the divisor However, it is impossible since D − T is non-zero and numerically trivial on a projective surface. Thus, c > 1, and hence b k > a k .
In particular, the divisor D µ is effective and its support does not contain the curve D k . Moreover, for every positive rational number ǫ, and hence D ǫ is not effective. This proves the second and the third assertions.
If both (S, T ) and (S, D µ ) are log canonical at P , then (S, D) must be log canonical at P because D = µ 1+µ T + 1 1+µ D µ and µ 1+µ + 1 1+µ = 1. Despite its naïve appearance, Lemma 2.2 is a very handy tool. To illustrate this, we here verify Theorem 1.12 for a del Pezzo surface of degree 1. This simple case also immediately follows from the proof of Proof. Let T be a curve in | − K S | that passes through the point P . Note that T is irreducible. If the log pair (S, T ) is log canonical at P , then it follows from Lemma 2.2 that there exists an effective anticanonical Q-divisor D ′ on the surface S such that the log pair (S, D ′ ) is not log canonical at P and Supp(D ′ ) does not contain the curve T . We then obtain 1 = T · D ′ mult P (D ′ ). This is impossible by Lemma 2.1. Thus, the log pair (S, T ) is not log canonical at the point P .
Moreover, the divisor T is singular at the point P . Therefore, the point P is not the base point of the pencil | − K S |. Consequently, such a divisor T is unique.
If the curve T is not contained in Supp(D), then 1 = T · D mult P (D). Therefore, the curve T must be contained in Supp(D) by Lemma 2.1.
The following is a ready-made Adjunction for our situation. See [24,Theorem 5.50] for a more general version.
Lemma 2.4. Suppose that the log pair (S, D) is not log canonical at P . If a component D j with a j 1 is smooth at the point P , then Proof. See [28, Lemma 2.5] for a characteristic-free proof in dimension 2.
Let f :S → S be the blow up of the surface S at the point P with the exceptional divisor E and letD be the proper transform of D by the blow up f . Then 3. Del Pezzo surfaces of degree 2 Let S be a del Pezzo surface of degree 2 with at most two ordinary double points. Then the linear system | − K S | is free from base points and induces a double cover π : S → P 2 ramified along a reduced quartic curve R ⊂ P 2 . Moreover, the curve R has at most two ordinary double points. In particular, the quartic curve R is irreducible. Proof. Suppose it is not true. Then we may write D = m 1 C 1 + Ω, where C 1 is an irreducible reduced curve, m 1 is a positive rational number strictly bigger than 1 and Ω is an effective Q-divisor whose support does not contain the curve C 1 . Since we have −K S · C 1 = 1. Then π(C 1 ) is a line in P 2 . Thus, there exists an irreducible reduced curve C 2 on S such that C 1 + C 2 ∼ −K S and π(C 1 ) = π(C 2 ). Note that C 1 = C 2 if and only if the line π(C 1 ) is an irreducible component of the branch curve R. Since the curve R is irreducible, this is not the case. Thus, we have C 1 = C 2 .
Note that C 2 1 = C 2 2 because C 1 and C 2 are interchanged by the biregular involution of S induced by the double cover π. Thus, we have 1 . Since C 1 and C 2 are smooth rational curves, we can easily obtain where k is the number of singular points of S that lie on the curve where m 2 is a non-negative rational number and Γ is an effective Q-divisor whose support contains neither the curve C 1 nor the curve C 2 . Then . The obtained two inequalities imply that m 1 1 and m 2 1 since C 2 1 = −1 + k 2 , k = 0, 1, 2. Since m 1 > 1 by assumption, it is a contradiction.
The following two lemmas can be verified in a similar way as that of [7,Lemma 3.5]. Nevertheless we present their proofs for reader's convenience. Proof. Suppose that (S, D) is not log canonical at a point P whose image by π lies outside R.
Let H be a general curve in | − K S | that passes through the point P . Since π(P ) ∈ R, the surface S is smooth at the point P . Then 2 = H · D mult P (H)mult P (D) mult P (D), and hence mult P (D) 2.
