Affine cones over smooth cubic surfaces

We show that affine cones over smooth cubic surfaces do not admit non-trivial $\mathbb{G}_a$ -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 by X = Spec n 0 H 0 (X, O X (nH)) .
Remark 1.1.The affine variety X 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 let Ŝd 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.
cone Ŝd 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 cone Ŝ3 is the affine hypersurface in A 4 defined by the equation F (x, y, z, w) = 0.For d = 2, the generalized cone Ŝ2 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 cone Ŝ1 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 variety Ŝd admits a non-trivial G a -action.The problem at the beginning is just a special case of this.
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-divisor1 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.
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. Let C 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 on C.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], [4, § 3.4] [10, Definition 3.1.1],[11, Appendix A], [38,Appendix 2]).The notation α(X, H) is due to G. Tian who defined α(X, H) in a different way ( [38,Appendix 2]).However, both the definitions coincide by [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].
Theorem 1.17.Let S d be a smooth del Pezzo surface of degree d.Then Remark 1.18.Theorem 1.12 also provides the exact values of the α-invariants for smooth del Pezzo surfaces of degrees 3. We here show how to extract the values from Theorem 1.12.Let µ be the value in Theorem 1.17 for the α-invariant of S d .From [32,Proposition 3.2] we can easily obtain an effective anticanonical divisor C on the surface S d such that (S d , µC) is log canonical but not Kawamata log terminal.This gives us α(S d ) µ.
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 .In addition, Supp(D) contains all the irreducible components of Supp(T ).
The log pair (S d , λT ) is log canonical ( [32, Proposition 3.2]).Put D ǫ = (1 + ǫ)D − ǫT for every non-negative rational number ǫ.Then D 0 = D and D ǫ is effective for 0 < ǫ ≪ 1 because Supp(D) contains all the irreducible components of Supp(T ).Choose the biggest ǫ such that D ǫ is still effective.Then Supp(D ǫ ) does not contain at least one irreducible component of Supp(T ).
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 (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: 3/4 if the tangent hyperplane section at P has a tacnode at the point P ; 5/6 if the tangent hyperplane section at P has a cusp at the point P ; 1 otherwise; there is an effective anticanonical divisor with a tacnode at the point P ; 5/6 if there is an effective anticanonical divisor with a cusp at the point P ; 1 otherwise; α S 1 (P ) = 5/6 if there is an effective anticanonical divisor with a cusp at the point P ; 1 otherwise.
By Lemma 1.21, Corollary 1.24 implies that Theorem 1.17 holds for smooth del Pezzo surfaces of degrees at most 3. Thus, it is quite natural that we should extend Corollary 1.24 to all smooth del Pezzo surfaces in order to obtain a functional generalisation of Theorem 1.17.This will be done in Section 6, where we prove Theorem 1.25.Let S d be a smooth del Pezzo surface of degree d 4. Then the α-function of S d is as follows: α S 7 (P ) = 1/3 if the point P lies on the −1-curve that intersects two other −1-curves; 1/2 otherwise; α S 6 (P ) = 1/2; α S 5 (P ) = 1/2 if there is −1-curve passing through the point P ; 2/3 if there is no −1-curve passing though the point P ; 3/4 if there is an effective anticanonical divisor that consists of two 0-curves intersecting tangentially at the point P ; 5/6 otherwise.
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.
Write D = r i=1 a i D i , where D i '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 D − T = r i=1 (a i − b i )D i is effective.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 . Put where b k a i − a k b i 0 by the choice of 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 [7, Lemma 3.1] or from the proof of [22, Proposition 5.1].
Lemma 2.3.Suppose that S is a smooth del Pezzo surface of degree 1 and D is an effective anticanonical Q-divisor on S. If the log pair (S, D) is not log canonical at the point P , then there exists a unique divisor T ∈ | − K S | such that (S, T ) is not log canonical at P .Moreover, the support of D contains all the irreducible components of T .
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 let D be the proper transform of D by the blow up f .Then The log pair (S, D) is log canonical at the point P if and only if the log pair ( S, D + (mult P (D)− 1)E) is log canonical along the curve E.
Remark 2.5.If the log pair (S, D) is not log canonical at P , then there exists a point Q on E at which the log pair ( S, D + (mult P (D) − 1)E) is not log canonical.Lemma 2.1 then implies If mult P (D) 2, then the log pair ( S, D + (mult P (D) − 1)E) is log canonical at every point of the curve E other than the point Q.Indeed, if the log pair ( S, D + (mult P (D) − 1)E) is not log canonical at another point O on E, then Lemma 2.4 generates an absurd inequality

