Delta invariants of smooth cubic surfaces

Abstract

We prove that $\delta$-invariants of smooth cubic surfaces are at least $\frac{6}{5}$.

All varieties are assumed to be projective and defined over $\mathbb {C}$.

1 Introduction

The existence of Kähler–Einstein metrics on Fano manifolds is an important problem in complex geometry. By the Yau–Tian–Donaldson conjecture (confirmed in [4, 21]), we know that all K-stable Fano manifolds are Kähler–Einstein. Moreover, we also know explicit criteria that can be used to verify K-stability in many cases. One such criterion has been found by Tian in [19] and later generalized by Fujita in [10]. It is the following

Theorem 1.1

([10, 19]) Let X be a Fano manifold of dimension $n\geqslant 2$. If $\alpha (X)\geqslant \frac{n}{n+1}$, then X is K-stable.

Here, $\alpha (X)$ is the $\alpha$-invariant defined in [19]. By [8, Theorem A.3], one has

In [5], the first author computed the $\alpha$-invariants of two-dimensional Fano manifolds, known as del Pezzo surfaces. Namely, if S be a smooth del Pezzo surface, then

In particular, if $K_S^2\leqslant 4$, then S is K-stable by Theorem 1.1, so that it is Kähler–Einstein. If $K_S^2=5$, then S is unique and . In this case, we have $\alpha _{\mathfrak {S}_5}(S)=2$ by [5], where $\alpha _{\mathfrak {S}_5}(S)$ is a $\mathfrak {S}_5$-invariant $\alpha$-invariant, which can be defined similarly to $\alpha (S)$. Now using an $\mathfrak {S}_5$-equivariant counterpart of Theorem 1.1 in [19], we conclude that the surface S is also Kähler–Einstein. All remaining del Pezzo surfaces are toric, so that they are Kähler–Einstein if and only if their Futaki characters vanish [22]. Together with Matsushima’s obstruction, this gives Tian’s celebrated theorem:

Theorem 1.2

([20]) A smooth del Pezzo surface admits a Kähler–Einstein metric if and only if it is not a blow-up of $\mathbb {P}^2$ at one or two points.

Note that smooth cubic surfaces form the hardest case in Tian’s original proof of this result, which requires Cheeger–Gromov theory, Hörmander $L^2$ estimates, partial $C^0$ estimates and the lower semi-continuity of log canonical thresholds. In this paper, we will give another proof of Theorem 1.2 in this case using a new criterion for K-stability, which has been recently discovered by Fujita and Odaka in [12]. They stated it in terms of the so-called $\delta$-invariant, which we describe now.

Fix a Fano manifold X. For a sufficiently large and sufficiently divisible integer k, consider a basis $s_1, \ldots , s_{d_k}$ of the vector space $H^0(\mathcal {O}_X(-kK_X))$, where $d_k=h^0(\mathcal {O}_X(-kK_X))$. For this basis, consider the $\mathbb {Q}$-divisor

Any $\mathbb {Q}$-divisor obtained in this way is called a k-basis type (anticanonical) divisor. Let

Then let

$$\begin{aligned} \delta (X)=\limsup _{k\in \mathbb {N}}\delta _k(X). \end{aligned}$$

By [2, Theorem A], one has

The number $\delta (X)$ is also referred to as the stability threshold (cf. [2, 3]), because of

Theorem 1.3

 ([2, Theorem B]) The following assertions hold:

• X is K-semistable if and only if $\delta (X)\geqslant 1$;

• X is uniformly K-stable if and only if $\delta (X)>1$.

How to compute or at least estimate $\delta (X)$ effectively? In general this is not very easy. In [17], Park and Won estimated the $\delta$-invariants of all smooth del Pezzo surfaces, which gave another proof of Tian’s Theorem 1.2. But it seems unclear to us how to generalize their approach for higher-dimensional Fano manifolds. Motivated by this, in our recent joint work with Yanir Rubinstein [7], we developed new geometric tools to estimate $\delta$-invariants of (log) del Pezzo surfaces, which enabled us to partially prove a conjecture proposed in [6]. In this paper, we will use the same methods to give a sharper estimate for the $\delta$-invaraints of smooth cubic surfaces. To be precise, we prove

Theorem 1.4

Let S be a smooth cubic surface in $\mathbb {P}^3$. Then $\delta (S)\geqslant \frac{6}{5}$.

Corollary 1.5

([17, 20]) All smooth cubic surfaces in $\mathbb {P}^3$ are uniformly K-stable, so that they are Kähler–Einstein.

For a smooth cubic surface S, it follows from [17, Theorem 4.9] that

Our bound $\delta (S)\geqslant \frac{6}{5}$ is slightly better. Moreover, the proof of Theorem 1.4 is completely different from the proof of [17, Theorem 4.9]. The essential ingredient in our proof is a vanishing order estimate for basis type divisors (see Theorem 2.9). This estimate combined with the techniques from [5] give us the desired lower bound for $\delta (S)$.

This paper is organized as follows. In Sect. 2, we present known results about divisors on smooth surfaces, and, as an illustration, we give a new proof of [17, Theorem 4.7]. In Sect. 3, we give various multiplicity estimates for basis type divisors on smooth cubic surfaces, which will be important to bound their $\delta$-invariants in the proof of Theorem 1.4. These estimates also imply that $\delta$-invariants of smooth cubic surfaces are at least $\frac{18}{17}$. In Sect. 4, we prove Theorem 1.4.

2 Basic tools

In this section, we collect some basic notions and tools that will be used throughout this article. Let S be a smooth surface, and let P be a point in S. Let D be an effective divisor on S. Suppose that $f=0$ is the local defining equation of D near the point P, then the multiplicity of D at P, is defined to be the vanishing order of f at P, which we denote by $\mathrm {mult}_P(D)$. Let $\pi :\widetilde{S}\rightarrow S$ be the blow-up of the point P, and let E be the exceptional curve of $\pi$. Denote by $\widetilde{D}$ the proper transform of D via $\pi$. Then we have

Definition 2.1

Let $C_1$ and $C_2$ be two irreducible curves on a surface S. Suppose that $C_1$ and $C_2$ intersect at P. Let $\mathcal {O}_{P}$ be the local ring of germs of holomorphic functions defined in some neighborhood of P. Then the local intersection number of $C_1$ and $C_2$ at the point P is defined by

where $f_1=0$ and $f_2=0$ are local defining functions of $C_1$ and $C_2$ around the point P. The global intersection number is defined by

This definition and the definition of $\mathrm {mult}_P(D)$ extend to $\mathbb {R}$-divisors by linearity. For instance, say we have a curve C and an $\mathbb {R}$-divisor $\Delta =\sum _ia_iZ_i$, where $Z_i$’s are distinct prime divisors and $a_i\in \mathbb {R}$. Then

where $(C.Z_i)_P=0$ if $Z_i$ does not pass through the point P.

In the following, let D be an effective $\mathbb {R}$-divisor on S. We will investigate how to express the singularity of the log pair (S, D) at the point P in terms of and .

Lemma 2.2

([14]) If (S, D) is not log canonical at P, then $\mathrm {mult}_P(D)>1$.

Let C be an irreducible curve on S. Write

$$\begin{aligned} D=aC+\Delta , \end{aligned}$$

where a is a non-negative real number that is also denoted as $\mathrm {ord}_C(D)$, and $\Delta$ is an effective $\mathbb {R}$-divisor on S whose support does not contain the curve C.

Lemma 2.3

([7, Proposition 3.3]) Suppose that $a\leqslant 1$, the curve C is smooth at the point P, and $\mathrm {mult}_P(\Delta )\leqslant 1$. If (S, D) is not log canonical at P, then

Corollary 2.4

If $a\leqslant 1$, the curve C is smooth at P, and the log pair (S, D) is not log canonical at P, then

Let $\pi :\widetilde{S}\rightarrow S$ be the blow-up of the point P, and let $E_1$ be the exceptional curve of $\pi$. Denote by $\widetilde{D}$ the proper transform of D via $\pi$. Then

This implies

Corollary 2.5

The log pair (S, D) is log canonical at P if and only if the log pair $(\widetilde{S},\widetilde{D}+(\mathrm {mult}_P(D)-1)E_1)$ is log canonical along the curve $E_1$.

Thus, using Lemma 2.2 and Corollary 2.5, we obtain the following simple criterion.