Let f :S → S be the blow up of the surface S at the point P . We have whereD is the proper transform of the divisor D on the surfaceS and E is the exceptional curve of the blow up f . Then the log pair (S,D + (mult P (D) − 1) E) is not log canonical at some point Q on E but log canonical at every point of E other than the point Q by Remark 2.5.
In addition, we have Since π(P ) ∈ R, there exists a unique reduced but possibly reducible curve C ∈ | − K X | such that the curve C passes through the point P and its proper transformC by the blow up f passes through the point Q. Note that the curve C is smooth at the point P . Since (S, C) is log canonical at the point P , Lemma 2.2 enables us to assume that the support of D does not contain at least one irreducible component of the curve C.
If the curve C is irreducible, then This contradicts (3.3). Thus, the curve C must be reducible. We may then write C = C 1 +C 2 , where C 1 and C 2 are irreducible smooth curves that intersect at two points. Without loss of generality we may assume that the curve C 1 is not contained in the support of D. The point P must belong to C 2 : otherwise we would have We put D = nC 2 + Ω, where n is a non-negative rational number and Ω is an effective Q-divisor whose support does not contain the curve C 2 . Then where k is the number of singular points of S on C 1 . On the other hand, the log pair (S, nC 2 + Ω + (mult P (D) − 1)E) is not log canonical at the point Q, whereC 2 andΩ are the proper transforms of C 2 and Ω, respectively, on the surfaceS, and we have n 1 by Lemma 3.1. We then obtain from Lemma 2.4. This is a contradiction. since T P is singular at the point P . Hence, T P must be reducible. We may then write T P = T 1 + T 2 , where T 1 and T 2 are smooth rational curves. Note that the point P is one of the intersection points of T 1 and T 2 . Without loss of generality, we may assume that the curve T 1 is not contained in the support of D.  Lemma 3.5. Suppose that the del Pezzo surface S is smooth. Let D be an effective anticanonical Q-divisor on S. Suppose that the log pair (S, D) is not log canonical at a point P . Then there exists a unique divisor T ∈ | − K S | such that (S, T ) is not log canonical at P . The support of the divisor D contains all the irreducible components of T . In case, the divisor T is either an irreducible rational curve with a cusp at P or a union of two −1-curves meeting tangentially at the point P .
Proof. By Lemma 3.2, the point π(P ) must lie on R. Then there exists a unique curve T ∈ |−K S | that is singular at the point P . By Lemma 3.4, the log pair (S, T ) is not log canonical at P .
Suppose that the support of D does not contain an irreducible component of T . Then the proof of Lemma 3.4 works verbatim to derive a contradiction.
Consequently, Lemma 3.5 shows that Theorem 1.12 holds for a smooth del Pezzo surface of degree 2.

Cubic surfaces
In the present section we prove Theorem 1.12. Lemma 2.3 and Lemma 3.5 show that Theorem 1.12 holds for del Pezzo surfaces of degrees 1 and 2, respectively. Thus, to complete the proof, let S be a smooth cubic surface in P 3 and let D be an effective anticanonical Q-divisor of the surface S. Proof. Suppose not. Then we may write D = mC + Ω, where C is an irreducible curve, m is a positive rational number strictly bigger than 1 and Ω is an effective Q-divisor whose support does not contain the curve C. Then It implies that the curve C is either a line or an irreducible conic.
Suppose that C is a line. Let Z be a general irreducible conic on S such that Z + C ∼ −K S . Since Z is general, it is not contained in the support of D. We then obtain 2 = Z · D = Z · (mC + Ω) = 2m + Z · Ω 2m.
It contradicts our assumption.
Suppose that C is an irreducible conic. Then there exists a unique line L on S such that L + C ∼ −K S . Write D = mC + nL + Γ, where n is a non-negative rational number and Γ is an effective Q-divisor whose support contains neither the conic C nor the line L. Then On the other hand, 2 = C · D = C · (mC + nL + Γ) = 2n + C · Γ 2n.
Combining two inequalities, we obtain 2m 1+n 2. This contradicts our assumption too.
For a point P on S, let T P be the tangent hyperplane section of the surface S at the point P . This is the unique anticanonical divisor that is singular at the point P . The curve T P is reduced but it may be reducible.
