On a conjecture of Tian

We study Tian's $\alpha$-invariant in comparison with the $\alpha_1$-invariant for pairs $(S_d,H)$ consisting of a smooth surface $S_d$ of degree $d$ in the projective three-dimensional space and a hyperplane section $H$. A conjecture of Tian asserts that $\alpha(S_d,H)=\alpha_1(S_d,H)$. We show that this is indeed true for $d=4$ (the result is well known for $d\leqslant 3$), and we show that $\alpha(S_d,H)<\alpha_1(S_d,H)$ for $d\geqslant 8$ provided that $S_d$ is general enough. We also construct examples of $S_d$, for $d=6$ and $d=7$, for which Tian's conjecture fails. We provide a candidate counterexample for $S_5$.


Introduction
In order to prove the existence of a Kähler-Einstein metric, known as the Calabi problem, on a smooth Fano variety, in [12] Gang Tian introduced a quantity, known as the α-invariant, that measures how singular pluri-anticanonical divisors on the Fano variety can be. There, he proved that a smooth Fano variety of dimension m admits a Kähler-Einstein metric provided that its α-invariant is bigger that m m+1 . Despite the fact that the Calabi problem for smooth Fano varieties has been solved (see [7,9,11,14]) this result of Tian is often the only way to prove the existence of the Kähler-Einstein metric for a given Fano.
In fact, the α-invariant turned out to have important applications in birational geometry as well; see for example [1]. Later, Tian generalised this invariant for arbitrary polarised pairs (X, L), where X is a smooth variety and L is an ample Cartier divisor on it. For the pair (X, L), it can be defined as α X, L = sup λ ∈ Q the log pair (X, λD) is log canonical for every effective Q-divisor D ∼ Q L ∈ R >0 .
This definition coincides with Tian's original definition in [12,13] by [6,Theorem A.3]. The number α(X, L) is often hard to compute but, in good situations, can be approximated by numbers that are much easier to control (see, for example, [5,Proposition 2.2]). For instance, if the linear system |nL| is not empty, Tian defined the n-th α-invariant of the pair (X, L) as α n X, L = sup λ ∈ Q the pair X, λ n D is log canonical for every D ∈ |nL| ∈ Q >0 . If the linear system |nL| is empty, one can simply put α n (X, L) = +∞. Then α(X, L) α n (X, L) and α X, L = inf n 1 α n X, L .
Then, Tian posed the following conjecture. . Suppose that L is very ample and defines a projectively normal embedding under its associated morphism, i.e., the graded algebra is generated by elements in H 0 (X, O X (L)). Then α(X, L) = α 1 (X, L).

Singularities of pairs
In this section we present local results about effective Q-divisors on smooth surfaces. Almost all these results can be found in [10, § 6] in much more general forms.
Let S be a smooth surface, let D be an effective non-zero Q-divisor on the surface S, and let P be a point in the surface S. Put D = r i=1 a i C i , where each C i is an irreducible curve on S, and each a i is a non-negative rational number. We assume here that all curves C 1 , . . . , C r are different. We call (S, D) a log pair.
Let π : S → S be a birational morphism such that S is also smooth. Then π is a composition of n blow ups of smooth points. 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, in addition, that r i=1 C i + n j=1 F j is a divisor with simple normal crossings.
Definition 2.1. 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 .
This definition is independent on the choice of birational morphism π : S → S provided that the surface S is smooth and r i=1 C i + n j=1 F j is a divisor with simple normal crossings. The log pair (S, D) is said to be log canonical if it is log canonical at every point of S.
Remark 2.2. Let R be any effective Q-divisor on S such that R ∼ Q D and R = D. Put for some rational number ǫ 0. Then D ǫ ∼ Q D. Moreover, there exists the greatest rational number ǫ 0 0 such that the divisor D ǫ 0 is effective. Then Supp(D ǫ 0 ) does not contain at least one irreducible component of Supp(R). Moreover, if (S, D) is not log canonical at P , and (S, R) is log canonical at P , then (S, D ǫ 0 ) is not log canonical at P by Definition 2.1, because The following result is well-known and is very easy to prove. Let π 1 : S 1 → S be a blow up of the point P , and let E 1 be the π 1 -exceptional curve. Denote by D 1 the proper transform of the divisor D on the surface S 1 via π 1 . Then Remark 2.4. The log pair (S, D) is log canonical at P if and only if (S 1 , D 1 + (mult P (D) − 1)E 1 ) is log canonical at every point of the curve E 1 .