Corollary 2.6

Suppose that

$$\begin{aligned} \mathrm {mult}_Q(\pi ^*(D))=\mathrm {mult}_P(D)+\mathrm {mult}_Q(\widetilde{D})\leqslant 2 \end{aligned}$$

for every point $Q\in E_1$. Then (S, D) is log canonical at P.

If D is a Cartier divisor, then its volume is the number

where the $\limsup$ can be replaced by a limit (see [15, Example 11.4.7]). Likewise, if D is a $\mathbb {Q}$-divisor, we can define its volume using the identity

for an appropriate $\lambda \in \mathbb {Q}_{>0}$. Then the volume only depends on the numerical equivalence class of the divisor D. Moreover, the volume function can be extended by continuity to $\mathbb {R}$-divisors. Furthermore, it is log-concave:

(2.1)

for any pseudoeffective $\mathbb {R}$-divisors $D_1$ and $D_2$ on the surface S. For more details about volumes of $\mathbb {R}$-divisors, we refer the reader to [15, 16].

If D is not pseudoeffective, then . If the divisor D is nef, then

This follows from the asymptotic Riemann–Roch theorem [15]. If the divisor D is not nef, its volume can be computed using its Zariski decomposition [13, 18]. Namely, if D is pseudoeffective, then there exists a nef $\mathbb {R}$-divisor N on the surface S such that

$$\begin{aligned} D\sim _{\mathbb {R}} N+\sum _{i=1}^r\, a_iC_i, \end{aligned}$$

where each $C_i$ is an irreducible curve on S with , each $a_i$ is a non-negative real number, and the intersection form of the curves $C_1,\ldots ,C_r$ is negative definite. Such decomposition is unique, and it follows from [1, Corollary 3.2] that

This immediately gives

Corollary 2.7

Let $Z_1,\ldots ,Z_s$ be irreducible curves on S such that for every i, and the intersection form of the curves $Z_1,\ldots ,Z_s$ is negative definite. Then

where $b_1,\ldots ,b_s$ are (uniquely defined) non-negative real numbers such that

for every j.

Corollary 2.8

Let Z be an irreducible curve on S such that and . Then

Let $\eta :\widehat{S}\rightarrow S$ be a birational morphism (possibly an identity) such that $\widehat{S}$ is smooth. Fix a (not necessarily $\eta$-exceptional) irreducible curve F in the surface $\widehat{S}$. Let

This is called the pseudo-effective threshold of F.

Theorem 2.9

Suppose that S is a smooth del Pezzo surface, and D is a k-basis type divisor with $k\gg 1$. Then

where $\varepsilon _k$ is a small constant depending on k such that $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$.

Proof

This is a very special case of [12, Lemma 2.2].$\square$

In [2, 3], the quantity

is also called the expected vanishing order of anticanonical sections along the divisor F.

Theorem 2.9 plays a crucial role in the proof of Theorem 1.4. As a warm up, let us show how to use Theorem 2.9 to estimate $\delta$-invariants of smooth del Pezzo surfaces of degree 1.

Theorem 2.10

([17, Theorem 4.7]) Let S be a smooth del Pezzo surface of degree 1. Then $\delta (S)\geqslant \frac{3}{2}$.

Proof

Fix some rational number $\lambda <\frac{3}{2}$. Let D be a k-basis type divisor with $k\gg 1$, and let P be a point in S. We have to show that the log pair $(S,\lambda D)$ is log canonical at P. By Lemma 2.2, it is enough to prove that

Applying Theorem 2.9 with $\widehat{S}=\widetilde{S}$, $\eta =\pi$ and $F=E_1$, we see that

where $\varepsilon _k$ is a constant depending on k such that $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$.

Let us compute $\tau (E_1)$. To do this, take a curve such that $P\in C$. Denote by $\widetilde{C}$ its proper transform on the surface $\widetilde{S}$. If C is smooth at P, then

$$\begin{aligned} \pi ^*(-K_S)\sim _{\mathbb {Q}} \widetilde{C}+E_1\quad \; \text {and}\quad \; \widetilde{C}^2=C^2-1=0, \end{aligned}$$

which implies that $\tau (E_1)=1$. In this case, we have

Therefore, if C is smooth at P, then the log pair $(S,\lambda D)$ is log canonical at P for $k\gg 1$.

To complete the proof, we may assume that C is singular at P. Then P is either nodal or cuspidial, so we have ${\text {mult}}_P(C)=2$ and

$$\begin{aligned} \pi ^*(-K_S)\sim \widetilde{C}+2E_1, \end{aligned}$$

so that $\tau (E_1)=2$, since $\widetilde{C}^2=-3$. Using Corollary 2.8, we see that

so that $\mathrm {mult}_P(D)\leqslant \frac{5}{6}+\varepsilon _k$. This gives $\delta (S)\geqslant \frac{6}{5}$. To get $\delta (S)\geqslant \frac{3}{2}$, we must work harder.

Fix a point $Q\in E_1$. By Corollary 2.6, to prove that $(S,\lambda D)$ is log canonical at P, it is enough to show that

Let $\sigma :\widehat{S}\rightarrow \widetilde{S}$ be the blow-up of the point Q. Denote by $E_2$ the exceptional curve of $\sigma$. Let . Applying Theorem 2.9 with $F=E_1$, we see that

Here, as above, the term $\varepsilon _k$ is a constant that depends on k such that $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$.

Let $\widehat{C}$ and $\widehat{E}_1$ be the proper transforms on $\widehat{S}$ of the curves C and $E_1$, respectively. Then the intersection form of the curves $\widehat{C}$ and $\widehat{E}_1$ is negative definite. If $Q\in \widetilde{C}$, then

$$\begin{aligned} \eta ^*(-K_S)\sim _\mathbb {Q}\widehat{C}+2\widehat{E}_1+3E_2, \end{aligned}$$

so that $\tau (E_2)=3$. In this case, using Corollary 2.8, we see that

provided that $0\leqslant x\leqslant \frac{2}{3}$. Likewise, if $\frac{2}{3}\leqslant x\leqslant 3$, then Corollary 2.7 gives

Thus, if $Q\in \widetilde{C}$, then

so that $\mathrm {mult}_Q(\pi ^*(D))\leqslant \frac{2}{\lambda }$ for $k\gg 1$, because

Likewise, if $Q\notin \widetilde{C}$, then

$$\begin{aligned} \eta ^*(-K_S)\sim _\mathbb {Q}\widehat{C}+2\widehat{E}_1+2E_2, \end{aligned}$$

so that $\tau (E_2)=2$. In this case, using Corollary 2.7, we deduce that

which implies that

so that $\mathrm {mult}_Q(\pi ^*(D))\leqslant \frac{2}{\lambda }$ for $k\gg 1$.$\square$

Remark 2.11

In the proof of Theorem 2.10, there is another way to treat the case when the curve C is singular at P, which relies on Lemma 2.3. Indeed, let S be a smooth del Pezzo surface of degree 1, let P be a point in S, and let C be a curve in that passes trough P. Suppose that

$$\begin{aligned} \mathrm {mult}_P(C)=2. \end{aligned}$$

Let D be any k-basis type divisor such that $D\sim -K_S$ with $k\gg 1$, and let $\lambda$ be a positive real number such that $\lambda <\frac{3}{2}$. Let us show that $(S,\lambda D)$ is log canonical at P. We argue by contradiction. Suppose that $(S,\lambda D)$ is not log canonical at P. Write

$$\begin{aligned} D=aC+\Delta , \end{aligned}$$

where $a\geqslant 0$ and $\Delta$ is an effective $\mathbb {Q}$-divisor whose support does not contain C. Note that

where $\varepsilon _k$ is a constant that depends on k such that $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$. Let $m=\mathrm {mult}_P(\Delta )$. Then

so that $m\leqslant \frac{1-a}{2}$. Let $\pi :\widetilde{S}\rightarrow S$ be the blow-up of the point P. Let E be the exceptional curve of $\pi$, and let $\widetilde{C}$ and $\widetilde{\Delta }$ be the proper transforms of C and $\Delta$ on $\widetilde{S}$, respectively. Then the log pair

$$\begin{aligned} \bigl (\widetilde{S},\lambda a\widetilde{C}+\lambda \widetilde{\Delta }+(\lambda (2a+m)-1)E\bigr ) \end{aligned}$$

is not log canonical at some point $Q\in E$. Note that $\lambda (2a+m)-1<1$. But

Thus, we have $Q\in E\cap \widetilde{C}$ by Corollary 2.4. On the other hand, for $k\gg 1$, we have

so that we can apply Lemma 2.3 to our pair at Q. This gives

so that $\lambda (1+4a)>4$, and hence

which is absurd for . This proves the desired log canonicity of our pair $(S,\lambda D)$.

