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European Journal of Mathematics

, Volume 5, Issue 3, pp 729–762 | Cite as

Delta invariants of smooth cubic surfaces

  • Ivan CheltsovEmail author
  • Kewei Zhang
Open Access
Research Article
  • 110 Downloads

Abstract

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

Keywords

Cubic surface Fano variety \(\delta \)-Invariant Stability threshold K-stability Kähler–Einstein metric 

Mathematics Subject Classification

14J45 14J26 32Q20 

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 hasIn [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, thenIn 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 Open image in new window . 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 \(\mathbb {Q}\)-divisorAny \(\mathbb {Q}\)-divisor obtained in this way is called a k-basis type (anticanonical) divisor. LetThen let
$$\begin{aligned} \delta (X)=\limsup _{k\in \mathbb {N}}\delta _k(X). \end{aligned}$$
By [2, Theorem A], one hasThe 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] thatOur 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 bywhere \(f_1=0\) and \(f_2=0\) are local defining functions of \(C_1\) and \(C_2\) around the point P. The global intersection number Open image in new window 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 \(a_i\in \mathbb {R}\). Thenwhere \((C.Z_i)_P=0\) 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 (SD) at the point P in terms of Open image in new window and Open image in new window .

Lemma 2.2

([14]) If (SD) 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 (SD) 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 (SD) 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 \(\pi \). ThenThis implies

Corollary 2.5

The log pair (SD) 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 (SD) is log canonical at P.
If D is a Cartier divisor, then its volume is the numberwhere 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 identityfor an appropriate \(\lambda \in \mathbb {Q}_{>0}\). Then the volume Open image in new window 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: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 Open image in new window . If the divisor D is nef, thenThis 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 Open image in new window , 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] thatThis immediately gives

Corollary 2.7

Let \(Z_1,\ldots ,Z_s\) be irreducible curves on S such that Open image in new window for every i, and the intersection form of the curves \(Z_1,\ldots ,Z_s\) is negative definite. Thenwhere \(b_1,\ldots ,b_s\) are (uniquely defined) non-negative real numbers such thatfor every j.

Corollary 2.8

Let Z be an irreducible curve on S such that Open image in new window and Open image in new window . 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 \(\widehat{S}\). LetThis 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 \(k\gg 1\). Thenwhere \(\varepsilon _k\) 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 quantityis 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 thatApplying Theorem 2.9 with \(\widehat{S}=\widetilde{S}\), \(\eta =\pi \) and \(F=E_1\), we see thatwhere \(\varepsilon _k\) 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 Open image in new window 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 haveTherefore, 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 thatso 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 thatLet \(\sigma :\widehat{S}\rightarrow \widetilde{S}\) be the blow-up of the point Q. Denote by \(E_2\) the exceptional curve of \(\sigma \). Let Open image in new window . Applying Theorem 2.9 with \(F=E_1\), we see thatHere, 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 thatprovided that \(0\leqslant x\leqslant \frac{2}{3}\). Likewise, if \(\frac{2}{3}\leqslant x\leqslant 3\), then Corollary 2.7 givesThus, if \(Q\in \widetilde{C}\), thenso that \(\mathrm {mult}_Q(\pi ^*(D))\leqslant \frac{2}{\lambda }\) for \(k\gg 1\), becauseLikewise, 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 thatwhich implies thatso 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 Open image in new window 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 thatwhere \(\varepsilon _k\) is a constant that depends on k such that \(\varepsilon _k\rightarrow 0\) as \(k\rightarrow \infty \). Let \(m=\mathrm {mult}_P(\Delta )\). Thenso 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 \(\lambda (2a+m)-1<1\). ButThus, we have \(Q\in E\cap \widetilde{C}\) by Corollary 2.4. On the other hand, for \(k\gg 1\), we haveso that we can apply Lemma 2.3 to our pair at Q. This givesso that \(\lambda (1+4a)>4\), and hencewhich is absurd for Open image in new window . 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 hasfor any \(\mu \in [0,\tau (F)]\).

Proof

The assertion follows from the fact that Open image in new window 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 hasfor 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 Open image in new window , we have \(\tau (F)=1\), so thatby 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 Open image in new window , thenApplying Theorem 2.9, we getLet \(T_P\) 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 computeThen 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 getThen 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 getNow 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 getNow 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 getThen 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 getThen 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 getThen 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 Open image in new window is free from base points and gives a morphism \(\phi :\widetilde{S}\rightarrow \mathbb {P}^2\). Taking its Stein factorization, we obtain a commutative diagramwhere \(\alpha \) 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 diagramwhere \(S^\prime \) 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 Open image in new window consisting of all curves having multiplicity at least 3 at the point P.

Let Open image in new window and Open image in new window . 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 Open image in new window . 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 thatprovided 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 Open image in new window .

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 Open image in new window , 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 Open image in new window and Open image in new window , 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 computeThus, 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 Open image in new window . Then Open image in new window , 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 getNow, direct computations and (3.1) give the required inequality.
To complete the proof the lemma, we have to consider the case Open image in new window . 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 computeThus, in particular, we have \(\tau (E_2)>\frac{5}{2}\), sinceAs in the previous cases, we can find \(\tau (E_2)\) and compute Open image in new window 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 thatThis 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 thatprovided that \(0\leqslant x\leqslant 2\). Since Open image in new window , we see that \(\tau (E_2)>2\).

Recall that \(\nu :\widetilde{S}\rightarrow S^\prime \) is the contraction of the curve \(\widetilde{C}\). Let Open image in new window 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 Open image in new window .