Corollary 2.5. If mult P (D) > 2, then (S, D) is not log canonical at P .
We can measure how far the pair (S, D) is from being log canonical at P by the positive rational number lct P S, D = sup λ ∈ Q | the log pair (S, λD) is log canonical at P .
This number has been introduced by Shokurov and is called the log canonical threshold of the pair (S, D) at the point P ∈ S. The log canonical threshold of the pair (S, D) is defined as By Lemma 2.3 and Corollary 2.5, we have The following theorem is a very special case of a much more general result known as Inversion of Adjunction (see, for example, [10, Theorem 6.29]). Theorem 2.7 ([10, Exercise 6.31], [3,Theorem 7]). Suppose that r 2. Put ∆ = r i=2 a i C i . Suppose that C 1 is smooth at P , a 1 1, and the log pair (S, D) is not log canonical at P . Then mult P (C 1 · ∆) > 1.

This theorem implies
Lemma 2.8. Suppose that (S, D) is not log canonical at P , and mult P (D) 2. Then there exists a unique point in E 1 such that (S 1 , D 1 + (mult P (D) − 1)E 1 ) is not log canonical at it.
Proof. If mult P (D) 2 and (S 1 , D 1 + (mult P (D) − 1)E 1 ) is not log canonical at two distinct points P 1 and P 1 of the curve E 1 , then by Theorem 2.7. By Remark 2.4, this proves the assertion.
A crucial role in the proof of Theorems 1.2 is played by Theorem 13]). Suppose that r 3. Put ∆ = r i=3 a i C i . Suppose that the curves C 1 and C 2 are smooth at P and intersect each other transversally at P , the log pair (S, D) is not log canonical at P , and mult P (∆) 1. Then either Recall that π is a composition of n blow ups of smooth points. We encourage the reader to prove both Theorems 2.7 and 2.9 using induction on n.

Smooth surfaces in P 3
In this section we collect global results about smooth surfaces in P 3 . These results will be used in the proof of Theorems  3 4 . Proof. Let us first consider the case d = 3. Then S 3 is a smooth cubic surface in P 3 . It is well-known that S 3 contains 27 lines. Taking hyperplane sections of the cubic surface S 3 passing through one of these lines L 1 , we see that either there exists a conic C in S 3 such that and L 1 is tangent to C, or S 3 contains two more lines L 2 and L 3 such that and all three lines L 1 , L 2 and L 3 intersect in a single point. In the former case, one has α 1 (S d , H) 3 4 by definition of α 1 (S d , H). Similarly, in the later case, one has α 1 (S d , H) 2 3 . We proved the required assertion in the case d = 3. Now let us prove it for d = 4. The proof is similar for higher degrees.
Let X ∼ = P 34 be the variety of all quartics in four variables, and suppose Y is the variety of all complete flag varieties in P 3 , hence Y is a projective variety of dimension 6. Consider the incidence variety Z ⊂ X × Y consisting of all pairs (X, Y ), where Y = (P, L, E), such that X ∩ E has an A 3 , or worse, singularity at P with tangent L. We claim that the fibres of the second projection are linear subspaces of codimension 6. To show this, we choose a coordinate system such that P , L and E are, respectively, defined by x = y = z = 0, x = y = 0 and x = 0. Then the fibre of Y is the set of quartics such that the coefficients of the monomials yzw 2 , yw 3 , z 3 w, z 2 w 2 , zw 3 , w 4 are equal to zero.
Therefore it follows that Z is irreducible and has dimension 34 + 6 − 6 = 34. In order to complete the proof, we need to show that the first projection is surjective. Since it is a projective map, the image W ⊂ X is closed. We claim that there exists a point X ∈ W with finite fibre. Then the generic fibre is finite and dim(W) = dim(Z) = 34.