The following (simple) result can be very handy.

Lemma 2.12

Under the assumptions and notations of Theorem  2.9, one has

for any $\mu \in [0,\tau (F)]$.

Proof

The assertion follows from the fact that is a non-increasing function on $x\in [0,\tau (F)]$.$\square$

Using (2.1), this result can be improved as follows:

Lemma 2.13

Under the assumptions and notations of Theorem 2.9, one has

for any $\mu \in [0,\tau (F)]$.

Proof

The required assertion follows from the proof of [11, Proposition 2.1].$\square$

We will apply both Lemmas 2.12 and 2.13 to estimate the integral in Theorem 2.9 in the cases when it is not easy to compute.

3 Multiplicity estimates

Let S be a smooth cubic surface in $\mathbb {P}^3$, and let D be a k-basis type divisor with $k\gg 1$. The goal of this section is to bound multiplicities of the divisor D using Theorem 2.9. As in Theorem 2.9, we denote by $\varepsilon _k$ a small number such that $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$.

Lemma 3.1

Let L be a line on S. Then

$$\begin{aligned} \mathrm{ord}_L(D)\leqslant \frac{5}{9}+\varepsilon _k. \end{aligned}$$

Proof

Let us use assumptions and notations of Theorem 2.9 with $\eta =\mathrm {Id}_S$ and $F=L$. Let H be a general hyperplane section of the surface S that contains L. Then $H=L+C$, where C is an irreducible conic. Since , we have $\tau (F)=1$, so that

by Theorem 2.9.$\square$

Fix a point $P\in S$. Let $\pi :\widetilde{S}\rightarrow S$ be the blow-up of this point. Denote by $E_1$ the exceptional divisor of $\pi$. Fix a point $Q\in E_1$. Let $\sigma :\widehat{S}\rightarrow \widetilde{S}$ be the blow-up of this point. Denote by $E_2$ the exceptional curve of $\sigma$. Let , then

Applying Theorem 2.9, we get

(3.1)

Let $T_P$ be the unique hyperplane section of the surface S that is singular at the point P. Then we have the following four possibilities:

• $T_P=L_1+L_2+L_3$, where $L_1$, $L_2$ and $L_3$ are lines such that $P=L_1\cap L_2\cap L_3$;

• $T_P=L_1+L_2+L_3$, where $L_1$, $L_2$ and $L_3$ are lines such that $L_3\not \ni P=L_1\cap L_2$;

• $T_P=L+C$, where L is a line and C is a conic such that $P\in C\cap L$;

• $T_P$ is an irreducible cubic curve.

We plan to bound the integral in (3.1) depending on the type of the curve $T_P$ and on the position of the point $Q\in E_1$. First, we deal with the cases when Q is contained in the proper transform of the curve $T_P$. We start with

Lemma 3.2

Suppose that $T_P=L_1+L_2+L_3$, where $L_1$, $L_2$ and $L_3$ are lines passing through P. Let $\widetilde{L}_1$, $\widetilde{L}_2$ and $\widetilde{L}_3$ be the proper transforms on $\widetilde{S}$ of the lines $L_1$, $L_2$ and $L_3$, respectively. Suppose that $Q\in \widetilde{L}_1\cap \widetilde{L}_2\cap \widetilde{L}_3$. Then

$$\begin{aligned} \mathrm {mult}_Q(\pi ^*(D))\leqslant \frac{17}{9}+\varepsilon _k. \end{aligned}$$

Proof

We may assume that $Q=\widetilde{L}_1\cap E_1$. Denote by $\widehat{L}_1$, $\widehat{L}_2$, $\widehat{L}_3$ and $\widehat{E}_1$ the proper transforms on $\widehat{S}$ of the curves $\widetilde{L}_1$, $\widetilde{L}_2$, $\widetilde{L}_3$ and $E_1$, respectively. Then the intersection form of the curves $\widehat{L}_1$, $\widehat{L}_2$, $\widehat{L}_3$ and $\widehat{E}_1$ is negative definite. Moreover, we have

$$\begin{aligned} \eta ^*(-K_S)\sim _{\mathbb {Q}} \widehat{L}_1+\widehat{L}_2+\widehat{L}_3+3\widehat{E}_1+4E_2. \end{aligned}$$

Thus, we conclude that $\tau (E_2)=4$. Now, using Corollary 2.7, we compute

Then the required result follows from (3.1).$\square$

Lemma 3.3

Suppose that $T_P=L_1+L_2+L_3$, where $L_1$, $L_2$ and $L_3$ are lines such that $P=L_1\cap L_2$ and $P\notin L_3$. Let $\widetilde{L}_1$ and $\widetilde{L}_2$ be the proper transforms on $\widetilde{S}$ of the lines $L_1$ and $L_2$, respectively. Suppose that $Q=\widetilde{L}_1\cap E_1$ or $\widetilde{L}_2\cap E_1$. Then

$$\begin{aligned} \mathrm {mult}_Q(\pi ^*(D))\leqslant \frac{49}{27}+\varepsilon _k. \end{aligned}$$

Proof

Denote by $\widehat{L}_1$, $\widehat{L}_2$, $\widehat{L}_3$ and $\widehat{E}_1$ the proper transforms on $\widehat{S}$ of the curves $L_1$, $L_2$, $L_3$ and $E_1$, respectively. Then

$$\begin{aligned} \eta ^*(-K_S)\sim _\mathbb {Q} \widehat{L}_1+\widehat{L}_2+\widehat{L}_3+2\widehat{E}_1+3E_2. \end{aligned}$$

Since the intersection form of the curves $\widehat{L}_1$, $\widehat{L}_2$, $\widehat{L}_3$ and $\widehat{E}_1$ is semi-negative definite, we conclude that $\tau (E_2)=3$. Then, using Corollary 2.7, we get

Then the required result follows from (3.1).$\square$

Lemma 3.4

Suppose that $T_P=L+C$, where L is a line, and C is an irreducible conic. Suppose that L and C meet transversally at P. Denote by $\widetilde{L}$ and $\widetilde{C}$ the proper transforms on $\widetilde{S}$ of the curves L and C, respectively. Suppose that $Q=\widetilde{L}\cap E_1$. Then

$$\begin{aligned} \mathrm {mult}_Q(\pi ^*(D))\leqslant \frac{9}{5}+\varepsilon _k. \end{aligned}$$

Proof

Denote by $\widehat{L}$, $\widehat{C}$ and $\widehat{E}_1$ the proper transforms on $\widehat{S}$ of the curves L, C and $E_1$, respectively. Then

$$\begin{aligned} \eta ^*(-K_S)\sim _\mathbb {Q} \widehat{L}+\widehat{C}+2\widehat{E}_1+3E_2. \end{aligned}$$

Since the intersection form of the curves $\widehat{L}$, $\widehat{C}$ and $\widehat{E}_1$ is negative definite, we conclude that $\tau (E_2)=3$. Moreover, using Corollary 2.7, we get

Now the required assertion follows from (3.1).$\square$

Lemma 3.5

Suppose that $T_P=L+C$, where L is a line, and C is an irreducible conic. Suppose that L and C meet transversally at P. Denote by $\widetilde{L}$ and $\widetilde{C}$ the proper transforms on $\widetilde{S}$ of the curves L and C, respectively. Suppose that $Q=\widetilde{C}\cap E_1$. Then

$$\begin{aligned} \mathrm {mult}_Q(\pi ^*(D))\leqslant \frac{5}{3}+\varepsilon _k. \end{aligned}$$

Proof

Denote by $\widehat{L}$, $\widehat{C}$ and $\widehat{E}_1$ the proper transforms on $\widehat{S}$ of the curves L, C and $E_1$, respectively. Then

$$\begin{aligned} \eta ^*(-K_S)\sim _\mathbb {Q} \widehat{L}+\widehat{C}+2\widehat{E}_1+3E_2. \end{aligned}$$

Since the intersection form of the curves $\widehat{L}$, $\widehat{C}$ and $\widehat{E}_1$ is negative definite, we conclude that $\tau (E_2)=3$. Moreover, using Corollary 2.7, we get

