Lowest log canonical thresholds of a reduced plane curve of degree d

Abstract

We describe the sixth worst singularity that a plane curve of degree $d\geqslant 5$ could have, using its log canonical threshold at the point of singularity. This is an extension of a result due to Cheltsov (J Geom Anal 27(3):2302–2338, 2017) wherein the five lowest values of log canonical thresholds of a plane curve of degree $d \geqslant 3$ were computed. These six small log canonical thresholds, in order, are 2 / d, $({2d-3})/{(d-1)^2}$, $({2d-1})/(d^2-d)$, $({2d-5})/({d^2-3d+1})$, $({2d-3})/(d^2-2d)$ and $({2d-7})/({d^2-4d+1})$. We give examples of curves with these values as their log canonical thresholds using illustrations.

1 Introduction

Let $C_d \subset {\mathbb {P}}^2$ be a reduced plane curve of degree d over ${\mathbb {C}}$ and P be a point on $C_d$. We aim to address the following question:

Question 1.1

Given a curve $C_d$ of fixed degree d, what is the worst singularity that the curve can have at the point P?

We can use various parameters to measure the singularity at the point P, such as multiplicity of the curve at P, $\mathrm{mult}_P(C_d)$, Milnor number, $\mu (P)$, or log canonical threshold of the curve at P, . In this paper, we will use to answer the above question. Recall that

By [4, Exercise 6.18] and [4, Lemma 6.35], we have

This implies that the smaller the value of , the worse the singularity of the curve $C_d$ at P.

In order to answer Question 1.1 for $d \leqslant 4$, the values of log canonical threshold of a given reduced curve $C_d$ at P were computed.

Example 1.2

If $d=1$ or $d=2$, then .

Example 1.3

If $d=3$, then is one of $\{ 1, {5}/{6}, {3}/{4}, {2}/{3}\}$. The worst singularity corresponds to and in this case, $C_d$ is a union of three lines intersecting at P (example of such a curve is $xy(x-y)=0$). Examples of curves with the given values of log canonical threshold (2 / 3, 3 / 4 , 5 / 6, 1, resp.) are illustrated in Fig. 1.

Fig. 1

(i) $C_3\,{:}\, xy(x-y)=0$; (ii) $C_3\,{:}\, y(x^2-y)=0$; (iii) $C_3\,{:}\, x^2-y^3=0;$ (iv) $C_3\,{:}\, xy(x-1)=0$

Example 1.4

(Erik Paemurru) Let $C_4$ be a quartic curve. Then is one of $\{ 1, {5}/{6}, {3}/{4}, {7}/{10}, {9}/{14}, {5}/{8}, {2}/{3}, {3}/{5}, {7}/{12}, {5}/{9}, {1}/{2} \}$. The worst singularity occurs when and in this case $C_4$ is a union of four lines passing through P (example of such a curve is $xy(x-y)(x+y)=0$) as illustrated in Fig. 2.

Fig. 2

$C_4\,{:}\,xy(x^2-y^2)=0$

For curves $C_d$ with $d \leqslant 3$, corresponds uniquely to a type of singularity of $C_d$ at the point P. When $d=4$, this is not the case. For example, both the curves illustrated in Fig. 3 have but have very different singularities at P.

Fig. 3

(i) $C_4\,{:}\, y(x^2-y^3)=0$; (ii) $C_4\,{:}\, (x+x^2+y^2)(x+y^2)=0$

All the three parameters mentioned earlier give the same answer to Question 1.1, since mult$_P(C_d) \leqslant d$, $\mu (P) \leqslant (d-1)^2$, with $\mathrm{mult}_P(C_d) = d$, $\mu (P)=(d-1)^2$ if and only if $C_d$ is a union of d lines. The following theorem proves that computing the log canonical threshold of the curve at P also gives the same answer to the above question.

Theorem 1.5

([2, Theorem 4.1]) One has and the equality holds if and only if $C_d$ is a union of d lines passing through P.

We can then ask the following question:

Question 1.6

What is the second worst singularity at the point P?

While examples given above answer this question for curves of degree $d\leqslant 4$, [1] answers this question for degree $d\geqslant 5$ curves. To present this answer, we introduce certain types of singularities in Sect. 2 and we call these types of singularities ${\mathbb {K}}_n, {\mathbb {T}}_n, \widetilde{{\mathbb {K}}}_n, \widetilde{{\mathbb {T}}}_n, {\mathbb {M}}_n, \widetilde{{\mathbb {M}}}_n, \widehat{{\mathbb {M}}}_n$, where $n=\mathrm {mult}_P(C_d)$. In [1], the following result was obtained.

Theorem 1.7

Suppose $d \geqslant 5$ and . Then the curve $C_d$ has singularities of type ${\mathbb {T}}_{d-1},{\mathbb {K}}_{d-1},\widetilde{{\mathbb {T}}}_{d-1},\widetilde{{\mathbb {K}}}_{d-1}$ at P and the values of their log canonical thresholds at P are $({2d-3})/{(d-1)^2}< ({2d-1})/(d^2-d)< ({2d-5})/({d^2-3d+1}) < ({2d-3})/(d^2-2d)$, respectively.