A quartic surface corresponds to a point X 0 ∈ W with finite fiber if it is nonsingular and the intersections with its tangent planes do not have triple points; equivalently, the rank of the hessian of the equation of the surface never drops to 2. An example of such a surface is given by the equation Arguing as in the proof of [5, Proposition 2.1], we get Proof. Similar as in the proof of Lemma 3.1, we define X ∼ = P ( d+3 3 )−1 , Y the variety of all complete flag varieties, and Z ⊂ X ×Y the incidence consisting of all pairs (X, Y ), where Y = (P, L, E), such that X∩E has an A 4 , or worse, singularity at P with tangent L. Now the fibers of the second projection have codimension 7 (defined by 6 linear and one quadratic equation). Since dim(Y) = 6, it follows that dim(Z) < dim(X ), hence the first projection cannot be surjective and the generic surface has no corresponding point in Z. This shows that its hyperplane sections have only singularities of type A 1 , A 2 , and A 3 .
The following result is due to Pukhlikov. Proof. Let X be a cone over the curve C i whose vertex is a sufficiently general point in P 3 . Then For an alternative proof of Pukhlikov's lemma, see the proof of [10, Lemma 5.36].
Since α(S 4 , H) < α 1 (S 4 , H), there exists an effective Q-divisor D such that D ∼ Q H and (S 4 , λD) is not log canonical for some λ < α 1 (S 4 , H). Since α 1 (S 4 , H) 3 4 by Lemma 3.1, we have By Lemma 3.3, the log pair (S 4 , λD) is log canonical outside of finitely many points. Let P be one of these points at which (S 4 , λD) is not log canonical. Consider the quartic curve T P that is cut out on S 4 by the hyperplane in P 3 that is tangent to S 4 at the point P . Then T P is a reduced plane quartic curve Lemma 3.3. It is singular at the point P by construction.
Lemma 4.2. The curve T P contains all lines in S 4 that passes through P .
Proof. If L is a line in S 4 that passes through P , then L is an irreducible component of the curve T P , because otherwise we would have which is absurd.
Put m = mult P (D). Then Lemma 2.3 and (4.1) imply Proof. If L is not contained in the support of D, then (4.3) gives which is absurd.
Let f : S 4 → S 4 be a blow up of the surface S at the point P . Denote by E the f -exceptional curve, and denote by D the proper transform of D on the surface S 4 . Then the log pair is not log canonical at some point Q ∈ E by Remark 2.4. Moreover, Lemma 2.8 implies because λ < 3 4 by (4.1). Let g : S 4 → S 4 be the blow up of the surface S 4 at the point Q, and let F be the exceptional curve of g. Denote by E and D the proper transforms of E and D, respectively. By Remark 2.4, the log pair and (4.5) is not log canonical at the point Q. Applying Lemma 2.8, we obtain m + m + m > 3 λ > 4, because λ < 3 4 by (4.1). Denote by T P the proper transform of the singular quartic curve T P on the surface S 4 . We have the following diagram: For the point Q, we have two mutually excluding possibilities: Q ∈ T P and Q ∈ T P . If Q ∈ T P , we can use geometry of the curve T P to derive a contradiction. If Q ∈ T P , then we often can obtain a contradiction using the following two lemmas.
which contradicts (4.10). Thus, we see that M ⋆ = M ′ . In particular, the curve M is not irreducible. Since M is smooth at P and P ∈ M ′ , then P ∈ M ⋆ . By Lemma 4.2, the curve M ′ is not a line, because Q ∈ T P by assumption. Hence, either M ′ is a conic or M ′ is a cubic curve. Therefore, we may have the following cases: where a is a non-negative rational number, and ∆ is an effective Q-divisor whose support does not contain M ′ . Then a 1 by Lemma 3.3. In fact, we can say more. Indeed, we have Denote by ∆ the proper transform of the divisor ∆ on the surface S 4 . Put n = mult P (∆) and n = mult Q ( ∆). Since O = E ∩ F and (4.8) is not log canonical at the point O, the log pair is also not log canonical at the point the point O. Applying Theorem 2.7 to this log pair, we obtain This gives M ′ · ∆ + n + n + 2a > 3 λ . On the other hand, we have Therefore, we obtain If M ′ is a conic, then (M ′ ) 2 = −2, so that that a > 1 2 by (4.13), which is impossible, because a 1 2 by (4.12). Thus, M ′ is a plane cubic curve. Then (M ′ ) 2 = 0. Now (4.13) gives a > 1 2 , which is impossible, since a 1 3 by (4.12).