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.Lemma 3.1.For an effective anticanonical Q-divisor D on S, the log pair (S, D) is log canonical outside finitely many points on S.
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 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 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 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 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.Lemma 3.2.For an effective anticanonical Q-divisor D on S, the log pair (S, D) is log canonical at every point outside the ramification divisor of the double cover π.
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 Let f : S → S be the blow up of the surface S at the point P .We have where D is the proper transform of the divisor D on the surface S 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 transform C 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, n C2 + Ω + (mult P (D) − 1)E) is not log canonical at the point Q, where C2 and Ω are the proper transforms of C 2 and Ω, respectively, on the surface S, and we have n 1 by Lemma 3.1.We then obtain . This is a contradiction.Proof.Suppose that (S, D) is not log canonical at the point P .Applying Lemma 2.2 to the log pairs (S, D) and (S, T P ), we may assume that Supp(D) does not contain at least one irreducible component of the curve T P .Thus, if the divisor T P is irreducible, then Lemma 2.1 gives an absurd inequality 2 = T P • D mult P (T P )mult P (D) 2mult P (D) > 2 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. Then 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

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, 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.Let L be a line passing through the point P that is not contained in the support of D. Then the inequality 1 = L • D mult P (D) implies that the log pair (S, D) is log canonical at the point P by Lemma 2.1.Lemma 4.3.Suppose that the tangent hyperplane section T P consists of three lines intersecting at the point P .If the support of D does not contain at least one of the three lines, then the log pair (S, D) is log canonical at the point P .
Proof.It immediately follows from Lemma 4.2.
From now on, let f : S → S be the blow up of the cubic surface S at the point P .In addition, let D be the proper transform of D by the blow up f and E be the exceptional curve of f .We then have (4.4) Note that the log pair (S, D) is log canonical at the point P if and only if the log pair ( S, D + (mult is log canonical along the exceptional divisor E. Lemma 4.5.Suppose that the tangent hyperplane section T P consists of a line and a conic intersecting tangentially at the point P .If the support of D does not contain both of the line and the conic, then the log pair (S, D) is log canonical at the point P .
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 We write D = n L + Ω, where Ω and L are the proper transforms of the divisor D and the line L, respectively, on the surface S. Let C be the proper transform of the conic C on the surface S. Note that the three curves L, C and E meet at one point.
The log pair ( S, n L+ 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 to C, and hence not to L either.Indeed, if so, then This contradicts the inequality from Remark 2.5.Let g : S → S be the contraction of the −2-curve L. Then S is a del Pezzo surface of degree 2 with one ordinary double point.In particular, the linear system | − K S | is free from base points and induces a double cover π : S → P2 ramified along an irreducible singular quartic curve R ⊂ P 2 .Note that the point g( L) is the ordinary double point of the surface S. Put Ω = g( Ω), Ē = g(E), C = g( C) and Q = g(Q).Then π( Ē) = π( C) since Ē + C is an anticanonical divisor on S. The point π( Q) lies outside R because the point Q lies outside C. Since the divisor Ω + mult P (D) − 1 Ē is Q-linearly equivalent to −K S by construction, Lemma 3.2 shows that the log pair ( S, Ω + (mult P (D) − 1) Ē) is log canonical at Q.However, it is not log canonical at the point Q 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 The surface S is a smooth del Pezzo surface of degree 2. The linear system | − K S | induces a double cover π : S → P 2 ramified along a smooth quartic curve R ⊂ P 2 .Let TP be the proper transform of the curve T P on the surface S. Then the integral divisor E + TP is linearly equivalent to −K S , and hence π(E) = π( TP ) is a line in P 2 .Moreover, the curve TP 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 and TP .
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 divisor D + (mult P (D) − 1)E is Q-linearly equivalent to −K S .The point Q therefore lies on the curve TP .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 surface S is a smooth del Pezzo surface of degree two.Since D + (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 surface S 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.Lemma 4.8.Suppose that the tangent hyperplane section T P consists of three lines one of which does not pass through the point P .Then the log pair (S, D) is log canonical at P .
Proof.The proof of this lemma is the central and the most beautiful part of the proof of Theorem 1.12.Since the proof is a bit lengthy, we present the proof in a separate section.See Section 5. Lemma 4.9.Suppose that the tangent hyperplane section T P consists of a line and a conic intersecting transversally.Then the log pair (S, D) is log canonical at the point P .
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 Let Ω, L and C 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-curve L. Let g : S → S be the contraction of the curve L. Then S is a del Pezzo surface of degree 2 with only one ordinary double point at the point g( L).In particular, the linear system | − K S | induces a double cover π : S → P 2 ramified along an irreducible singular quartic curve R ⊂ P 2 .