This result and Theorem 1.5 give the five worst singularities of the curve $C_d$. In this paper, we describe the sixth worst one. To be precise we prove

Theorem 1.8

Suppose $d \geqslant 5$ and

Then the curve $C_d$ has singularity of type ${\mathbb {M}}_{d-1}$, $\widetilde{{\mathbb {M}}}_{d-1}$ or $\widehat{{\mathbb {M}}}_{d-1}$ at the point P with .

In the case of $d=5$, one can hope to determine all possible values of log canonical threshold of quintic curves, like in the case of $d=4$ this was done in Example 1.4.

In Sect. 3 we present preliminary results used in the Proof of Theorem 1.8, while the proof itself is given in Sect. 4.

2 Cusps and other singularities

Let C be a reduced curve on a smooth surface S and P be a point on C. We are interested in singularities of the curve C at the point P. In this section, we introduce various types of singularities which we denote by ${\mathbb {T}}_n,{\mathbb {K}}_n,\widetilde{{\mathbb {T}}}_n,\widetilde{{\mathbb {K}}}_n,{\mathbb {M}}_n,\widetilde{{\mathbb {M}}}_n$ and $\widehat{{\mathbb {M}}}_n$, where $n=\mathrm {mult}_P(C)$. We aim to describe geometric properties of the curve C having one of these types of singularities at P.

Let $f_1:S_1 \rightarrow S$ be the blow-up of S at the point P. Let $C^1$ be the proper transform in $S_1$ of the curve C and $E_1$ be the exceptional divisor of the blow-up.

Fig. 4

C with a ${\mathbb {K}}_n$ singularity at P and its blow-up at P

2.1 Singularities of type ${\mathbb {K}}_n$ (cusps)

A curve C having singularity of type ${\mathbb {K}}_n$ can be defined with the help of its geometric properties as given below. These singularities are also called cusps.

• $\mathrm {mult}_P(C)=n \geqslant 2$,

• ,

• $C^1$ intersects $E_1$ tangentially at $P_1$ and is smooth at this point (Fig. 4).

Recall from [5, Theorem 1.1] that the log canonical threshold of a cuspidal curve is

Remark 2.1

Suppose $S={\mathbb {P}}^2$. Let C be a curve of degree $d \geqslant 3$ having a ${\mathbb {K}}_{n}$ singularity at P. Then $n \leqslant d-1$. If $n=d-1$, then the curve C is irreducible. Such curves do exist. For example, the curve given by $zx^{d-1}+y^d=0$ has singularity of type ${\mathbb {K}}_{d-1}$ at the point .

2.2 Singularities of type ${\mathbb {T}}_{n}$

A curve C having singularity of type ${\mathbb {T}}_{n}$ at P can be defined using the following geometric properties:

• $\mathrm{mult}_{P}(C)=n \geqslant 3$,

• ,

• the point $P_1$ is an ordinary double point of $C^1$ (Fig. 5).

Fig. 5

C with a ${\mathbb {T}}_n$ singularity at P and its blow-up at P

Remark 2.2

Suppose $S={\mathbb {P}}^2$ and C is a curve of degree d. Let L be a line in ${\mathbb {P}}^2$ passing through P, whose proper transform $L^1$ in $S_1$ passes through $P_1$. If the curve C has singularity of type ${\mathbb {T}}_n$, then $C=L+Z$, where Z is an irreducible curve of degree $d-1$ that does not contain L as an irreducible component. Since if not, then

$$\begin{aligned} d \geqslant L.C = d-1+2=d+1 \end{aligned}$$

which is absurd. Thus, $C=L+Z$ and $L \cap Z=P$ where Z has singularity of type ${\mathbb {K}}_{d-2}$ at the point P.

2.3 Singularities of type $\widetilde{{\mathbb {T}}}_{n}$

A curve C having singularity of type $\widetilde{{\mathbb {T}}}_{n}$ at P can be defined using the following geometric properties:

• $\mathrm{mult}_{P}(C)=n \geqslant 4$,

• ,

• the point $P_1$ is an ordinary double point of $C^1$,

• $C^1$ intersects $E_1$ transversally at $Q_1$ and is smooth at this point (Fig. 6).

Fig. 6

C with a $\widetilde{{\mathbb {T}}}_{n}$ singularity at P and its blow-up at P

Remark 2.3

Suppose $S={\mathbb {P}}^2$ and C is a curve of degree d. Let L be a line in ${\mathbb {P}}^2$ passing through the point P, whose proper transform $L^1$ in $S_1$ passes through the point $P_1$. Similar computations as in Remark 2.2 imply $C=Z+L$ so that $L \cap Z=P$, where Z is an irreducible curve of degree $d-1$ that does not contain L as an irreducible component and Z has singularity of type $\widetilde{{\mathbb {K}}}_{d-2}$ at the point P, which is introduced in the next subsection.