by Theorem 2.7. On the other hand, we have , which contradicts (4.1). Recall that T P is a reduced plane quartic curve that is singular at the point P . This implies that there are twelve possibilities for the curve T P as follows. (A) mult P (T P ) = 4, hence T P consists of four lines that intersect at P . (B) mult P (T P ) = 3 and T P (B1) consists of four lines and three of them intersect at P , or (B2) it is an irreducible quartic with a singular point P of multiplicity 3, or (B3) it consists of a conic and two lines, all intersecting at P , or (B4) it consists of a cubic curve with a singular point P of multiplicity 2 and a line passing through P . (C) mult P (T P ) = 2 and T P (C1) consists of four lines, two of which pass through P , or (C2) it consist of a conic and two lines, and the two lines intersect at P and P does not lie on the conic, or (C3) it consist of a conic and two lines and P is the intersection point of the conic with one of the lines, or (C4) it consists of a cubic curve and a line and P is the intersection of the two at a smooth point of the cubic curve, or (C5) it consists of a cubic curve and a line and P is singular point of the cubic curve with multiplicity 2 and does not lie on the line, or (C6) it consists of two conics and they intersect at P , or (C7) it is an irreducible quartic curve with a singular point P of multiplicity 2. In the rest of this section, we eliminate all these possibilities case by case using Lemmas 4.11 and 4.14.
To succeed in doing this, we also need Lemma 4.15. We may assume that the support of the divisor D does not contain at least one irreducible component of the plane quartic curve T P .
Proof. Note that (S 4 , λT P ) is log canonical at P , because λ < α 1 (S 4 , H). Thus, it follows from Remark 2.2 that there exists an effective Q-divisor D ′ on the surface S 4 such that D ′ ∼ Q H, the log pair (S 4 , λD ′ ) is not log canonical at P , and the support of D ′ does not contain at least one irreducible component of the curve T P . Replacing D by D ′ , we obtain the required assertion.
We denote by C ⋆ the irreducible component of the curve T P that is not contained in the support of the divisor D. By Lemma 4.4, if P ∈ C ⋆ , then C ⋆ is not a line. This gives Now we are going to deal with the cases (B1), (B2), (B3), and (B4). In these four cases, λ < 2 3 . Indeed, one has lct P (S 4 , T P ) 2 mult P (T P ) by (2.6). Thus, we have Proof. Suppose that we are in the case (B1). Then mult P (T P ) = 3 and T P consists of four lines L 1 , L 2 , L 3 , and L 4 such that the first three intersect at P , and L 4 does not pass through P . Thus, we have the following picture: By Lemma 4.4, the lines L 1 , L 2 , and L 3 are contained in the support of D, and C ⋆ = L 4 . Hence, we put D = a 1 L 1 + a 2 L 2 + a 3 L 3 + Ω, where a 1 , a 2 , and a 3 are positive rational numbers, and Ω is an effective Q-divisor whose support does not contain the lines L 1 , L 2 , L 3 , and L 4 . Put n = mult P (Ω). Then m = n + a 1 + a 2 + a 3 .
Denote by Ω the proper transform of the divisor Ω on the surface S 4 . Also denote the proper transforms of the lines L 1 , L 2 , and L 3 on the surface S 4 by L 1 , L 2 , and L 3 , respectively. Then we can rewrite the log pair (4.8) as On the surface S 4 , one has L 2 1 = −2. Thus, we have Similarly, we see that a 1 − 2a 2 + a 3 + n 1 and a 1 + a 2 − 2a 3 + n 1. Adding these three inequalities together, we get n 1. On the other hand, we have 1 = D · L 4 = a 1 L 1 + a 2 L 2 + a 3 L 3 + Ω · L 4 = a 1 + a 2 + a 3 + Ω · L 4 a 1 + a 2 + a 3 , which gives a 1 + a 2 + a 3 1. In particular, we have m = n + a 1 + a 2 + a 3 2. Then Lemmas 4.11 and 4.14 imply that Q is contained in one of the curves L 1 , L 2 , and L 3 . Without loss of generality, we may assume that Q ∈ L 1 .
Proof. Suppose that we are in the case (B3). Then mult P (T P ) = 3 and T P consists of a conic C 1 and two lines L 1 and L 2 , all intersecting at the point P . Thus, we have the following picture: By Lemma 4.4, both lines L 1 and L 2 are contained in the support of the divisor D. Hence we can write D = a 1 L 1 + a 2 L 2 + Ω, where a 1 and a 2 are positive rational numbers, and Ω is an effective Q-divisor whose support does not contain the lines L 1 and L 2 . Recall that the support of Ω does not contain the curve C ⋆ by assumption. In our case, the curve C ⋆ is the conic C 1 .