Now the required assertion follows from (3.1).$\square$

Lemma 3.6

Suppose that $T_P=L+C$, where L is a line and C is an irreducible conic. Suppose that L and C meet tangentially at P. Denote by $\widetilde{L}$ and $\widetilde{C}$ the proper transforms on $\widetilde{S}$ of the curves L and C, respectively. Suppose that $Q=E_1\cap \widetilde{L}\cap \widetilde{C}$. Then

$$\begin{aligned} \mathrm {mult}_Q(\pi ^*(D))\leqslant \frac{17}{9}+\varepsilon _k. \end{aligned}$$

Proof

Denote by $\widehat{L}$, $\widehat{C}$ and $\widehat{E}_1$ the proper transforms on $\widehat{S}$ of the curves $\widetilde{L}$, $\widetilde{L}$ and $E_1$, respectively. Then

$$\begin{aligned} \eta ^*(-K_S)\sim _\mathbb {Q} \widehat{L}+\widehat{C}+2\widehat{E}_1+4E_2. \end{aligned}$$

Since the intersection form of the curves $\widehat{L}$, $\widehat{C}$ and $\widehat{E}_1$ is negative definite, we conclude that $\tau (E_2)=4$. Moreover, using Corollary 2.7, we get

Then the required result follows from (3.1).$\square$

Lemma 3.7

Suppose that $T_P$ is an irreducible cubic. Let $\widetilde{C}$ be the proper transform of the curve C on the surface $\widetilde{S}$. Suppose that $Q\in \widetilde{C}$. Then

$$\begin{aligned} \mathrm {mult}_Q(\pi ^*(D))\leqslant \frac{5}{3}+\varepsilon _k. \end{aligned}$$

Proof

Denote by $\widehat{C}$ and $\widehat{E}_1$ the proper transforms on $\widehat{S}$ of the curves $\widetilde{C}$ and $E_1$, respectively. Then

$$\begin{aligned} \eta ^*(-K_S)\sim _\mathbb {Q}\widehat{C}+2\widehat{E}_1+3E_2. \end{aligned}$$

This gives $\tau (E_2)=3$, because the intersection form of the curves $\widehat{C}$ and $\widehat{E}_1$ is negative definite. Using Corollary 2.7, we get

Then the required result follows from (3.1).$\square$

Now we consider the cases when Q is not contained in the proper transform of the singular curve $T_P$ on the surface $\widetilde{S}$. We start with

Lemma 3.8

Suppose that $T_P=L_1+L_2+L_3$, where $L_1$, $L_2$ and $L_3$ are lines passing through P. Let $\widetilde{L}_1$, $\widetilde{L}_2$ and $\widetilde{L}_3$ be the proper transforms on $\widetilde{S}$ of the lines $L_1$, $L_2$ and $L_3$, respectively. Suppose that $Q\notin \widetilde{L}_1\cup \widetilde{L}_2\cup \widetilde{L}_3$. Then

$$\begin{aligned} \mathrm {mult}_Q(\pi ^*(D))\leqslant \frac{5}{3}+\varepsilon _k. \end{aligned}$$

Proof

Denote by $\widehat{L}_1$, $\widehat{L}_2$, $\widehat{L}_3$ and $\widehat{E}_1$ the proper transforms on $\widehat{S}$ of the curves $\widetilde{L}_1$, $\widetilde{L}_2$, $\widetilde{L}_3$ and $E_1$, respectively. Then

$$\begin{aligned} \eta ^*(-K_S)\sim _\mathbb {Q} \widehat{L}_1+\widehat{L}_2+\widehat{L}_3+3\widehat{E}_1+3E_2. \end{aligned}$$

This gives $\tau (E_2)=3$, because the intersection form of the curves $\widehat{L}_1$, $\widehat{L}_2$, $\widehat{L}_3$ and $\widehat{E}_1$ is negative definite. Using Corollary 2.7, we get

Then the required result follows from (3.1).$\square$

In the remaining cases, the pseudoeffective threshold $\tau (E_2)$ is not (always) easy to compute. There is a (birational) reason for this. To explain it, recall from [9] that the linear system is free from base points and gives a morphism $\phi :\widetilde{S}\rightarrow \mathbb {P}^2$. Taking its Stein factorization, we obtain a commutative diagram

where $\alpha$ is a birational morphism, $\beta$ is a double cover branched over a (possibly singular) quartic curve, and $\rho$ is a linear projection from the point P. Here, the surface $\overline{S}$ is a (possibly singular) del Pezzo surface of degree 2. Note that the morphism $\alpha$ is biregular if and only if the curve $T_P$ is irreducible. Moreover, if $T_P$ is reducible, then $\alpha$-exceptional curves are proper transforms of the lines on S that pass through P.

Let $\iota$ be the Galois involution of the double cover $\beta$. Then its action lifts to $\widetilde{S}$. On the other hand, this action does not always descent to a (biregular) action of the surface S. Nevertheless, we can always consider $\iota$ as a birational involution of the surface S. This involution is known as Geiser involution (see [9]). It is biregular if and only if P is an Eckardt point of the surface. In this case, the curve $E_1$ is $\iota$-invariant. However, if P is not an Eckardt point, then $\iota (E_1)$ is the proper transform of the (unique) irreducible component of the curve $T_P$ that is not a line passing through P. In both cases, there exists a commutative diagram

where $S^\prime$ is a smooth cubic surface in $\mathbb {P}^3$, which is isomorphic to the surface S via the involution $\tau$, the morphism $\nu$ is the contraction of the curve $\iota (E_1)$, and $\psi$ is a birational map given by the linear subsystem in consisting of all curves having multiplicity at least 3 at the point P.

Let and . Denote by $T^\prime _Q$ the unique hyperplane section of the cubic surface $S^\prime$ that is singular at $Q^\prime$. If P is not an Eckardt point and Q is not contained in the proper transform of the curve $T_P$, then . In this case, the number $\tau (E_2)$ can be computed using $T^\prime _Q$. This explains why the remaining cases are (slightly) more complicated.

Lemma 3.9

Suppose that $T_P=L_1+L_2+L_3$, where $L_1$, $L_2$ and $L_3$ are lines such that $P=L_1\cap L_2$ and $P\notin L_3$. Let $\widetilde{L}_1$, $\widetilde{L}_2$ and $\widetilde{L}_3$ be the proper transforms on $\widetilde{S}$ of the lines $L_1$, $L_2$ and $L_3$, respectively. Suppose that $Q\notin \widetilde{L}_1\cup \widetilde{L}_2$. Then

$$\begin{aligned} \mathrm {mult}_Q(\pi ^*(D))\leqslant \frac{5}{3}+\varepsilon _k. \end{aligned}$$

Proof

Denote by $\widehat{L}_1$, $\widehat{L}_2$, $\widehat{L}_3$ and $\widehat{E}_1$ the proper transforms on $\widehat{S}$ of the curves $L_1$, $L_2$, $L_3$ and $E_1$, respectively. Then

$$\begin{aligned} \eta ^*(-K_S)\sim _\mathbb {Q} \widehat{L}_1+\widehat{L}_2+\widehat{L}_3+2\widehat{E}_1+2E_2, \end{aligned}$$

which implies that $\tau (E_2)\leqslant 2$. Using Corollary 2.8, we see that

provided that $0\leqslant x\leqslant 2$. However, we have $\tau (E_2)>2$, because the intersection form of the curves $\widehat{L}_1$, $\widehat{L}_2$, $\widehat{L}_3$ and $\widehat{E}_1$ is not semi-negative definite. This also follows from the fact that .

Recall that $\nu :\widetilde{S}\rightarrow S^\prime$ is the contraction of the curve $\widetilde{L}_3$. We let $L_1^\prime =\nu (\widetilde{L}_1)$, $L_2^\prime =\nu (\widetilde{L}_2)$ and $E_1^\prime =\nu (E_1)$. Then $L^\prime _1$, $L^\prime _2$ and $E_1^\prime$ are coplanar lines on $S^\prime$.

Since , the line $E_1^\prime$ is an irreducible component of the curve $T^\prime _Q$. Thus, either $T^\prime _Q$ consists of three lines, or $T^\prime _Q$ is a union of the line $E_1^\prime$ and an irreducible conic.