2.4 Singularities of type $\widetilde{{\mathbb {K}}}_{n}$

A curve C with singularity of type $\widetilde{{\mathbb {K}}}_{n}$ can be defined using the following geometric properties:

• $\mathrm{mult}_{P}(C)=n\geqslant 3$,

• ,

• $C^1$ intersects $E_1$ tangentially at the point $P_1$ and is smooth at this point,

• $C^1$ is smooth at $Q_1$ and intersects $E_1$ transversally at this point (Fig. 7).

Fig. 7

C with a $\widetilde{{\mathbb {K}}}_{n}$ singularity at P and its blow-up at P

Remark 2.4

Suppose $S={\mathbb {P}}^2$ and C is a curve of degree d. Then C with a $\widetilde{{\mathbb {K}}}_{d-1}$ singularity at P exists. Such a curve can be reducible, for example, $y(x^{d-2}-y^{d-1})=0$ or can be irreducible, for example, $x^{d-2}y+y^d+x^d=0$. If C is reducible, then $C=L+Z$ where Z is a curve of degree $d-1$ which does not contain L as an irreducible component and has singularity of type ${\mathbb {K}}_{d-2}$ at the point P.

2.5 Singularities of type ${\mathbb {M}}_{n}$

A curve C with singularity of type ${\mathbb {M}}_{n}$ at P can be defined using the following geometric properties:

• $\mathrm{mult}_P(C)=n\geqslant 5$,

• ,

• $C^1$ is smooth at the points $Q_1$ and $R_1$ where it intersects transversally with $E_1$,

• the point $P_1$ is an ordinary double point of $C^1$ (Fig. 8).

Fig. 8

C with an ${\mathbb {M}}_n$ singularity at P and its blow-up at P

Remark 2.5

Suppose $S={\mathbb {P}}^2$ and C is a curve of degree d. A curve having singularity of type ${\mathbb {M}}_{d-1}$ at P exists, for example, $x(x^2-y^2)(x^{d-4}-y^{d-3})=0$. It is reducible and thus $C=Z+L$ where L is a line in S that contains the point P so that its proper transform $L^1$ in $S_1$ contains the point $P_1$ and Z is an irreducible curve of degree $d-1$ which does not contain L as an irreducible component.

2.6 Singularities of type $\widetilde{{\mathbb {M}}}_n$

A curve C with singularity of type $\widetilde{{\mathbb {M}}}_n$ at P can be defined using the following geometric properties:

• $\mathrm {mult}_P(C)=n \geqslant 5$,

• ,

• $P_1$ is an ordinary double point of $C^1$ with ,

• $C^1$ intersects $E_1$ tangentially at the point $Q_1$ with and is smooth at this point (Fig. 9).

Fig. 9

C with an $\widetilde{{\mathbb {M}}}_n$ singularity at P and its blow-up at P

Remark 2.6

Suppose $S={\mathbb {P}}^2$ and C is a curve of degree d. Then $n=d-1$ is possible and a curve with singularity of type $\widetilde{{\mathbb {M}}}_{d-1}$ exists. For example, $y(zx^2+y^3)(zy^{d-4}+x^{d-3})=0$ has an $\widetilde{{\mathbb {M}}}_{d-1}$ singularity at the point . In this case, C is reducible and thus, $C=L+Z$ where L is the line in S containing P such that its proper transform $L^1$ passes through the point $P_1$ in $S_1$ and Z is a $d-1$ degree irreducible curve that does not contain L as an irreducible component.

2.7 Singularities of type $\widehat{{\mathbb {M}}}_n$

A curve C with singularity of type $\widehat{{\mathbb {M}}}_n$ at P can be defined using the following geometric properties:

• $\mathrm {mult}_P(C)=n \geqslant 5$,

• ,

• $P_1$ is an ordinary double point of $C^1$ with $(C_1.E_1)_{P_1}=n-2$,

• $Q_1$ is an ordinary double point of $C^1$ with $(C_1.E_1)_{Q_1}=2$ (Fig. 10).

Fig. 10

C with an $\widehat{{\mathbb {M}}}_n$ singularity at P and its blow-up at P

Remark 2.7

Suppose $S={\mathbb {P}}^2$ and C is a curve of degree d with singularity of type $\widehat{{\mathbb {M}}}_n$ at the point P. Then $n=d-1$ is possible, for example, C given by $x(zx^{d-4}+y^{d-3})(z^2y^2+x^4)=0$ has an $\widehat{{\mathbb {M}}}_{d-1}$ singularity at the point . That is, $C=L+Z$ where L is a line in S that passes through the point P whose proper transform contains the point $P_1$ and Z is an irreducible curve in S of degree $d-1$ which does not contain L as an irreducible component.