In order to prove Theorem 1.12 we must show that (S, D) is log canonical at the point P provided that one of the following two conditions is satisfied: • the log pair (S, T P ) is log canonical at P ; • the log pair (S, T P ) is not log canonical at P but Supp(D) does not contain at least one irreducible component of the curve T P . The log pair (S, T P ) is log canonical at P if and only if the point P is an ordinary double point of the curve T P . Thus, (S, T P ) is log canonical at P if and only if T P is one of the following curves: an irreducible cubic curve with one ordinary double point, a union of three coplanar lines that do not intersect at one point, a union of a line and a conic that intersect transversally at two points.
Overall, we must consider the following cases: (a) T P is a union of three lines that intersect at the point P (Eckardt point); (b) T P is a union of a line and a conic that intersect tangentially at the point P ; (c) T P is an irreducible cubic curve with a cusp at the point P ; (d) T P is an irreducible cubic curve with one ordinary double point; (e) T P is a union of three coplanar lines that do not intersect at one point; (f) T P is a union of a line and a conic that intersect transversally at two points. We consider these cases one by one in separate lemmas, i.e., Lemmas 4.3, 4.5, 4.6, 4.7, 4.8 and 4.9. We however present the detailed proof of Lemma 4.8 in Section 5 to improve the readability of this section. These lemmas altogether imply Theorem 1.12.   Proof. Suppose that the log pair (S, D) is not log canonical at the point P . Let L and C be the line and the conic, respectively, such that T P = L + C. By Lemma 4.2, we may assume that the conic C is not contained but the line L is contained in the support of D. We write D = nL + Ω, where n is a positive rational number and Ω is an effective Q-divisor whose support contains neither the line L nor the conic C. We have mult P (D) C · D = 2. We writeD = nL +Ω, whereΩ andL are the proper transforms of the divisor D and the line L, respectively, on the surfaceS. LetC be the proper transform of the conic C on the surfacẽ S. Note that the three curvesL,C and E meet at one point.
The log pair (S, nL+Ω+(mult P (D)−1)E) is not log canonical at some point Q on E. However, it is log canonical at every point on E except the point Q by Remark 2.5 since mult P (D) 2. We also obtain mult P (D) + mult Q (D) > 2 from Remark 2.5. This implies that the point Q does not belong toC, and hence not toL either. Indeed, if so, then 2 − mult P (D) =C · nL +Ω n + mult Q (Ω) = mult Q (D).
This contradicts the inequality from Remark 2.5. Let g :S →S be the contraction of the −2-curveL. ThenS is a del Pezzo surface of degree 2 with one ordinary double point. In particular, the linear system | − KS| is free from base points and induces a double cover π :S → P 2 ramified along an irreducible singular quartic curve R ⊂ P 2 . Note that the point g(L) is the ordinary double point of the surfaceS. PutΩ = g(Ω), E = g(E),C = g(C) andQ = g(Q). Then π(Ē) = π(C) sinceĒ +C is an anticanonical divisor onS. The point π(Q) lies outside R because the point Q lies outsideC. Since the divisor Ω + mult P (D) − 1 Ē is Q-linearly equivalent to −KS by construction, Lemma 3.2 shows that the log pair (S,Ω + (mult P (D) − 1)Ē) is log canonical atQ. However, it is not log canonical at the pointQ since g is an isomorphism in a neighborhood of the point Q. It is a contradiction. Lemma 4.6. Suppose that the tangent hyperplane section T P is an irreducible cubic curve with a cusp at the point P . If the curve T P is not contained in the support of D, then the log pair (S, D) is log canonical at the point P .
Proof. First, from the inequality 3 = T P · D mult P (T P )mult P (D) = 2mult P (D), we obtain mult P (D) shows that it is log canonical at every point on E except the point Q since mult P (D) 3 2 . We also obtain mult P (D) + mult Q (D) > 2 from Remark 2.5.