Put n = mult P (Ω). Let us show that n 6 5 . We have Similarly, we see that n 1 − a 1 + 2a 2 . Finally, we have which implies that a 1 + a 2 1 − n 2 . Adding these three inequalities together, we get n 6 5 .
By (4.17), we have λ < 2 3 . Since n 6 5 , we see that λn 1. Thus, we can apply Theorem 2.9 to the log pair (S 4 , a 1 L 1 + a 2 L 2 + Ω). This gives λΩ · L 1 > 2(1 − λa 2 ) or λΩ · L 2 > 2(1 − λa 1 ). Without loss of generality, we may assume that the former inequality holds. Then which implies that 2a 1 + a 2 > 2 λ − 1. Since λ < 2 3 , we have 2a 1 + a 2 > 2, which is impossible since we already proved that a 1 + a 2 1 − n 2 1. Proof. Suppose that we are in the case (B4). Then mult P (T P ) = 3 and T P consists of a cubic curve C 1 with a singular point P of multiplicity 2 and a line L passing through P . Thus, we have the following picture: P C 1 L By Lemma 4.4, the line L is contained in the support of the divisor D. Hence, C ⋆ = C 1 , and we can write D = aL + Ω, where a is a positive rational number, and Ω is an effective Q-divisor whose support does not contain the line L. Put n = mult P (Ω). Then which implies that a + n 3 2 . On the other hand, λ < 2 3 by (4.17), so that n + a > 3 2 by Lemma 2.3. The contradiction is clear. Proof. Suppose that we are either in the case (C1) or in the case (C2). Then T P consists of two lines L 1 and L 2 , and a possibly reducible conic C 1 , where P is the intersection point of the lines L 1 and L 2 , and P is not contained in the conic C 1 . If we are in the case (C1), then the conic C 1 splits as a union of two different lines L 3 and L 4 , which implies that we have the following picture: If we are in the case (C2), then the conic C 1 is irreducible, so that we have the following picture: By Lemma 4.4, both lines L 1 and L 2 are contained in the support of the divisor D. In particular, C ⋆ = L 1 and C ⋆ = L 2 . Write D = Ω + a 1 L 1 + a 2 L 2 , where a 1 and a 2 are positive rational numbers, and Ω is an effective Q-divisor whose support does not contain the lines L 1 and L 2 . Put n = mult P (Ω). Then Similarly, we see that n 1 − a 1 + 2a 2 . Finally, we have which implies that a 1 + a 2 1. Adding these three inequalities together, we get n 3 2 . Recall that m = n + a 1 + a 1 . We see that m 5 2 , because a 1 + a 2 1 and n 3 2 . In particular, λm < 15 8 , because λ < 3 4 by (4.1). Denote by Ω the proper transform of the divisor Ω on the surface S 4 . Similarly, denote by L 1 and L 2 the proper transform of the lines L 1 and L 2 on the surface S 4 , respectively. Then we can rewrite the log pair (4.5) as Since λm < 15 8 , this log pair is log canonical at every point of E that is different from Q by Corollary 4.6. Put n = mult Q ( Ω). Then n n.
Denote by Ω the proper transform of the divisor Ω on the surface S 4 . Since Q ∈ L 1 ∪ L 2 , the log pair is not log canonical at the point O and is log canonical at every point of F that is different from O. Applying Theorem 2.7 to this log pair and the curve E, we get λ a 1 + a 2 + 2n) − 2 = λ n − n + λ(a 1 + a 2 + n + n) − 2 = = λΩ · E + λ(a 1 + a 2 + n + n) − 2 = λΩ + λ(a 1 + a 2 + n + n) − 2 F · E > 1 which implies that a 1 + a 2 + 2n > 3 λ > 4, because λ < 3 4 by (4.1). This is a contradiction, since we already proved that a 1 + a 2 1 and n 3 2 . Lemma 4.23. The case (C3) is impossible.