Suppose that $T^\prime _Q=E_1^\prime +Z^\prime$, where $Z^\prime$ is an irreducible conic on $S^\prime$. Then and , which implies that the conic $Z^\prime$ does not meet the lines $L^\prime _1$ and $L^\prime _2$. Denote by $\widehat{Z}$ the proper transform of the conic $Z^\prime$ on the surface $\widehat{S}$. We have

$$\begin{aligned} \eta ^*(-K_S)\sim _\mathbb {Q} \frac{1}{2}\,\bigl (\widehat{Z}+\widehat{L}_1+\widehat{L}_2\bigr )+2\widehat{E}_1+\frac{5}{2}\,E_2. \end{aligned}$$

This implies that $\tau (E_2)=\frac{5}{2}$, because the intersection form of the curves $\widehat{Z}$, $\widehat{L}_1$, $\widehat{L}_2$ and $\widehat{E}_1$ is semi-negative definite. Using this $\mathbb {Q}$-rational equivalence and Corollary 2.7, we compute

Thus, a direct computation and (3.1) give

$$\begin{aligned} \mathrm {mult}_Q(\pi ^*(D))\leqslant \frac{59}{36}+\varepsilon _k<\frac{5}{3}+\varepsilon _k, \end{aligned}$$

which gives the required assertion.

To complete the proof, we may assume that $T^\prime _Q=E_1^\prime +M^\prime +N^\prime$, where $M^\prime$ and $N^\prime$ are two lines on $S^\prime$ such that . Then , which implies that the lines $M^\prime$ and $N^\prime$ do not meet the lines $L^\prime _1$ and $L^\prime _2$. Denote by $\widehat{M}$ and $\widehat{N}$ the proper transforms on the surface $\widehat{S}$ of the lines $M^\prime$ and $N^\prime$, respectively.

Suppose that $Q^\prime$ is also contained in the line $N^\prime$. This simply means that $Q^\prime$ is an Eckardt point of the surface $S^\prime$. Then

$$\begin{aligned} \eta ^*(-K_S)\sim _\mathbb {Q}\frac{1}{2}\,\bigl (\widehat{M}+\widehat{N}+\widehat{L}_1+\widehat{L}_2\bigr )+2\widehat{E}_1+3E_2. \end{aligned}$$

This gives $\tau (E_2)\geqslant 3$. In fact, we have $\tau (E_2)=3$ here, because the intersection form of the curves $\widehat{M}$, $\widehat{N}$, $\widehat{L}_1$, $\widehat{L}_2$, $\widehat{E}_1$ is negative definite. Using Corollary 2.7, we get

Now, direct computations and (3.1) give the required inequality.

To complete the proof the lemma, we have to consider the case . Then

$$\begin{aligned} \eta ^*(-K_S)\sim _\mathbb {Q}\frac{1}{2}\,\bigl (\widehat{M}+\widehat{N}+\widehat{L}_1+\widehat{L}_2\bigr )+2\widehat{E}_1+\frac{5}{2}\,E_2. \end{aligned}$$

In particular, we see that $\tau (E_2)\geqslant \frac{5}{2}$. Using this $\mathbb {Q}$-rational equivalence and Corollary 2.7, we compute

Thus, in particular, we have $\tau (E_2)>\frac{5}{2}$, since

As in the previous cases, we can find $\tau (E_2)$ and compute for $x>\frac{5}{2}$. However, we can avoid doing this. Namely, note that the divisor $\widehat{E}_1+2\widehat{N}+\widehat{M}$ is nef and

$$\begin{aligned} \bigl (\widehat{E}_1+2\widehat{N}+\widehat{M}\bigr )\cdot (\eta ^*(-K_S)-xE_2)=6-2x, \end{aligned}$$

so that $\tau (E_2)\leqslant 3$. Therefore, using (3.1) and Lemma 2.12, we see that

This finishes the proof of the lemma.$\square$

Lemma 3.10

Suppose that $T_P=L+C$, where L is a line and C is an irreducible conic. Denote by $\widetilde{L}$ and $\widetilde{C}$ the proper transforms on $\widetilde{S}$ of the curves L and C, respectively. Suppose that $Q\notin \widetilde{L}\cup \widetilde{C}$. Then

$$\begin{aligned} \mathrm {mult}_Q(\pi ^*(D))\leqslant \frac{5}{3}+\varepsilon _k. \end{aligned}$$

Proof

Denote by $\widehat{L}$, $\widehat{C}$ and $\widehat{E}_1$ the proper transforms on $\widehat{S}$ of the curves L, $\widetilde{C}$ and $E_1$, respectively. Then

$$\begin{aligned} \eta ^*(-K_S)\sim _\mathbb {Q}\widehat{L}+\widehat{C}+2\widehat{E}_1+2E_2, \end{aligned}$$

so that $\tau (E_2)\geqslant 2$. Using Corollary 2.8, we see that

provided that $0\leqslant x\leqslant 2$. Since , we see that $\tau (E_2)>2$.

Recall that $\nu :\widetilde{S}\rightarrow S^\prime$ is the contraction of the curve $\widetilde{C}$. Let and $E_1^\prime =\nu (E_1)$. Then $L^\prime$ is a line and $E^\prime _1$ is a conic on $S^\prime$ such that .

First, we suppose that $T^\prime _Q$ is irreducible. Denote by the proper transform of the cubic $T^\prime _Q$ on the surface $\widehat{S}$. Then and

Since $\widehat{L}^2=\widehat{E}_1^2=-2$ and $\widehat{T}_Q^2=-1$, we see that the intersection form of the curves $\widehat{L}$, $\widehat{T}_Q$ and $\widehat{E}_1$ is negative definite. On the other hand, we have

$$\begin{aligned} \eta ^*(-K_S)\sim _\mathbb {Q} \frac{1}{2}\,(\widehat{T}_Q+\widehat{L})+\frac{3}{2}\,\widehat{E}_1+\frac{5}{2}\,E_2. \end{aligned}$$

This shows that $\tau (E_2)=\frac{5}{2}$. Hence, using Corollary 2.7, we get

Then a direct calculation and (3.1) give

$$\begin{aligned} \mathrm {mult}_Q(\pi ^*(D))\leqslant \frac{103}{63}+\varepsilon _k<\frac{5}{3}+\varepsilon _k. \end{aligned}$$

Now we suppose that $T^\prime _Q=\ell ^\prime +Z^\prime$, where $\ell ^\prime$ is a line, and $Z^\prime$ is an irreducible conic. Denote by $\widehat{\ell }$ and $\widehat{Z}$ the proper transforms on $\widehat{S}$ of the curves $\ell ^\prime$ and $Z^\prime$, respectively. We get

$$\begin{aligned} \eta ^*(-K_S)\sim _\mathbb {Q}\frac{1}{2}\,\bigl (\widehat{\ell }+\widehat{Z}+\widehat{L}\bigr )+\frac{3}{2}\,\widehat{E}_1+\frac{5}{2}\,E_2 \end{aligned}$$

which implies that $\tau (E_2)\geqslant \frac{5}{2}$. Using Corollary 2.7, we get

In particular, we have

which implies that $\tau (E_2)>\frac{5}{2}$. Observe that the divisor $\widehat{\ell }+2\widehat{Z}+\widehat{L}$ is nef and

$$\begin{aligned} \bigl (\widehat{\ell }+2\widehat{Z}+\widehat{L}\bigr )\cdot (\eta ^*(-K_S)-xE_2)=9-3x, \end{aligned}$$

which implies that $\tau (E_2)\leqslant 3$. Thus, using (3.1) and Lemma 2.12, we get

To complete the proof of the lemma, we may assume that $T^\prime _Q=\ell ^\prime +M^\prime +N^\prime$, where $\ell ^\prime$, $M^\prime$ and $N^\prime$ are lines such that . Since $E^\prime _1$ is a conic passing through $Q^\prime$, we conclude that $Q^\prime$ is not contained in the line $\ell ^\prime$. Note that , and the lines $\ell ^\prime$, $M^\prime$ and $N^\prime$ do not pass through $P^\prime$.