The surfaceS is a smooth del Pezzo surface of degree 2. The linear system | − KS| induces a double cover π :S → P 2 ramified along a smooth quartic curve R ⊂ P 2 . LetT P be the proper transform of the curve T P on the surfaceS. Then the integral divisor E +T P is linearly equivalent to −KS, and hence π(E) = π(T P ) is a line in P 2 . Moreover, the curveT P tangentially meet the curve E at a single point. Thus the point π(Q) lies on R if and only if the point Q is the intersection point of E andT P .
Applying Lemma 3.2 to the log pair (S,D + mult P (D) − 1 E), we see that the point π(Q) belongs to R because the log pair (S,D + (mult P (D) − 1)E) is not log canonical at the point Q and the divisorD + (mult P (D) − 1)E is Q-linearly equivalent to −KS. The point Q therefore lies on the curveT P . Then This contradicts Lemma 2.1.
For the remaining three cases, we show that the hypothesis of Theorem 1.12 is never fulfilled, so that Theorem 1.12 is true.
Lemma 4.7. If the tangent hyperplane section T P is an irreducible cubic curve with a node at the point P , then the log pair (S, D) is log canonical at the point P .
Proof. Suppose that (S, D) is not log canonical at P . The surfaceS is a smooth del Pezzo surface of degree two. SinceD + (mult P (D) − 1)E ∼ Q −K Y and the log pair (S,D + (mult P (D) − 1)E) is not log canonical at some point Q on E, it follows from Lemma 3.5 that there must be an anticanonical divisor H on the surfaceS such that has either a tacnode or a cusp at the point Q.
If the divisor H has a tacnode at the point Q, then it consists of the exceptional divisor E and another −1-curve L meeting E tangentially at Q. Then the divisor f (H) is an effective anticanonical divisor on S such that it has a cusp at the point P and it is distinct from the divisor T P . This is impossible.
If the divisor H has a cusp at the point Q, then it must be irreducible. However, it is impossible since H is singular at the point Q and E · H = 1.  Proof. We write T P = L + C, where L is a line and C is an irreducible conic that intersect L transversally at the point P . Suppose that (S, D) is not log canonical at the point P .
By Lemmas 2.2 and 4.2, we may assume that the conic C is not contained but the line L is contained in the support of D. We write D = nL + Ω, where n is a positive rational number and Ω is an effective Q-divisor whose support contains neither the line L nor the conic C.
The log pair (S,D + (mult P (D) − 1)E) is not log canonical at some point Q on E. However, Remark 2.5 shows that it is log canonical at every point on E except the point Q since mult P (D) D · C = 2.
LetΩ,L andC be the proper transforms of the divisor Ω, the line L and the conic C by the blow up f , respectively.
Suppose that the point Q does not belong to the −2-curveL. Let g :S →S be the contraction of the curveL. ThenS is a del Pezzo surface of degree 2 with only one ordinary double point at the point g(L). In particular, the linear system | − KS| induces a double cover π :S → P 2 ramified along an irreducible singular quartic curve R ⊂ P 2 . PutΩ = g(Ω),Ē = g(E),C = g(C) andQ = g(Q). Then π(Ē) = π(C) sinceĒ +C is an anticanonical divisor onS. The point π(Q) lies on R if and only if the point Q lies onC. The log pair (S,Ω + (mult P (D) − 1)Ē) is not log canonical atQ since g is an isomorphism in a neighborhood of the point Q. Since the divisorΩ + mult P (D) − 1 Ē is Q-linearly equivalent to −KS by construction, Lemma 3.2 shows that the point Q belongs toC.
Note thatC +Ē is the unique curve in |− KS| that is singular at the pointQ. But the log pair (S,C +Ē) is log canonical at the pointQ. Hence, it follows from Lemma 3.4 that the log pair (S,Ω + (mult P (D) − 1)Ē) is log canonical at the pointQ. This is a contradiction. Therefore, the point Q belongs to the −2-curveL. On the other hand, the divisorĽ +Č +Ě is an anticanonical divisor of the surfaceŠ. Since the pointP is the intersection point ofĽ andĚ and the divisorĎ is Q-linearly equivalent to −KŠ, Lemma 4.8 implies that (Š,Ď) is log canonical at the pointP . This is a contradiction.