Proof. Suppose that we are in the case (C3). Then mult P (T P ) = 2, the curve T P consist of a conic curve C 1 and two lines L 1 and L 2 , and the point P is the intersection point of the conic with the line L 1 . Thus, we have the following picture: By Lemma 4.4, the line L 1 is contained in the support of the divisor D. In particular, C ⋆ = L 1 . Thus, either C ⋆ = L 2 of C ⋆ = C 1 . Write D = Ω + aL 1 + bC 1 , where a is a positive rational number, b is a non-negative rational number, and Ω is an effective Q-divisor whose support does not contain the curves L 1 and C 1 . If b > 0, then the support of Ω does not contain the line L 2 , which implies that Hence, either b = 0 or a + 2b 1 (or both), so that a + 2b 1, because a 1 by Lemma 3.3.
Denote by Ω the proper transform of the divisor Ω on the surface Ω. Similarly, denote by L 1 and C 1 the proper transform of the curves L 1 and C 1 on the surface Ω, respectively. Then we can rewrite the log pair (4.5) as S 4 , λa L 1 + λb C 1 + λ Ω + λ(a + b + n) − 1 E .
Since m < 2 λ , this log pair is log canonical at every point of E that is different from Q by Corollary 4.6. Put n = mult Q ( Ω). Then n n.
Let us show that Q ∈ L 1 . Suppose that Q ∈ L 1 . Then which implies that 2 n n + n 1 + 2a − 2b. But we already know that n n 2 − 2a + 2b. Adding these two inequalities together, we get n 1. If Q ∈ C 1 , then we also have n Ω · C 1 = Ω · C 1 − n = 2 − 2a + 2b − n, which implies that 2 n n + n 2 − 2a + 2b. Thus, if Q ∈ C 1 , then Keeping in mind that a + 2b 1, we conclude that n + b Thus, we have 2b + n > 8 3 . One the other hand, we already know that n + 2b − 2a 1, n + 2b − 2a 2, and a + 2b 1, so that which is a contradiction. This shows that Q ∈ C 1 . Denote by Ω the proper transform of the divisor Ω on the surface S 4 . Recall that the log pair (4.8) is not log canonical at the point O ∈ F . Moreover, it is log canonical at every point of F that is different from O by Corollary 4.9, because m + m = a + b + n + n a + 2b + 2n 4 < 3 λ , since a + 2b 1, n 3 2 and λ < 3 4 . Then O = F ∩ E by Lemma 4.11. Since Q ∈ L 1 ∪ C 1 , we see that the log pair is not log canonical at the point O ∈ F and is log canonical in all other points of the curve F . Applying Theorem 2.7 to this log pair and the curve E, we get Proof. Suppose that we are in the case (C4). Then mult P (T P ) = 2 and T P consists of a cubic curve C 1 and a line L, and P is their intersection at a smooth point of the cubic curve. Thus, we have the following picture: P C 1 L By Lemma 4.4, the line L 1 is contained in the support of the divisor D, so that C ⋆ = C 1 . Write D = Ω + aL 1 , where a is a positive rational number, and Ω is an effective Q-divisor whose support does not contain the line L 1 . Put n = mult P (Ω). Then n Ω · L 1 = H − aL 1 · L 1 = 1 + 2a, which gives n − 2a 1. Similarly, we obtain n + 3a 3, because n Ω · C 1 = H − aL 1 · C 1 = 3 − 3a.
We see that n + a = 2 5 (n − 2a) + 3 5 (n + 3a) 11 5 , which implies that m = n + a < 2 λ , because λ > 3 4 . Thus, it follows from Corollary 4.6 that the log pair (4.5) is log canonical at every point of E that is different from Q.
Note that a 1 by Lemma 3.3. This also follows from n + 3a 3. We also know that a > 0. In fact, one can show that a > 1 6 . Indeed, we have λ 1 + 2a = λΩ · L 1 > 1 by Theorem 2.7. This gives a > 1 6 , since λ > 3 4 . Denote by Ω the proper transform of the divisor Ω on the surface Ω. Similarly, denote by L 1 the proper transform of the line L 1 on the surface Ω. Then we can rewrite the log pair (4.5) as ( S 4 , λa L 1 + λ Ω + (λ(a + n) − 1)E). Put n = mult Q ( Ω). Then n n.
Thus, we have n 3 2 . Then m = n + 2a < 2 λ , because λ > 3 4 by (4.1). Thus, it follows from Corollary 4.6 that the log pair (4.5) is log canonical at every point of E that is different from Q.