First, we suppose that \(T^\prime _Q\) is irreducible. Denote by Open image in new window the proper transform of the cubic \(T^\prime _Q\) on the surface \(\widehat{S}\). Then Open image in new window andSince \(\widehat{L}^2=\widehat{E}_1^2=-2\) 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 getThen 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 getIn particular, we havewhich 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 getTo 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 Open image in new window . 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 Open image in new window , 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 computeso that, in particular, we haveObserve 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 getThe 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 Open image in new window provided that \(0\leqslant x\leqslant 2\).

Recall that \(\nu :\widetilde{S}\rightarrow S^\prime \) is the contraction of the curve \(\widetilde{C}\). Let Open image in new window . 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 Open image in new window the proper transform of the curve \(T^\prime _Q\) on the surface \(\widehat{S}\). We get \(\widehat{E}_1^2=-2\), Open image in new window , Open image in new window andwhich implies that \(\tau (E_2)=\frac{5}{2}\). Using Corollary 2.7, we getThen (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 getHence, 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 Open image in new window . 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 giveswhich 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 getIn particular, we haveObserve 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 getThis completes the proof of the lemma.\(\square \)

Using Corollary 2.6 and Lemmas 3.23.11, 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 \(\pi \). ThenBy 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 thatLet \(\sigma :\widehat{S}\rightarrow \widetilde{S}\) 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 Open image in new window the proper transform of the curve \(T_P\) on the surface \(\widetilde{S}\). Then Open image in new window . 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. 1.

    \(T_P\) is a union of three lines passing through P;

     
  2. 2.

    \(T_P\) is a union of three lines and only two of them pass through P;

     
  3. 3.

    \(T_P\) is a union of a line and a conic that intersect transversally at P;

     
  4. 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 \(L_3\). ThenDenote 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 havewhere \(\varepsilon _k\) is a small constant depending on k such that \(\varepsilon _k\rightarrow 0\) as \(k\rightarrow \infty \). Thus, applying Corollary 2.4, we see thatwhich gives Open image in new window . Combining this with (4.2), we getLet \(\widetilde{m}=\mathrm {mult}_Q(\widetilde{\Omega })\). Then by Lemma 3.2, we havewhere \(\varepsilon _k\) 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 thatDenote 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 giveswhich 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) givewhich 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 givesso that Open image in new window . Using (4.2) we get \(\lambda +4a_1>4\). Using (4.3), we getwhich 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 \(L_2\). ThenDenote 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 havewhere \(\varepsilon _k\) is a small constant depending on k such that \(\varepsilon _k\rightarrow 0\) as \(k\rightarrow \infty \). Thus, using Corollary 2.4, we obtain Open image in new window . Then, using (4.7), we deduceLet \(\widetilde{m}=\mathrm {mult}_Q(\widetilde{\Omega })\). By Lemma 3.3, we havewhere \(\varepsilon _k\) 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 deduceDenote 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 getwhich 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 deducebecause \(\lambda <\frac{6}{5}\) and \(k\gg 1\). Then we may apply Lemma 2.3 to getso that Open image in new window . Using (4.7) we get \(\lambda +4a_1>4\). Then, by (4.8), we havewhich 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 uswhere \(\varepsilon _k\) is a small constant depending on k such that \(\varepsilon _k\rightarrow 0\) as \(k\rightarrow \infty \). And also, we haveDenote 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 Open image in new window . Combining this with (4.13), we have \(\lambda +2a-b>2\), so thatLet \(\widetilde{m}=\mathrm {mult}_Q(\widetilde{\Omega })\). Then Lemma 3.4 giveswhere \(\varepsilon _k\) 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 thatDenote 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) givewhich 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 obtainbecause \(\lambda <\frac{6}{5}\) and \(\varepsilon _k\rightarrow 0\) as \(k\rightarrow \infty \). Thus, applying Lemma 2.3, we getwhich gives Open image in new window . Using (4.13), we get \(\lambda +4a>4+b\geqslant 4\), so that (4.12) implieswhich 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 thatwhere \(\varepsilon _k\) is a small constant depending on k such that \(\varepsilon _k\rightarrow 0\) as \(k\rightarrow \infty \). And also, we haveDenote 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 givesCombining this with (4.19), we get \(\lambda +2a>2\), so thatLet \(\widetilde{m}=\mathrm {mult}_Q(\widetilde{\Omega })\). Then Lemma 3.6 giveswhere \(\varepsilon _k\) 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 thatDenote 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) givewhich 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) givesCombining this with (4.17), we see that Open image in new window , so that (4.20) giveswhich 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 thatsince \(\lambda <\frac{6}{5}\) and \(k\rightarrow \infty \). Then we may apply Lemma 2.3 to getwhich gives Open image in new window . Using (4.19), we see that \(\lambda +4a>4\), so that (4.18) giveswhich is impossible, since \(\lambda <\frac{6}{5}\) and \(\varepsilon _k\rightarrow 0\) as \(k\rightarrow \infty \).

The proof of Theorem 1.4 is complete.

Notes

Acknowledgements

The authors thank Yanir Rubinstein for many helpful discussions. This paper was finished during the authors’ visit to the Department of Mathematics at the University of Maryland, College Park. The authors appreciate its excellent environment and hospitality.

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Authors and Affiliations

  1. 1.School of MathematicsThe University of EdinburghEdinburghUK
  2. 2.Laboratory of Algebraic GeometryNational Research University Higher School of EconomicsMoscowRussia
  3. 3.Beijing International Center for Mathematical ResearchPeking UniversityBeijingPeople’s Republic of China

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