As we already mentioned, Theorem 1.12 follows from Lemmas 4.3, 4.5, 4.6, 4.7, 4.8 and 4.9. Thus Theorem 1.12 has been proved under the assumption that Lemma 4.8 is valid. This will be shown in the following section.

The proof of Lemma 4.8
To prove Lemma 4.8, we keep the notations used in Section 4. We write T P = L + M + N , where L, M , and N are three coplanar lines on S. We may assume that the point P is the intersection point of the two lines L and M , while it does not lie on the line N . We also write D = a 0 L + b 0 M + c 0 N + Ω 0 , where a 0 , b 0 , and c 0 are non-negative rational numbers and Ω 0 is an effective Q-divisor on S whose support contains none of the lines L, M and N . Put m 0 = mult P (Ω 0 ).
Suppose that the log pair (S, D) is not log canonical at the point P . Let us seek for a contradiction.
By Lemma 4.1, the log pair (S, D) is log canonical outside finitely many points. In particular, we have 0 a 0 , b 0 , c 0 1. Also, Lemma 2.1 implies m 0 + a 0 + b 0 > 1. Proof. Since the log pair (S, a 0 L + b 0 M + Ω 0 ) is not log canonical at the point P either, it follows from Lemma 2.4 that The log pair (S, L + M + N ) is log canonical. Since the log pair (S, a 0 L + b 0 M + c 0 N + Ω 0 ) is not log canonical at P , it follows from Lemma 2.2 that the log pair is not log canonical at the point P . Then Lemma 2.1 shows It verifies m 0 + a 0 + b 0 > c 0 + 1.
Since the rational numbers a 0 , b 0 , c 0 are at most 1 and the log pair (S, L + M + N ) is log canonical, the effective Q-divisor Ω 0 cannot be the zero-divisor. Let r be the number of the irreducible components of the support of the Q-divisor Ω 0 . Then we write where e i 's are positive rational numbers and C i0 's are irreducible reduced curves of degrees d i0 on the surface S. We then see Denote byL,M andÑ the proper transforms of the lines L, M and N , respectively, on the surfaceS . For each i, denote byC i0 the proper transform of the curve C i0 on the surfaceS. Then Recall that a 0 + b 0 + m 0 = mult P (D). Proof. It immediately follows from the three inequalities The log pair is not log canonical at some point Q on E. Since mult P (D) = a 0 + b 0 + m 0 2, it follows from Remark 2.5 that the log pair (5.4) is log canonical at every point of the curve E other than the point Q.
Let g :S →S be the contraction of the −2-curvesL andM . ThenS is a del Pezzo surface of degree 2 with two ordinary double points at the points g(L) and g(M ). The linear system | − KS| induces a double cover π :S → P 2 ramified along an irreducible singular quartic curve R ⊂ P 2 .
Lemma 5.5. The point Q on the exceptional curve E belongs to either the −2-curveL or the −2-curveM .
Proof. Suppose that the point Q lies on neitherL norM . PutĒ = g(E),N = g(Ñ ) and Q = g(Q). In addition, we putC i0 = g(C i0 ) for each i. Then π(Ē) = π(N ). The point π(Q) lies outside R since the pointQ is a smooth point of the anticanonical divisorĒ +N onS.
Since g is an isomorphism in a neighborhood of the point Q, the log pair is not log canonical at the pointQ. The divisor c 0N + (a 0 + b 0 + m 0 − 1)Ē + r i=1 e iCi0 is an effective anticanonical Q-divisor on the surfaceS. Hence, we are able to apply Lemma 3.2 to the log pair (5.6) to obtain a contradiction.
From now on we may assume that the point Q is the intersection point of the −2-curveL and the −1-curve E without loss of generality.
Let ρ : S P 2 be the linear projection from the point P . Then ρ is a generically 2-to-1 rational map. Thus the map ρ induces a birational involution τ P of the cubic surface S. The involution τ P is classically known as the Geiser involution associated to the point P (see [27]).
Remark 5.7. By construction, the involution τ P is biregular outside the union L ∪ M ∪ N . In fact, one can show that τ P is biregular outside the point P and the line N . Moreover, one can show that τ P (L) = L and τ P (M ) = M .