Denote by Ω the proper transform of the divisor Ω on the surface Ω. Similarly, denote by C 1 the proper transform of the curve L 1 on the surface Ω. Then we can rewrite the log pair (4.5) as ( S 4 , λa C 1 + λ Ω + (λ(n + 2a) − 1)E). Put n = mult Q ( Ω). Then n n. If Q ∈ C 1 , then m = n. If Q ∈ C 1 , then m = n + a.
Denote by Ω the proper transform of the divisor Ω on the surface S 4 , and denote by C 1 the proper transform of the curve C 1 on the surface S 4 . Then we can rewrite the log pair (4.8) as (S 4 , λaC 1 + λΩ + (λ(n + 2a) − 1)E + (λ(n + 2a + m) − 2)F ). This log pair is not log canonical at the point O ∈ F by construction. Moreover, we have m + m = n + 2a + n + a 2n + 3a 3 + 3a 4 < 3 λ , since λ < 3 4 . Thus, it follows from Corollary 4.9 that the log pair (4.8) is log canonical at every point of the curve F that is different from the point O.
Let us show that O = F ∩ E. Suppose that O = F ∩ E. If O ∈ C 1 , then Theorem 2.7 applied to the log pair (4.8) and the curve E gives λ 3a + 2n) − 2 λ 2a + 2n + m − n) − 2 = λ n − n + λ(n + 2a + m) − 2 = = λΩ · E + λ(n + 2a + m) − 2 = λΩ + λ(n + 2a + m) − 2 F · E > 1 which implies that 3a + 2n > 3 λ . This is impossible, because a  . Thus, we see that O ∈ C 1 . In particular, Q ∈ C 1 , m = n + a, and C 1 has a cuspidal singularity at the point P . Now we apply Theorem 2.7 to the log pair (4.8) and the curve C 1 at the point O. This gives , which is impossible, because we already proved that a 1 3 . Thus, we see that O = F ∩ E. We already know that m < 2 λ and m + m < 3 λ . Thus, if Q ∈ C 1 , then we can apply Lemma 4.11 to obtain O = F ∩ E, which is not the case. Hence, we conclude that Q ∈ C 1 , so that m = n + a. If O ∈ C 1 , then the log pair (S 4 , λΩ + (λ(n + 2a + m) − 2)F ) is not log canonical at the point O as well, which implies that n = Ω · F > 1 λ > 4 3 by Theorem 2.7. On the other hand, we have 3 = Ω · C 1 − 2n = Ω · C 1 n, which implies that 3 n 2n + n 3, so that n 1. This shows that O ∈ C 1 .
Since O = F ∩ E and O ∈ C 1 , we conclude that P is an ordinary double point of the curve C 1 . Hence, the curves C 1 and E intersect transversally at the point Q. Thus, applying Theorem 2.7 to the log pair (4.5) and the curve E, we get λn = λ Ω · E > 1 − λa, which implies a + n > 1 λ > 4 3 . Similarly, applying Theorem 2.7 to the log pair (4.5) and the curve C 1 , we get which implies that 2a > n + 2 λ − 3 > n − 1 3 . Thus, we have 2a > n − 1 3 > ( 4 3 − a) − 1 3 = 1 − a, which implies that a > 1 3 . But we already proved that a 1 3 . This is a contradiction. Lemma 4.26. The case (C6) is impossible.
Proof. Suppose that we are in the case (C6). Then mult P (T P ) = 2 and T P consists of two conic curves and they intersect at P . Thus, we have the following picture: Without loss of generality, we may assume that C 1 = C ⋆ . This gives 2 = C 1 · D m. Then m 2 λ and m + m 3 λ by Lemma 4.14. Hence, Corollary 4.6 implies that the log pair (4.5) is log canonical at every point of the curve E that is different from Q. Moreover, Corollary 4.9 implies that the log pair (4.8) is log canonical at every point of the curve F that is different from O. Furthermore, Lemma 4.14 implies that O = E ∩ F . Denote by C 1 and C 2 the proper transforms on the surface S 4 of the conics C 1 and C 2 , respectively. By Lemma 4.11, we see that Q ∈ C 1 ∪ C 2 . If Q ∈ C 1 , then 2 − m = D · C 1 m which implies that m + m 2. On the other hand, we have m + m > 2 λ > 8 3 by (4.7). Hence, we see that Q ∈ C 1 and Q ∈ C 2 .