Denote by $\widehat{\ell }$, $\widehat{M}$ and $\widehat{N}$ the proper transforms on $\widehat{S}$ of the lines $\ell ^\prime$, $M^\prime$ and $N^\prime$, respectively. We get

$$\begin{aligned} \eta ^*(-K_S)\sim _{\mathbb {Q}}\frac{1}{2}\,\bigl (\widehat{\ell }+\widehat{M}+\widehat{N}+\widehat{L}\bigr )+\frac{3}{2}\,\widehat{E}_1+\frac{5}{2}\,E_2, \end{aligned}$$

which implies that $\tau (E_2)\geqslant \frac{5}{2}$. In fact, we have $\tau (E_2)>\frac{5}{2}$, because the intersection form of the curves $\widehat{\ell }$, $\widehat{M}$, $\widehat{N}$, $\widehat{L}$ and $\widehat{E}_1$ is not semi-negative definite. Nevertheless, we can use Corollary 2.7 to compute

so that, in particular, we have

Observe that the divisor $2\widehat{\ell }+\widehat{M}+\widehat{N}$ is nef and

$$\begin{aligned} \bigl (2\widehat{\ell }+\widehat{M}+\widehat{N}\bigr )\cdot (\eta ^*(-K_S)-xE_2)=6-2x, \end{aligned}$$

which implies that $\tau (E_2)\leqslant 3$. Thus, using (3.1) and Lemma 2.13, we get

The proof is complete.$\square$

Lemma 3.11

Suppose that $T_P$ is an irreducible cubic curve. Let $\widetilde{C}$ be its proper transform on the surface $\widetilde{S}$. Suppose that $Q\notin \widetilde{C}$. Then

$$\begin{aligned} \mathrm {mult}_Q(\pi ^*(D))\leqslant \frac{5}{3}+\varepsilon _k. \end{aligned}$$

Proof

Denote by $\widehat{C}$ and $\widehat{E}_1$ the proper transforms on $\widehat{S}$ of the curves $\widetilde{C}$ and $E_1$, respectively. Then

$$\begin{aligned} \eta ^*(-K_S)\sim _\mathbb {Q}\widehat{C}+2\widehat{E}_1+2E_2. \end{aligned}$$

Thus, using Corollary 2.8, we get provided that $0\leqslant x\leqslant 2$.

Recall that $\nu :\widetilde{S}\rightarrow S^\prime$ is the contraction of the curve $\widetilde{C}$. Let . Then $E_1^\prime$ is an irreducible cubic curve that is singular at $P^\prime$. Thus, the curve $E_1^\prime$ is smooth at the point $Q^\prime$, so that $T^\prime _Q\ne E_1^\prime$. One can easily check that $T^\prime _Q$ does not contain $P^\prime$.

Suppose that $T^\prime _Q$ is an irreducible cubic. Denote by the proper transform of the curve $T^\prime _Q$ on the surface $\widehat{S}$. We get $\widehat{E}_1^2=-2$, , and

which implies that $\tau (E_2)=\frac{5}{2}$. Using Corollary 2.7, we get

Then (3.1) and direct calculations give

$$\begin{aligned} \mathrm {mult}_Q(\pi ^*(D))\leqslant \frac{49}{30}+\varepsilon _k<\frac{5}{3}+\varepsilon _k. \end{aligned}$$

Now we suppose that $T^\prime _Q=\ell ^\prime +Z^\prime$, where $\ell ^\prime$ is a line and $Z^\prime$ is an irreducible conic. Denote by $\widehat{\ell }$ and $\widehat{Z}$ the proper transforms on $\widehat{S}$ of the curves $\ell ^\prime _Q$ and $Z^\prime$, respectively. We get

$$\begin{aligned} \eta ^*(-K_S)\sim _\mathbb {Q}\frac{1}{2}\,(\widehat{\ell }+\widehat{Z})+\frac{3}{2}\,\widehat{E}_1+\frac{5}{2}\,E_2. \end{aligned}$$

Since the intersection form of the curves $\widehat{\ell }$, $\widehat{Z}$ and $\widehat{E}_1$ is semi-negative definite, we conclude that $\tau (E_2)=\frac{5}{2}$. Using Corollary 2.7, we get

Hence, using (3.1), we see that

$$\begin{aligned} \mathrm {mult}_Q(\pi ^*(D))\leqslant \frac{59}{36}+\varepsilon _k<\frac{5}{3}+\varepsilon _k. \end{aligned}$$

To complete the proof, we may assume that $T^\prime _Q=\ell ^\prime +M^\prime +N^\prime$, where $\ell ^\prime$, $M^\prime$ and $N^\prime$ are lines such that . Denote by $\widehat{\ell }$, $\widehat{M}$ and $\widehat{N}$ the proper transforms on $\widehat{S}$ of the lines $\ell ^\prime$, $M^\prime$ and $N^\prime$, respectively. If $Q^\prime$ is contained in the line $\ell ^\prime$, then

$$\begin{aligned} \eta ^*(-K_S)\sim _\mathbb {Q}\frac{1}{2}\,\bigl (\widehat{\ell }+\widehat{M}+\widehat{N}\bigr )+\frac{3}{2}\,\widehat{E}_1+3E_2, \end{aligned}$$

and the intersection form of the curves $\widehat{\ell }$, $\widehat{M}$, $\widehat{N}$ and $\widehat{E}_1$ is negative definite, which implies that $\tau (E_2)=3$. In this case, Corollary 2.7 gives

which implies the required inequality by (3.1).

To complete the proof, we may assume that $Q^\prime$ is not contained in $\ell ^\prime$. Then the intersection form of the curves $\widehat{\ell }$, $\widehat{M}$, $\widehat{N}$ and $\widehat{E}_1$ is not semi-negative definite. Since

$$\begin{aligned} \eta ^*(-K_S)\sim _\mathbb {Q}\frac{1}{2}\,\bigl (\widehat{\ell }+\widehat{M}+\widehat{N}\bigr )+\frac{3}{2}\,\widehat{E}_1+\frac{5}{2}\,E_2, \end{aligned}$$

we conclude that $\tau (E_2)>\frac{5}{2}$. Moreover, using Corollary 2.7, we get

In particular, we have

Observe that the divisor $2\widehat{\ell }+\widehat{M}+\widehat{N}$ is nef and

$$\begin{aligned} \bigl (2\widehat{\ell }+\widehat{M}+\widehat{N}\bigr )\cdot (\eta ^*(-K_S)-xE_2)=6-2x, \end{aligned}$$

which implies that $\tau (E_2)\leqslant 3$. Thus, using (3.1) and Lemma 2.12, we get

This completes the proof of the lemma.$\square$

Using Corollary 2.6 and Lemmas 3.2–3.11, we immediately get

Corollary 3.12

We have $\delta (S)\geqslant \frac{18}{17}$.

4 Proof of the main result

In this section, we prove Theorem 1.4. Let S be a smooth cubic surface. We have to prove that $\delta (S)\geqslant \frac{6}{5}$. Fix a positive rational number $\lambda <\frac{6}{5}$. Let D be a k-basis type divisor. To prove Theorem 1.4, it is enough to show that, the log pair $(S,\lambda D)$ is log canonical for $k\gg 1$. Suppose that this is not the case. Then there exists a point $P\in S$ such that $(S,\lambda D)$ is not log canonical at P for $k\gg 1$. Let us seek for a contradiction using results obtained in Sect. 3.

Let $\pi :\widetilde{S}\rightarrow S$ be the blow-up of the point P, and let $E_1$ be the exceptional divisor of the blow-up $\pi$. Denote by $\widetilde{D}$ the proper transform of D via $\pi$. Then

By Corollary 2.5, the log pair $(\widetilde{S},\lambda \widetilde{D}+(\lambda \mathrm {mult}_P(D)-1)E_1)$ is not log canonical at some point $Q\in E_1$. Thus, using Lemma 2.2, we see that

(4.1)

Let $\sigma :\widehat{S}\rightarrow \widetilde{S}$ be the blow-up of the point Q, and let $E_2$ be the exceptional curve of $\sigma$. Denote by $\widehat{D}$ and $\widehat{E}_1$ the proper transforms on $\widehat{S}$ of the divisors $\widetilde{D}$ and $E_1$, respectively. By Corollary 2.5, the log pair

$$\begin{aligned} \bigl (\widehat{S},\lambda \widehat{D}+(\lambda \mathrm {mult}_P(D)-1)\widehat{E}_1+(\lambda \mathrm {mult}_P(D)+\lambda \mathrm {mult}_Q(\widetilde{D})-2)E_2\bigr ) \end{aligned}$$

is not log canonical at some point $O\in E_2$.

Let $T_P$ be the hyperplane section of the surface S that is singular at P. Then $T_P$ must be reducible. This follows from (4.1) and Lemmas 3.7 and 3.11.