For each i, put C i1 = τ P (C i0 ) and denote by d i1 the degree of the curve C i1 . We then employ new effective Q-divisors Let ι P be the biregular involution of the surfaceS induced by the double cover π. Then ι P induces a biregular involution υ P of the surfaceS since the surfaceS is the minimal resolution of singularities of the surfaceS. Thus, we have a commutative diagram On the other hand, we have υ P (E) =Ñ since π • g(E) = π • g(Ñ ). This means that there exists an isomorphism σ : S → S ′ that makes the diagram Since h is an isomorphism locally around the point Q, the log pair Now we are able to replace the original effective Q-divisor D by the new effective Q-divisor D 1 . By Lemma 5.8, both the Q-divisors have the same properties that we have been using so far. However, the new Q-divisor Ω 1 is slightly better than the original one Ω 0 in the sense of the following lemma.
Lemma 5.9. The degree of the Q-divisor Ω 1 is strictly smaller than the degree of Ω 0 , i.e., Proof. Since D 1 ∼ Q −K S by Lemma 5.8, we obtain On the other hand, we have a 0 + b 0 + c 0 + r i=1 e i d i0 = 3 by (5.2). Thus, we obtain because a 0 + b 0 + m 0 − 1 − c 0 > 0 by Lemma 5.1.
Repeating this process, we can obtain a sequence of the effective anticanonical Q-divisors on the surface S such that each log pair (S, D k ) is not log canonical at one of the three intersection points L ∩ M , L ∩ N and M ∩ N . Note that where C ik 's are irreducible reduced curves of degrees d ik . We then obtain a strictly decreasing sequence of rational numbers by Lemma 5.9. This is a contradiction since the subset r i=1 e i n i n 1 , n 2 , . . . , n r ∈ N ⊂ Q is discrete and bounded below. It completes the proof of Lemma 4.8.

α-functions on smooth del Pezzo surfaces
In this section, we prove Theorem 1.25. Let S d be a smooth del Pezzo surface of degree d.
Before we proceed, we here make a simple but useful observation. Proof. This is obvious.
We already show that the α-function α P 2 of the projective plane is the constant function with the value 1 3 (see Example 1.22) and the α-function α P 1 ×P 1 of the quadric surface is the constant function with the value 1 2 (see Example 1.23).
Lemma 6.2. The α-function α F 1 on the blow-up F 1 of P 2 at one point is the constant function with the value 1 3 . Proof. Let P be a given point on F 1 . Let π : F 1 → P 1 be the P 1 -bundle morphism onto P 1 . Let C be its section with C 2 = −1 and let L P be the fiber of the morphism π over the point π(P ). Since 2C + 3L P ∼ −K F 1 , we have α F 1 (P ) The surface S 7 is the blow-up of P 2 at two distinct points Q 1 and Q 2 . Let E be the proper transform of the line passing through the points Q 1 and Q 2 by the two-point blow up f : S 7 → P 2 with the exceptional curves E 1 and E 2 .
Lemma 6.3. The α-function on the del Pezzo surface S 7 of degree 7 has the following values Proof. Let P be a point on S. Then α S 7 (P ) α(S) = 1 3 by Theorem 1.17 and Lemma 1.21. If the point P belongs to E, then α S 7 (P ) 1 3 since 2E 1 + 2E 2 + 3E ∼ −K S . Therefore, α S 7 (P ) = 1 3 . Suppose that the point P lies outside E. Let L be a line on P 2 whose proper transform by the blow up f passes through the point P . Since f * (2L) + E is an effective anticanonical divisor passing through the point P , we have α S 7 (P ) 1 2 . Let g : S → P 1 × P 1 be the birational morphism obtained by contracting the −1-curve E. Then this morphism is an isomorphism around the point P . Then α S 7 (P ) α P 1 ×P 1 (g(P )) by Lemma 6.1. Since α P 1 ×P 1 is the constant function with the value 1 2 , we obtain α S 7 (P ) = 1 2 . Lemma 6.4. The α-function α S 6 on the del Pezzo surface S 6 of degree 6 is the constant function with the value 1 2 .