2.8 Defining equations

In this section, we describe a curve C having any of the above types of singularities using local equations. These descriptions actually are not essential to prove Theorem 1.8. Up to analytic change of coordinates, the equations of the curve C with the respective singularities are given below:

• $\mathbb {K}_n$: $\displaystyle x^n+y^{n+1}+\sum _{{i=1}}^{n+1}\, a_ix^iy^{n+1-i}+\mathrm {H.O.T.}=0$,

• $\mathbb {T}_n$: $\displaystyle x\biggl (x^{n-1}-y^{n}+\sum _{{i=2}}^{n+1}\, a_i x^{i-1}y^{n+1-i}+\mathrm {H.O.T}\biggr )=0$,

• $\widetilde{T}_n$: $\displaystyle x\biggl (y(y^{n-1}-x^{n-2})+\sum _{{i=2}}^{n+1}\,a_i x^{i-1} y^{n+1-i}+\mathrm {H.O.T}\biggr )=0$,

• $\widetilde{K}_n$: $\displaystyle y(x^{n-1}-y^{n})+\sum _{{i=1}}^{n+1}\,a_ix^{i-1}iy^{n+1-i}+\mathrm {H.O.T}=0$,

• $\mathbb {M}_n$:$\displaystyle x\biggl ((x^2-y^2)(x^{n-3}-y^{n-2})+\!\!\!\sum _{{i=2, i \ne 3}}^{n} \!\!\! a_i x^i y^{n+1-i}+\mathrm {H.O.T}\biggr )=0$,

• $\widetilde{\mathbb {M}}_n$: $\displaystyle y\biggl ((x^2+y^3)(y^{n-3}+x^{n-2})+\sum _{{i=1}}^{n-1}\, a_i x^i y^{n-i}+\!\!\!\! \sum _{{i=0 , i \ne n-2}}^{n+1}\!\!\!\!\!\! b_i x^i y^{n+1-i}+\mathrm {H.O.T}\biggr )=0$,

• $\widehat{\mathbb {M}}_n$: $\displaystyle x\biggl ((x^{n-3}+y^{n-2})(y^2+x^4)+\!\!\!\sum _{{i=0 , i\ne 1}}^{n} \!\!\! a_i x^{i-1} y^{n+1-i}+ \sum _{{i=0}}^{n}\, b_i x^{i-1} y^{n+2-i}+\!\!\! \sum _{{i=0 , i \ne 5}}^{n+2}\!\!\!c_i x^{i-1} y^{n+3-i}+\mathrm {H.O.T}\biggr )=0$.

The above set of equations comprise an exhaustive list of curves C of a given degree with the various types of singularities, up to analytic change of coordinates and include the curves missing from the list in [1, Definition 1.9], as pointed out by the referee.

3 Preliminaries

Let S be a smooth surface and P be a point in S. Let D be an effective non-zero ${\mathbb {Q}}$-divisor on the surface S. Then,

$$\begin{aligned} D=\sum _{i=1}^{r}\,a_iC_i \end{aligned}$$

where each $C_i$ is an irreducible curve on S and $a_i \in {\mathbb {Q}}_{\geqslant 0}$.

Let $\pi :{\widetilde{S}} \rightarrow S$ be a birational morphism such that ${\widetilde{S}}$ is smooth. One can then conclude that $\pi$ is a composition of n blow-ups of points. For each $C_i$, we denote its proper transform by $\widetilde{C_i}$ and the exceptional curves of the blow-up by $F_1, F_2,\dots , F_n$. Then,

where $b_i \in {\mathbb {Q}}$. Suppose is a divisor with simple normal crossings (SNC).

Definition 3.1

([4, Definition 6.16]) The log pair (S, D) is said to be log canonical at P if the following conditions are satisfied:

• $a_i \leqslant 1$ for every $C_i$ such that $P \in C_i$,

• for every such that .

Similarly, the log pair (S, D) is said to be Kawamata log terminal at P if

• $a_i < 1$ for every $C_i$ such that $P \in C_i$,

• for every such that .

Let $\pi _1:S_1 \rightarrow S$ be the blow-up of S at the point P and $E_1$ be the exceptional curve of the blow-up. Let $D^1$ be the proper transform of the divisor D on the surface $S_1$ after blow-up. Let

$$\begin{aligned} D^{S_1} = D^1+(\mathrm {mult}_P(D)-1)E_1. \end{aligned}$$

This is called the log pull-back of the log pair (S, D). Observe that

$$\begin{aligned} K_{S_1}+D^1+(\mathrm {mult}_P(D)-1)E_1 \sim _{{\mathbb {Q}}} \pi _1^*(K_S+D). \end{aligned}$$

This implies that the log pair (S, D) is not log canonical at P if $\mathrm {mult}_P(D)>2$, and is not Kawamata log terminal if $\mathrm {mult}_P(D) \geqslant 2$.

Remark 3.2

The log pair (S, D) is log canonical at the point P if and only if ($S_1,D^{S_1})$ is log canonical at every point in $E_1$. Similarly, the log pair (S, D) is Kawamata log terminal at the point P if and only if ($S_1,D^{S_1}$) is Kawamata log terminal at every point in $E_1$.

Lemma 3.3

([4, Exercise 6.18]) Suppose (S, D) is not log canonical at P, then $\mathrm {mult}_P(D)>1$. Similarly, if (S, D) is not Kawamata log terminal at P, then $\mathrm {mult}_P(D) \geqslant 1$.

Let Z be an irreducible curve on S that contains the point P and is smooth at P. Suppose that Z is not contained in . Let $\mu$ be a non-negative rational number.