Denote by the proper transform of the curve $T_P$ on the surface $\widetilde{S}$. Then . This follows from (4.1) and Lemmas 3.9 and 3.10.

In the remaining part of this section, we will deal with the following four cases:

• 1. $T_P$ is a union of three lines passing through P;

• 2. $T_P$ is a union of three lines and only two of them pass through P;

• 3. $T_P$ is a union of a line and a conic that intersect transversally at P;

• 4. $T_P$ is a union of a line and a conic that intersect tangentially at P.

We will treat each of them in a separate subsection. We start with

4.1 Case 1

We have $T_P=L_1+L_2+L_3$, where $L_1$, $L_2$ and $L_3$ are lines passing through the point P. We write

$$\begin{aligned} \lambda D=a_1L_1+a_2L_2+a_3L_3+\Omega , \end{aligned}$$

where $a_1$, $a_2$ and $a_3$ are non-negative rational numbers, and $\Omega$ is an effective $\mathbb {Q}$-divisor whose support does not contain $L_1$, $L_2$ or $L_3$. Then

(4.2)

Denote by $\widetilde{L}_1$, $\widetilde{L}_2$ and $\widetilde{L}_3$ the proper transforms on $\widetilde{S}$ of the lines $L_1$, $L_2$ and $L_3$, respectively. We know that $Q\in \widetilde{L}_1\cup \widetilde{L}_2\cup \widetilde{L}_3$, so that we may assume that $Q=\widetilde{L}_1\cap E_1$. Let $\widetilde{\Omega }$ be the proper transform of the divisor $\Omega$ on the surface $\widetilde{S}$, and let $m=\mathrm {mult}_P(\Omega )$. Then the log pair

$$\begin{aligned} \bigl (\widetilde{S},a_1\widetilde{L}_1+\widetilde{\Omega }+(a_1+a_2+a_3+m-1)E_1\bigr ) \end{aligned}$$

is not log canonical at the point Q.

By Lemma 3.1, we have

(4.3)

where $\varepsilon _k$ is a small constant depending on k such that $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$. Thus, applying Corollary 2.4, we see that

which gives . Combining this with (4.2), we get

(4.4)

Let $\widetilde{m}=\mathrm {mult}_Q(\widetilde{\Omega })$. Then by Lemma 3.2, we have

(4.5)

where $\varepsilon _k$ is a small constant depending on k such that $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$. Then using (4.4) and $m\geqslant \widetilde{m}$, we deduce that

(4.6)

Denote by $\widehat{L}_1$ and $\widehat{\Omega }$ the proper transforms on $\widehat{S}$ of the divisors $\widetilde{L}_1$ and $\widetilde{\Omega }$, respectively. Then the log pair

$$\begin{aligned} \bigl (\widehat{S},a_1\widehat{L}_1+\widehat{\Omega }+(a_1+a_2+a_3+m-1)\widehat{E_1}+(2a_1+a_2+a_3+m+\widetilde{m}-2)E_2\bigr ) \end{aligned}$$

is not log canonical at the point O.

We claim that $O\in \widehat{L}_1\cup \widehat{E}_1$. Indeed, we have $(2a_1+a_2+a_3+m+\widetilde{m}-2)<1$ by (4.5). Thus, if $O\not \in \widehat{L}_1\cup \widehat{E}_1$, then Corollary 2.4 gives

which is impossible by (4.6). Thus, we have $O\in \widehat{L}_1\cup \widehat{E}_1$.

If $O\in \widehat{E}_1$, then the log pair

$$\begin{aligned} \bigl (\widehat{S},\widehat{\Omega }+(a_1+a_2+a_3+m-1)\widehat{E_1}+(2a_1+a_2+a_3+m+\widetilde{m}-2)E_2\bigr ) \end{aligned}$$

is not log canonical at the point O. Then Corollary 2.4 gives $a_1+a_2+a_3+m+\widetilde{m}>2$, so that (4.4) and (4.5) give

which is impossible, since $\lambda <\frac{6}{5}$ and $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$.

Thus, we see that $O\in \widehat{L}_1$. Then the log pair

$$\begin{aligned} \bigl (\widehat{S},a_1\widehat{L}_1+\widehat{\Omega }+(2a_1+a_2+a_3+m+\widetilde{m}-2)E_2\bigr ) \end{aligned}$$

is not log canonical at the point O. Now, using (4.5) and (4.6), we have

$$\begin{aligned} \mathrm {mult}_O\bigl (\widehat{\Omega }+(2a_1+a_2+a_3+m+\widetilde{m}-2)E_2\bigr )&=2a_1+a_2+a_3+m+2\widetilde{m}-2\\&<\biggl (\frac{10}{3}+\frac{3\varepsilon _k}{2}\biggr )\lambda -3<1, \end{aligned}$$

since $\lambda <\frac{6}{5}$ and $k\gg 1$. Thus, Lemma 2.3 gives

so that . Using (4.2) we get $\lambda +4a_1>4$. Using (4.3), we get

which is impossible, since $\lambda <\frac{6}{5}$ and $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$.

4.2 Case 2

We have $T_P=L_1+L_2+L_3$, where $L_1$, $L_2$ and $L_3$ are coplanar lines such that $P=L_1\cap L_2$ and $P\notin L_3$. As in the previous case, we write

$$\begin{aligned} \lambda D=a_1L_1+a_2L_2+\Omega , \end{aligned}$$

where $a_1$ and $a_2$ are non-negative rational numbers, and $\Omega$ is an effective $\mathbb {Q}$-divisor whose support does not contain the lines $L_1$ and $L_2$. Then

(4.7)

Denote by $\widetilde{L}_1$ and $\widetilde{L}_2$ the proper transforms on $\widetilde{S}$ of the lines $L_1$ and $L_2$, respectively. We know that $Q\in \widetilde{L}_1\cup \widetilde{L}_2$, so that we may assume that $Q=\widetilde{L}_1\cap E_1$. Let $\widetilde{\Omega }$ be the proper transform of the divisor $\Omega$ on the surface $\widetilde{S}$, and let $m=\mathrm {mult}_P(\Omega )$. Then the log pair

$$\begin{aligned} \bigl (\widetilde{S},a_1\widetilde{L}_1+\widetilde{\Omega }+(a_1+a_2+a_3+m-1)E_1\bigr ) \end{aligned}$$

is not log canonical at the point Q.

By Lemma 3.1, we have

(4.8)

where $\varepsilon _k$ is a small constant depending on k such that $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$. Thus, using Corollary 2.4, we obtain . Then, using (4.7), we deduce

(4.9)

Let $\widetilde{m}=\mathrm {mult}_Q(\widetilde{\Omega })$. By Lemma 3.3, we have

(4.10)

where $\varepsilon _k$ is a small constant depending on k such that $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$. Thus, using (4.9) and $\widetilde{m}\leqslant m$, we deduce

(4.11)

Denote by $\widehat{L}_1$ and $\widehat{\Omega }$ the proper transforms on $\widehat{S}$ of the divisors $\widetilde{L}_1$ and $\widetilde{\Omega }$, respectively. Then the log pair

$$\begin{aligned} \bigl (\widehat{S},a_1\widehat{L}_1+\widehat{\Omega }+(a_1+a_2+m-1)\widehat{E}_1+(2a_1+a_2+m+\widetilde{m}-2)E_2\bigr ) \end{aligned}$$

is not log canonical at the point O. Then $2a_1+a_2+m+\widetilde{m}-2<1$ by (4.10). Thus, using (4.11) and arguing as in Sect. 4.1, we see that $O\in \widehat{L}_1\cup \widehat{E}_1$.

If $O\in \widehat{E}_1$, then the log pair

$$\begin{aligned} \bigl (\widehat{S},\widehat{\Omega }+(a_1+a_2+m-1)\widehat{E}_1+(2a_1+a_2+m+\widetilde{m}-2)E_2\bigr ) \end{aligned}$$

is not log canonical at the point O, so that $a_1+a_2+m+\widetilde{m}>2$ by Corollary 2.4. Hence, using (4.9) and (4.10), we get

which is impossible, since $\lambda <\frac{6}{5}$ and $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$.

We see that $O\in \widehat{L}_1$. Then the log pair

$$\begin{aligned} \bigl (\widehat{S},a_1\widehat{L}_1+\widehat{\Omega }+(2a_1+a_2+m+\widetilde{m}-2)E_2\bigr ) \end{aligned}$$

is not log canonical at the point O. Now, using (4.10) and (4.11), we deduce

because $\lambda <\frac{6}{5}$ and $k\gg 1$. Then we may apply Lemma 2.3 to get

so that . Using (4.7) we get $\lambda +4a_1>4$. Then, by (4.8), we have

which is impossible, since $\lambda <\frac{6}{5}$ and $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$.