Proof. Let P be a given point on the del Pezzo surface S 6 . One can easily check α S 6 (P ) 1 2 . One the other hand, we have a birational morphism h : S 6 → S 7 , where S 7 is a del Pezzo surface of degree 7, such that the morphism h is an isomorphism around the point P and the point h(P ) is not on the −1-curve of S 7 connected to two different −1-curves. Then α S 6 (P ) Proof. Let P be a point on S 5 . Suppose that P lies on a −1-curve. Then there exists an effective anticanonical divisor not reduced at P . Thus, α S 5 (P ) 1 2 . Meanwhile, we have 1 2 = α(S 5 ) α S 5 (P ) by Lemma 1.21 and Theorem 1.17. Therefore, α S 5 (P ) = 1 2 . Suppose that the point P is not contained in any −1-curve. Then there exist exactly five irreducible smooth rational curves C 1 , . . . , C 5 passing through the point P with −K S · C i = 2 for each i (cf. the proof of [7, Lemma 5.8]). Moreover, for every C i , there are four irreducible smooth rational curves belongs to the bi-anticanonical linear system | − 2K S 5 | (cf. Remark 1.14). Therefore, α S 5 (P ) 2 3 . Suppose that α S 5 (P ) < 2 3 . Then there is an effective anticanonical Q-divisor D such that (S, λD) is not log canonical at the point P for some positive rational number λ < 2 3 . Then mult P (D) > 1 λ by Lemma 2.1. Let f : S 4 → S 5 be the blow up of the surface S 5 at the point P with the exceptional curve E and letD be the proper transform of the divisor D on the surface S 4 . Then the surface S 4 is a smooth del Pezzo surface of degree 4. We have which implies that the log pair (S 4 , λD + (λmult P (D) − 1)E) is not log canonical.
Lemma 6.6. The α-function on a del Pezzo surface S 4 of degree 4 has the following values if there is an effective anticanonical divisor that consists of two 0-curves intersecting tangentially at P ; 5/6 otherwise.
Proof. Let P be a point on S 4 . If the point P lies on a −1-curve L, then there are mutually disjoint five −1-curves E 1 , . . . , E 5 that intersect L. Let h : S 4 → P 2 be the contraction of all E i 's.
Since h(L) is a conic in P 2 , we see that 3L+ 1 i 5 E i is a member in the linear system |−2K S 4 | (cf. Remark 1.14). This means that α S 4 (P ) 2 3 . Therefore, α S 4 (P ) = 2 3 since α(S 4 ) α S 4 (P ) by Lemma 1.21 and α(S 4 ) = 2 3 by Theorem 1.17. Suppose that the point P does not lie on a −1-curve. Put ω = 3 4 in the case when there is an effective anticanonical divisor that consists of two 0-curves intersecting tangentially at the point P and put ω = 5 6 otherwise. One can easily find an effective anticanonical divisor F on the surface S 4 such that (S 4 , λF ) is not log canonical at the point P for every positive rational number λ > ω (see [32,Proposition 3.2]). This shows that α S 4 (P ) ω. Moreover, it is easy to check that the log pair (S 4 , ωC) is log canonical at the point P for each C ∈ | − K S 4 |.
Suppose α S 4 (P ) < ω. Then there is an effective anticanonical Q-divisor D such that (S, ωD) is not log canonical at the point P . Note that there are only finitely many effective anticanonical divisors C 1 , . . . , C k such that each (S 4 , C i ) is not log canonical at the point P . Applying Lemma 2.2, we may assume that for each i at least one irreducible component of Supp(C i ) is not contained in the support of D.
Let f : S 3 → S 4 be the blow up of the surface S 4 at the point P with the exceptional curve E and letD be the proper transform of the divisor D on the surface S 3 . Then S 3 is a smooth cubic surface in P 3 and the curve E is a line in S 3 . Moreover, the log pair (S 3 ,D + (mult P (D) − 1)E) is not log canonical at some point Q on E because the log pair (S 4 , D) is not log canonical at the point P .