Theorem 3.4

([3, Theorem 7], [4, Exercise 6.31], [6, Corollary 3.12]) Suppose the log pair $(S, \mu Z+D)$ is not log canonical (not Kawamata log terminal, resp.) at P and $\mu \leqslant 1$ $(\mu <1$, resp.). Then $\mathrm {mult}_P(D.Z)>1$.

Lemma 3.5

If (S, D) is not log canonical at P and $\mathrm {mult}_P(D) \leqslant 2$, then there exists a unique point in $E_1$ such that $(S_1,D^{S_1})$ is not log canonical at it. Similarly, if (S, D) is not Kawamata log terminal at P, and $\mathrm {mult}_P(D)<2$, then there exists a unique point in $E_1$ such that $(S_1,D^{S_1})$ is not Kawamata log terminal at it.

Proof

Suppose (S, D) is not log canonical at P and $\mathrm {mult}_P(D) \leqslant$ 2 and suppose there exist two distinct points $P_1$ and $P_2$ in $E_1$ at which $(S_1,D^{S_1})$ is not log canonical. Then,

by Theorem 3.4. Thus, Remark 3.2 proves the first assertion. Similarly we can prove the second assertion.$\square$

Lemma 3.6

([1, Lemma 2.14]) Suppose (S, D) is not Kawamata log terminal at P, and (S, D) is Kawamata log terminal in a punctured neighbourhood of the point P, then $\mathrm {mult}_P(D)>1$.

Proof

Suppose $\mathrm {mult}_P(D) \leqslant 1$. Let us seek for a contradiction. Since (S, D) is not Kawamata log terminal at P, we have that is not Kawamata log terminal at some point $P_1\in E_1$. From Lemma 3.5 we have that this point $P_1$ is unique. This implies that $\mathrm {mult}_{P}(D)>1$, by Lemma 3.3, which in turn contradicts our assumption.$\square$

Let $Z_1$ and $Z_2$ be irreducible curves on the surface S such that $Z_1$ and $Z_2$ are not contained in and $P \in Z_1 \cap Z_2$. Also, suppose that $Z_1$ and $Z_2$ are smooth at P and intersect transversally at P. Let $\mu _1$ and $\mu _2$ be non-negative rational numbers.

Theorem 3.7

([3, Theorem 13]) If the log pair $(S, \mu _1Z_1+ \mu _2Z_2+D)$ is not log canonical at the point P, and $\mathrm {mult}_P(D) \leqslant 1$, then $\mathrm {mult}_P(D.Z_1) >2(1-\mu _2)$ or $\mathrm {mult}_P(D.Z_2) >2(1-\mu _1)$ (or both). Similarly, if the log pair $(S, \mu _1Z_1+ \mu _2Z_2+D)$ is not Kawamata log terminal at the point P, and $\mathrm {mult}_P(D)<1$, then $\mathrm {mult}_P(D.Z_1) \geqslant 2(1-\mu _2)$ or $\mathrm {mult}_P(D.Z_2) \geqslant 2(1-\mu _1)$ (or both).

4 Proof of the main result

Let us now prove the main result of the paper. Let $C_d$ be a reduced curve of degree $d \geqslant 5$ on a smooth surface S such that $P \in C_d$ and let $m_0=\mathrm {mult}_P(C_d)$. Suppose $({2d-3})/(d^2-2d)<\mathrm {lct}_P(S,C_d) < ({2d-7})/({d^2-4d+1})$. This means that there exists $\lambda < ({2d-7})/({d^2-4d+1})$ such that $(S,\lambda C_d)$ is not Kawamata log terminal at P. Let us also assume that $m_0 \ne d$ and thus $C_d$ is not a union of d lines. We want to show that the curve $C_d$ has singularity of type ${\mathbb {M}}_{d-1},\widetilde{{\mathbb {M}}}_{d-1}$ or $\widehat{{\mathbb {M}}}_{d-1}$ at the point P. It is important to notice that the arguments presented below are very similar to the arguments in [1]. Also, these are local arguments, i.e., it is not necessary for the curve $C_d$ to be smooth everywhere outside of P. We assume that the respective divisors on the surface S at various levels are Kawamata log terminal (or log canonical) at a punctured neighbourhood of P.

Lemma 4.1

The following inequalities are used in the proof of the main result:

• (i) $\lambda < {2}/({d-1})$,

• (ii) $\lambda < ({2k+1})/{kd}$ , for $k\in {\mathbb {Z}}_{>0}$ such that $k \leqslant d-3$,

• (iii) $\lambda < ({2k+1})/({kd+1})$ for $k\in {\mathbb {Z}}_{>0}$ such that $k\leqslant d-5$,

• (iv) $\lambda <{3}/{d}$,

• (v) $\lambda <{2}/({d-2})$,

• (vi) $\lambda <{6}/({3d-4})$,

• (vii) $\lambda < {5}/{2d}$.

The proof is straightforward.