Put Ω = g( Ω), Ē = g(E), C = g( C) and Q = g(Q).Then π( Ē) = π( C) since Ē + C is an anticanonical divisor on S. The point π( Q) lies on R if and only if the point Q lies on C. The log pair ( S, Ω + (mult P (D) − 1) Ē) is not log canonical at Q since g is an isomorphism in a neighborhood of the point Q.Since the divisor Ω + mult P (D) − 1 Ē is Q-linearly equivalent to −K S by construction, Lemma 3.2 shows that the point Q belongs to C.
Note that C + Ē is the unique curve in |− K S | that is singular at the point Q.But the log pair ( S, C + Ē) is log canonical at the point Q.Hence, it follows from Lemma 3.4 that the log pair ( S, Ω + (mult P (D) − 1) Ē) is log canonical at the point Q.This is a contradiction.Therefore, the point Q belongs to the −2-curve L. Now we can apply [8,Theorem 1.28] to the log pair ( S, n L+(mult P (D)−1)E + Ω) at the point Q to obtain a contradiction immediately.Indeed, it is enough to put M = 1, A = 1, N = 0, B = 2, and α = β = 1 in [8,Theorem 1.28] and check that all the conditions of [8, Theorem 1.28] are satisfied.However, there is a much simpler way to obtain a contradiction.Let us take this simpler way.
There exists another line M on the surface S that intersects L at a point.The line M does not intersect the conic C since 1 = T P • M = (L + C) • M = L • M .In particular, the point P does not lie on the line M .Let h : S → Š be the contraction of the proper transform of the line M on the surface S. Since M is a −1-curve and the point P does not lie on the line M , the surface Š is a smooth cubic surface in P 3 .
Put Ω = h( Ω), Ě = h(E), Ľ = h( L), Č = h( C), P = h(Q) and Ď = h( D).Then ( Š, Ď) is not log canonical at the point P since h is an isomorphism in a neighborhood of the point Q.On the other hand, the divisor Ľ + Č + Ě is an anticanonical divisor of the surface Š.Since the point P is the intersection point of Ľ and Ě and the divisor Ď is Q-linearly equivalent to −K Š , Lemma 4.8 implies that ( Š, Ď) is log canonical at the point P .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.
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 are positive rational numbers and C i0 's are irreducible reduced curves of degrees d i0 on the surface S. We then see Denote by L, M and Ñ the proper transforms of the lines L, M and N , respectively, on the surface S .For each i, denote by Ci0 the proper transform of the curve C i0 on the surface S. Then Recall that a 0 + b 0 + m 0 = mult P (D).
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-curves L and M .Then S is a del Pezzo surface of degree 2 with two ordinary double points at the points g( L) and g( M ).The linear system | − K S | 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-curve L or the −2-curve M .
Proof.Suppose that the point Q lies on neither L nor M .Put Ē = g(E), N = g( Ñ ) and Q = g(Q).In addition, we put Ci0 = g( Ci0 ) for each i.Then π( Ē) = π( N ).The point π( Q) lies outside R since the point Q is a smooth point of the anticanonical divisor Ē + N on S.
Since g is an isomorphism in a neighborhood of the point Q, the log pair is not log canonical at the point Q.The divisor c 0 N + (a 0 + b 0 + m 0 − 1) Ē + r i=1 e i Ci0 is an effective anticanonical Q-divisor on the surface S. 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-curve L 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 Lemma 5.8.The divisor D 1 is an effective anticanonical Q-divisor on the surface S. The log pair (S, D 1 ) is not log canonical at the intersection point of L and N .
Proof.Let h : S → S ′ be the contraction of the −1-curve Ñ .Then S ′ is a smooth cubic surface in P Then the integral divisor L ′ + M ′ + E ′ is an anticanonical divisor of the cubic surface S ′ .In particular, the curves L ′ , M ′ and E ′ are coplanar lines on S ′ .Moreover, the point Q ′ is the intersection point of L ′ and E ′ by the assumption right after Lemma 5.5.It does not lie on the line M ′ .
Let ι P be the biregular involution of the surface S induced by the double cover π.Then ι P induces a biregular involution υ P of the surface S since the surface S is the minimal resolution of singularities of the surface S. Thus, we have a commutative diagram S f g & & ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ υ P / / Sf g x x q q q q q q q q q q q q q S ρ / / This shows τ P = f • υ P • f −1 .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., 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  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 by Di its proper transform on the surface W . Then every curve Di lies in a fiber of the morphism g.Lemma A.3.For every effective anticanonical Q-divisor H with Supp(H) ⊆ Supp(D), the log pair (S, H) is not log canonical at the point P .
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 put H = 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, and M. Zaidenberg in 2003 proposed the following question ( [15, Question 2.22]):

Corollary 1 . 19 .
Let S d be a smooth del Pezzo surface of degree d 3.If d = 3, suppose, in addition, that S 3 does not contain an Eckardt point.Then S d admits a Kähler-Einstein metric.

Lemma 3 . 4 .
For a smooth point P of S with π(P ) ∈ R, let T P be the unique divisor in | − K S | that is singular at the point P .If the log pair (S, T P ) is log canonical at P , then for an effective anticanonical Q-divisor D on S the log pair (S, D) is log canonical at the point P .

Lemma 4 . 1 .
The log pair (S, D) is log canonical outside finitely many points.

Lemma 4 . 2 .
If the support of D does not contain a line passing through the point P , then the log pair (S, D) is log canonical at the point P .

2 =
−K S • L = D • L = r i=1 a i D i • L = a r D r • L,and hence a r = 2.It implies α(S)