4.3 Case 3

We have $T_P=L+C$, where L is a line and C is an irreducible conic such that they intersect transversally at P. As in the previous cases, we write

$$\begin{aligned} \lambda D=aL+bC+\Omega , \end{aligned}$$

where a and b are non-negative rational numbers, and $\Omega$ is an effective $\mathbb {Q}$-divisor whose support does not contain the curves L and C. Then Lemma 3.1 gives us

(4.12)

where $\varepsilon _k$ is a small constant depending on k such that $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$. And also, we have

(4.13)

Denote by $\widetilde{L}$ and $\widetilde{C}$ the proper transforms on $\widetilde{S}$ of the curves L and C, respectively. We know that $Q\in \widetilde{L}\cup \widetilde{C}$. Moreover, using (4.1) and Lemma 3.5, we see that $Q=\widetilde{L}\cap E_1$.

Denote by $\widetilde{\Omega }$ the proper transforms on $\widetilde{S}$ of the divisor $\Omega$. Let $m=\mathrm {mult}_P(\Omega )$. Then the log pair

$$\begin{aligned} \bigl (\widetilde{S},a\widetilde{L}+\widetilde{\Omega }+(a+b+m-1)E_1\bigr ) \end{aligned}$$

is not log canonical at Q. Since $a<1$, we can apply Corollary 2.4 to this log pair and the curve $\widetilde{L}$. This gives . Combining this with (4.13), we have $\lambda +2a-b>2$, so that

(4.14)

Let $\widetilde{m}=\mathrm {mult}_Q(\widetilde{\Omega })$. Then Lemma 3.4 gives

(4.15)

where $\varepsilon _k$ is a small constant depending on k such that $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$. Thus, using (4.14) and $\widetilde{m}\leqslant m$, we deduce that

(4.16)

Denote by $\widehat{L}$ and $\widehat{\Omega }$ the proper transforms on $\widehat{S}$ of the divisors $\widetilde{L}$ and $\widetilde{\Omega }$, respectively. Then the log pair

$$\begin{aligned} \bigl (\widehat{S},a\widehat{L}+\widehat{\Omega }+(a+b+m-1)\widehat{E}_1+(2a+b+m+\widetilde{m}-2)E_2\bigr ) \end{aligned}$$

is not log canonical at the point O. Note that $2a+b+m+\widetilde{m}-2<1$ by (4.15). Thus, using (4.16) and arguing as in Sect. 4.1, we see that $O\in \widehat{L}\cup \widehat{E_1}$.

If $O\in \widehat{E_1}$, then the log pair

$$\begin{aligned} \bigl (\widehat{S},\widehat{\Omega }+(a+b+m-1)\widehat{E}_1+(2a+b+m+\widetilde{m}-2)E_2\bigr ) \end{aligned}$$

is not log canonical at O. Applying Corollary 2.4 again, we obtain $a+b+m+\widetilde{m}>2$, so that (4.14) and (4.15) give

which is impossible, since $\lambda <\frac{6}{5}$ and $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$.

We see that $O\in \widehat{L}$. Then the log pair

$$\begin{aligned} \bigl (\widehat{S},a\widehat{L}+\widehat{\Omega }+(2a+b+m+\widetilde{m}-2)E_2\bigr ) \end{aligned}$$

is not log canonical at the point O. Now using (4.15) and (4.16), we obtain

because $\lambda <\frac{6}{5}$ and $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$. Thus, applying Lemma 2.3, we get

which gives . Using (4.13), we get $\lambda +4a>4+b\geqslant 4$, so that (4.12) implies

which is impossible, since $\lambda <\frac{6}{5}$ and $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$.

4.4 Case 4

We have $T_P=L+C$, where L is a line, and C is an irreducible conic that tangents L at the point P. We write

$$\begin{aligned} \lambda D=aL+bC+\Omega , \end{aligned}$$

where a and b are non-negative rational numbers, and $\Omega$ is an effective $\mathbb {Q}$-divisor whose support does not contain L and C. Let $m=\mathrm {mult}_P(\Omega )$. Then

$$\begin{aligned} a+b+m>1 \end{aligned}$$

(4.17)

by Lemma 2.2. Meanwhile, it follows from Lemma 3.1 that

(4.18)

where $\varepsilon _k$ is a small constant depending on k such that $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$. And also, we have

(4.19)

Denote by $\widetilde{L}$ and $\widetilde{C}$ the proper transforms on $\widetilde{S}$ of the curves L and C, respectively. We know that $Q=\widetilde{L}\cap \widetilde{C}$. Denote by $\widetilde{\Omega }$ the proper transforms on $\widetilde{S}$ of the divisor $\Omega$. Then the log pair

$$\begin{aligned} \bigl (\widetilde{S},a\widetilde{L}+b\widetilde{C}+\widetilde{\Omega }+(a+b+m-1)E_1\bigr ) \end{aligned}$$

is not log canonical at the point Q. Since $a<1$ by (4.18), we may apply Corollary 2.4 to this log pair at Q with respect to the curve $\widetilde{L}$. This gives

Combining this with (4.19), we get $\lambda +2a>2$, so that

(4.20)

Let $\widetilde{m}=\mathrm {mult}_Q(\widetilde{\Omega })$. Then Lemma 3.6 gives

(4.21)

where $\varepsilon _k$ is a small constant depending on k such that $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$. Thus, using (4.20) and $\widetilde{m}\leqslant m$, we deduce that

(4.22)

Denote by $\widehat{L}$, $\widehat{C}$ and $\widehat{\Omega }$ the proper transforms on $\widehat{S}$ of the divisors $\widetilde{L}$, $\widetilde{C}$ and $\widetilde{\Omega }$, respectively. Then the log pair

$$\begin{aligned} \bigl (\widehat{S},a\widehat{L}+b\widehat{C}+\widehat{\Omega }+(a+b+m-1)\widehat{E}_1+(2a+2b+m+\widetilde{m}-2)E_2\bigr ) \end{aligned}$$

is not log canonical at O. Moreover, it follows from (4.21) that $2a+2b+m+\widetilde{m}-2<1$. Thus, using (4.22) and arguing as in Sect. 4.1, we see that $O\in \widehat{L}\cup \widehat{C}\cup \widehat{E_1}$.

If $O\in \widehat{E_1}$, then the log pair

$$\begin{aligned} \bigl (\widehat{S},\widehat{\Omega }+(a+b+m-1)\widehat{E}_1+(2a+2b+m+\widetilde{m}-2)E_2\bigr ) \end{aligned}$$

is not log canonical at O. In this case, Corollary 2.4 applied to this log pair (and the curve $E_2$) gives $a+b+m+\widetilde{m}>2$, so that (4.20) and (4.15) give

which is impossible, since $\lambda <\frac{6}{5}$ and $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$.

If $O\in \widehat{C}$, then the log pair

$$\begin{aligned} \bigl (\widehat{S},b\widehat{C}+\widehat{\Omega }+(2a+2b+m+\widetilde{m}-2)E_2\bigr ) \end{aligned}$$

is not log canonical at O. In this case, if we apply Corollary 2.4 to this log pair with respect to $E_2$, we get $b+\widetilde{m}>1$, so that (4.21) gives

Combining this with (4.17), we see that , so that (4.20) gives

which is impossible, since $\lambda <\frac{6}{5}$ and $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$.

We see that $O\in \widehat{L}$. Then the log pair

$$\begin{aligned} \bigl (\widehat{S},a\widehat{L}+\widehat{\Omega }+(2a+2b+m+\widetilde{m}-2)E_2\bigr ) \end{aligned}$$

is not log canonical at the point O. Now using (4.21), (4.22) and $\lambda <\frac{6}{5}$, we deduce that

since $\lambda <\frac{6}{5}$ and $k\rightarrow \infty$. Then we may apply Lemma 2.3 to get

which gives . Using (4.19), we see that $\lambda +4a>4$, so that (4.18) gives

which is impossible, since $\lambda <\frac{6}{5}$ and $\varepsilon _k\rightarrow 0$ as $k\rightarrow \infty$.

The proof of Theorem 1.4 is complete.