We will now introduce some notations. Let $S=S_0={\mathbb {P}}^2$ and . Consider a sequence of blow-ups $f_i:S_i \rightarrow S_{i-1}$ such that $f_1$ is the blow-up of $P_0=P$, $f_2$ is the blow-up of the point $P_1$, and so on, i.e., $f_i$ is the blow-up of the point $P_{i-1}\in S_{i-1}$. We have

$$\begin{aligned} \cdots \rightarrow S_{k+1} \xrightarrow {\,f_{k+1}\,}{S_k}\rightarrow \cdots \xrightarrow {\,f_4\,} {S_3}\xrightarrow {\,f_3\,}{S_2} \xrightarrow {\,f_2\,}{S_1}\xrightarrow {\,f_1\,}{S_0}. \end{aligned}$$

Also, let $f:S_{k+1} \rightarrow S$ be the composition of the blow-ups, i.e., $f=f_1\circ f_2 \circ \cdots \circ f_{k+1}$. The $f_i$-exceptional divisor during each blow-up is denoted by $E_i$. The proper transform of the exceptional divisors in $S_i$ is denoted by for all $j < i$. Also, after the $f_i$ blow-up, the curve $C_d$ is denoted by $C_d^i$ in $S_i$. The divisors comprising of the curve and the exceptional curves on every floor $S_i$ are together denoted by $D^{S_i}$. We will explicitly describe how each of these points of blow-up are chosen.

Since $(S,\lambda C_d$) is not Kawamata log terminal at the point $P\in C_d$, by Remark 3.2 one has that $(S_1,\lambda C_d^1+(\lambda m_0-1)E_1)$ is not Kawamata log terminal at some point in $E_1$. Let this point be $P_1$.

Lemma 4.2

$\lambda m_0 <2$.

Proof

Since $m_0 \leqslant d-1$, we have $\lambda m_0 \leqslant \lambda (d-1)$. Using Lemma 4.1 (i), we get $\lambda m_0 <2$.$\square$

From Lemma 3.5 this implies that the point $P_1$ is a unique point on $E_1$ at which $(S_1,D^{S_1})$, that is, $(S_1,\lambda C_d^1+(\lambda m_0-1)E_1)$ is not Kawamata log terminal.

Let L be the line in ${\mathbb {P}}^2$ whose proper transform, $L^1$ in $S_1$, contains the point $P_1$.

Lemma 4.3

Suppose $m_0=d-1$. Then L is an irreducible component of $C_d$.

Proof

Observe that

$$\begin{aligned} C_d^1\sim _{{\mathbb {Q}}}f_1^*(C_d)-m_0E_1. \end{aligned}$$

If L is not an irreducible component of the curve $C_d$, then we have

$$\begin{aligned} m_1 \leqslant C_d^1.L^1= C_d.L-m_0 E_1.L^1=d-m_0, \end{aligned}$$

(1)

where $m_1=\hbox {mult}_{P_1}(C_d^1)$. Thus we have $m_0+m_1 \leqslant d$. $P_1 \in C_d^1$, since if not, then $(S_1,(\lambda m_0-1)E_1)$ is not log canonical at the point $P_1$, which is not possible since $\lambda m_0-1<1$ from Lemma 4.2. Since $m_0=d-1$, $m_1=1$. Therefore, $C_d^1$ is smooth at $P_1$.

Let $k=\mathrm{mult}_{P_1}(C_d^1.E_1)$. We claim that $k>d-3$. Indeed, suppose $k \leqslant d-3$. Since $(S_1, D^{S_1})$ is not Kawamata log terminal at the point $P_1 \in E_1$, we have that $(S_2,D^{S_2})$ is not Kawamata log terminal at some point in $E_2$, where

$$\begin{aligned} D^{S_2}=\lambda C_d^2+(\lambda (d-1)-1)E_1^2+(\lambda d-2)E_2. \end{aligned}$$

Let this point be $P_2$. Note that all the coefficients of the curves in $D^{S_2}$ are less than 1. In particular, we have

$$\begin{aligned} \lambda d-2 =\frac{d-2}{d^2-4d+1}<1. \end{aligned}$$

Thus, by Definition 3.1, $(S_3,D^{S_3})$ is not Kawamata log terminal at some point in $E_3$. Let this point be $P_3$. Thus, $f_2$ is the blow-up of $P_1 \in C^1 \cap E_1$, $f_3$ is the blow-up of $P_2 \in C^2 \cap E_2$, and so on. One then has the sequence as mentioned in (i).

We require $k+1$ blow-ups to ensure simple normal crossing of the elements of the divisor over the point P. Here the points of blow-up are such that $P_i=C^i \cap E_i$ and using the notations described earlier, we have

$$\begin{aligned} \begin{aligned} K_{S_{k+1}}&+\lambda C_d^{k+1}+(\lambda (d-1)-1)E_1^{k+1}\\ {}&+(\lambda d-2)E_2^{k+1}+\dots +(\lambda kd-2k)E_{k+1} \sim _{{\mathbb {Q}}} f^*(K_S+\lambda C_d). \end{aligned} \end{aligned}$$

(2)

Let the coefficients of $E_i^{k+1}$ in (2) be denoted by $b_i$. Then since

$$\begin{aligned} \lambda C^{k+1}+ \sum _{i=1}^{k}\,b_i E_i^{k+1}+b_{k+1}E_{k+1} \end{aligned}$$

is a divisor with simple normal crossings over P, at least one of $b_i >1$ or $b_{k+1}>1$. But the coefficients $b_i$ are such that for all $j<i$ and we have

$$\begin{aligned} \lambda kd -2k \leqslant \frac{d^2-5d+6}{d^2-4d+1}<1 \end{aligned}$$

from Lemma 4.1 (ii). That is, in particular for all $j<k+1$.