Let T Q be the tangent hyperplane section of the cubic surface S 3 at the point Q. Note that the divisor T Q contains the line E. SinceD + (mult P (D) − 1)E is an effective anticanonical Q-divisor on S 3 , it follows from Corollary 1.13 that the log pair (S 3 , T Q ) is not log canonical at the point Q and the support ofD contains all the irreducible components of T Q . In fact, it follows that the divisor T Q is either a union of three lines meeting at the point Q or a union of a line and a conic intersecting tangentially at the point Q. The divisor f (T Q ) is an effective anticanonical divisor on S 4 such that the log pair (S 4 , f (T Q )) is not log canonical at the point P . This contradicts our assumption since the support of D contains all the irreducible components of the divisor f (T Q ).
Consequently, Theorem 1.25 follows from Examples 1.22 and 1.23, and Lemmas 6.2, 6.3, 6.4, 6.5 and 6.6. Proof. This immediately follows from the exact sequence To prove Lemma 1.10, we must show that there exists a point P ∈ S such that • the log pair (S, D) is not log canonical at the point P ; • if there exists a unique divisor T in the anticanonical linear system | − K S | such that the log pair (S, T ) is not log canonical at the point P , then there is an effective anticanonical Q-divisor D ′ on the surface S such that the log pair (S, D ′ ) is not log canonical at the point P ; -the support of D ′ does not contain at least one irreducible component of the support of the divisor T . The natural projection U ∼ = Z × A 1 → Z induces a rational map π : S P 1 given by a pencil L on the surface S. Then either L is base-point-free or its base locus consists of a single point. Proof. Suppose that the pencil L is base-point-free. Then π is a morphism, which implies that there exists exactly one irreducible component of Supp(D) that does not lie in a fiber of π. Moreover, this component is a section. Without loss of generality, we may assume that this component is D r . Let L be a sufficiently general curve in L. Then and hence a r = 2. It implies α(S) 1 2 . However, it contradicts Theorem 1.17 since the degree of the surface S is at most 4.
Denote the unique base point of the pencil L by P . Let us show that the point P is the point we are looking for. Resolving the base locus of the pencil L, we obtain a commutative diagram where f is a composition of blow ups at smooth points over the point P and g is a morphism whose general fiber is a smooth rational curve. Denote by E 1 , . . . , E n the exceptional curves of the birational morphism f . Then there exists exactly one curve among them that does not lie in the fibers of the morphism g. Without loss of generality, we may assume that this curve is E n . Then E n is a section of the morphism g.
For every D i , denote byD i its proper transform on the surface W . Then every curveD i lies in a fiber of the morphism g. Proof. Write H = k i=1 ǫ i ∆ i , where each ǫ i is a non-negative rational number and each ∆ i is an irreducible reduced curve. Denote by∆ i the proper transform of ∆ i on the surface W for each i and putH = k i=1 ǫ i∆i . Then for some rational numbers δ 1 , . . . , δ n . For a sufficiently general fiber L of the morphism g, because E n is a section of the morphism g, every curve∆ i lies in a fiber of the morphism g and every curve E i with i < n also lies in a fiber of the morphism g. Hence, the log pair (S, H) is not log canonical at the point P .
Applying Lemma A.3 to (S, D), we see that the log pair (S, D) is not log canonical at P . Thus, if there exists no anticanonical divisor T such that (S, T ) is not log canonical at P , then we are done. Hence, to complete the proof of Lemma 1.10, we assume that there exists a unique divisor T ∈ | − K S | such that (S, T ) is not log canonical at P . Then Lemma 1.10 follows from the lemma below. Note that U = S \ Supp(D ′ ), which implies that the number of the irreducible components of Supp(D ′ ) may be less than rk Pic(S). Because of this, we can apply Lemma 2.2 only once here. This shows that we really need to use the uniqueness of the divisor T in the anticanonical linear system | − K S | such that (S, T ) is not log canonical at P in the proof of Lemma A.4. Indeed, if there is another divisor T ′ in | − K S | such that (S, T ′ ) is not log canonical at P either, then we would not be able to apply Lemma 2.2 since we may have D ′ = T ′ .