This contradiction implies that $k > d-3$. We also know that

$$\begin{aligned} k=\mathrm {mult}_{P_1}(C_d^1.E_1)\leqslant (C_d^1.E_1)=m_0=d-1. \end{aligned}$$

Therefore, these inequalities imply $k=d-1$ or $d-2$. Thus, when

• $k=d-1$, then $C_d$ has singularity of type ${\mathbb {K}}_{d-1}$ at P (see Sect. 2.1).

• $k=d-2$, then $C_d$ has singularity of type $\widetilde{{\mathbb {K}}}_{d-1}$ at P (see Sect. 2.4).

If the curve $C_d$ has either of the above singularities at the point P, then or , respectively. Since we assume , neither of these values for k are possible. Therefore, this contradiction implies that L is an irreducible component of the curve $C_d$.$\square$

Lemma 4.4

Suppose $m_0=d-1$. Then $C_d$ has singularity of type ${\mathbb {M}}_{d-1}$, $\widetilde{{\mathbb {M}}}_{d-1}$ or $\widehat{{\mathbb {M}}}_{d-1}$ at the point P.

Proof

From Lemma 4.3 we know that L is an irreducible component of the curve $C_d$, i.e., we have $C_d=C_{d-1}+L$ where $C_{d-1}$ is an irreducible curve of degree $d-1$ which does not contain L as an irreducible component. Let $n_0=\mathrm {mult}_{P}(C_{d-1})$. Since $m_0=\mathrm {mult}_P(C_d)=d-1$, we have $n_0=m_0-1=d-2$ .

Let $f_1:S_1 \rightarrow S$ be the blow-up at the point P and $n_1=\mathrm {mult}_{P_1}(C_{d-1}^1)$. We have $n_1=m_1-1$. We also have $P_1 \in C_{d-1}^1$ since if not, it would mean that $(S_1, \lambda L^1+(\lambda (d-1)-1)E_1)$ is not log canonical at the point $P_1$ which is a contradiction since $\lambda <1$ and $\lambda (d-1)-1<1$ and are SNC divisors over $P_1$. Thus, $n_1 \geqslant 1$.

Consider

that is, $n_0+n_1 \leqslant d-1$ and since $n_0=d-2$, we have $n_1=1$. Thus the curve $C_{d-1}^1$ is smooth at $P_1$.

Let $k=\mathrm {mult}_{P_1}(C_{d-1}^1.E_1)$. We claim $k> d-5$. Instead, suppose $k \leqslant d-5$, then using similar computations as in Lemma 4.3, after $k+1$ blow-ups, we get

where $(S_{k+1},D^{S_{k+1}})$ is not Kawamata log terminal at some point in $E_{k+1}$, which we take to be $P_{k+1}$. Here again, f is a composition of $k+1$ blow-ups and $b_i$ are the coefficients of $E_i^{k+1}$ in the above equation.

Since the curves in the divisor

$$\begin{aligned} \lambda C_{d-1}^{k+1} +\lambda L^{k+1} +\sum _{i=1}^{k}\,b_i E_i^{k+1}+b_{k+1}E_{k+1} \end{aligned}$$

intersect at simple normal crossing at the point P after $k+1$ blow-ups, one of these coefficients should be such that $b_i > 1$ or $b_{k+1}>1$ but we have

from Lemma 4.1 (iii) and the coefficients are such that for all $j<i$. In particular, for all $j<k+1$. This contradiction implies $k>d-5$. We also know that

$$\begin{aligned} k=\mathrm {mult}_{P_1}(C_{d-1}^1.E_1)\leqslant (C_{d-1}^1.E_1)=\mathrm {mult}_{P}(C_{d-1})=n_0=d-2. \end{aligned}$$

Thus, these inequalities imply that $k=d-2$ or $d-3$ or $d-4$. Thus, when

• $k=d-2$, $C_d$ has singularity of type ${\mathbb {T}}_{d-1}$ at P (see Sect. 2.2),

• $k=d-3$, $C_d$ has singularity of type $\widetilde{{\mathbb {T}}}_{d-1}$ at P (see Sect. 2.3).

If $C_d$ has either one of the above singularities at the point P, then or , respectively. Since we assume that $\mathrm {lct}_P({\mathbb {P}}^2,C_d) >({2d-3})/(d^2-2d)$, these values of k are not possible. Thus $k=d-4$, i.e., $C_d$ has singularity of type ${\mathbb {M}}_{d-1}$, $\widetilde{{\mathbb {M}}}_{d-1}$ or $\widehat{{\mathbb {M}}}_{d-1}$ at P.$\square$

Observe that Lemmas 4.3 and  4.4 complete the proof of the main result if $m_0=d-1$. In the remaining part of the section, we will prove that $m_0 \leqslant d-2$ is not possible. In particular, we prove the following proposition.

Proposition 4.5

If $m_0 \leqslant d-2$, then $\mathrm {lct}_P(S, C_d) \geqslant {2}/({d-1})$.

This in turn proves that for our choice of $\lambda$ and the assumption that $(S,\lambda C_d)$ is not Kawamata log terminal at P, $m_0 \leqslant d-2$ is not possible, since $\lambda <{2}/({d-1})$. Let us prove this proposition by the method of contradiction.

Proof

Suppose $m_0 \leqslant d-2$ and . Let $\mu ={2}/({d-1})$. Then $(S,\mu C_d$) is not log canonical, in particular, is not Kawamata log terminal at a point, say P. Let us now obtain the necessary contradiction.

Claim 1

The line L is not an irreducible component of the curve $C_d$.

Proof

We shall prove this by contradiction. Suppose L is an irreducible component of the curve $C_d$. Then $C_d=L+C_{d-1}$, where $C_{d-1}$ is an irreducible curve of degree $d-1$ in ${\mathbb {P}}^2$ and does not contain L as an irreducible component. Let $f_1:S_1 \rightarrow S$ be the blow-up at the point P in $C_d$. Let $n_0=\mathrm {mult}_P(C_{d-1})$.

Since $(S, \mu C_{d-1}+\mu L)$ is not log canonical at P, we have that $(S_1, \mu C_{d-1}^1 + \mu L^1 +(\mu (n_0+1)-1)E_1)$ is not log canonical at some point in $E_1$. We choose this point to be $P_1$. Let $n_1=\mathrm {mult}_{P_1}(C_{d-1}^1)$. Consider

$$\begin{aligned} d-1-n_0=C_{d-1}^1.L^1\geqslant n_1 \end{aligned}$$

which implies that $n_0+n_1 \leqslant d-1$. But $n_0=m_0-1\leqslant d-3$, using our assumption.

Also, $2n_1 \leqslant n_0+n_1$ which implies $2n_1 \leqslant d-1$. We can then conclude that $\mu n_1 \leqslant 1$. We also have $L^1$ and $E_1$ smooth at $P_1$ and intersecting transversally at $P_1$. Thus applying Theorem 3.7, we get

(3)

which implies that $\mu (d-1) > 2$ or

(4)

which implies that $\mu (n_0+2)> 2$.

The two inequalities in (3) and (4) imply that $\mu (d-1) > 2$ which is absurd. Thus, L is not an irreducible component of $C_d$. $\blacksquare$

Since L is not an irreducible component of the curve $C_d$, from the computations in (1) we can also assume that $m_0+m_1 \leqslant d$.

Since $(S,\mu C_d)$ is not log canonical at the point P and since $\mu <1$, we have that $(S_1,\mu C_d^1+ (\mu m_0-1)E_1)$ is not log canonical at some point in $E_1$, say $P_1$. We also have

$$\begin{aligned} \mu m_0-1 \leqslant \mu (d-2)-1=\frac{2}{d-1}\,(d-2)-1=\frac{d-3}{d-1}<1, \end{aligned}$$

for $d\geqslant 5$. Thus, from Lemma 3.5 there exists a unique point in $E_2$, say $P_2$, such that $(S_2, \mu C_d^2 +(\mu m_0-1)E_1^2+(\mu (m_0+m_1)-2)E_2)$ is not log canonical at $P_2$.

We know that $P_2 \in C_d^2$, since if not, it would imply $(S_2,(\mu m_0-1)E_1^2+(\mu (m_0+m_1)-2)E_2)$ is not log canonical at the point $P_2$. This is not possible since $\mu m_0-1<1$, $\mu (m_0 +m_1)-2 \leqslant \mu d -2 <1$, and $E_1^2,E_2$ are SNC divisors at $P_2$.

Claim 2

$P_2 \notin E_1^2$.

Proof

Suppose $P_2 \in E_1^2$. Observe that

$$\begin{aligned} C_d^2 \sim f_2^*(C_d^1)-m_1E_2 \end{aligned}$$

so that $C_d^2.E_1^2=C_d^1.E_1-m_1 E_2.E_1^2=m_0-m_1$.

Also, since $P_2 \in C_d^2 \cap E_1^2 \cap E_2$, we have $m_2=\mathrm {mult}_{P_2}(C_d^2) \leqslant (C_d^2.E_1^2)$. Therefore, $m_2 \leqslant m_0-m_1$. Since $m_2 \leqslant m_1$, we have $2m_2 \leqslant m_1+m_2 \leqslant m_0$ which implies

From Lemma 4.1 (v), we have

$$\begin{aligned} \mu m_2 \leqslant \mu \, \frac{m_0}{2} \leqslant \mu \,\frac{d-2}{2}<1. \end{aligned}$$

We also know that $E_1^2$ and $E_2$ are smooth at $P_2$ and intersect transversally at $P_2$.

Since $(S_2, \mu C_d^2 + (\mu m_0-1)E_1^2+(\mu (m_0+m_1)-2)E_2)$ is not log canonical at the point $P_2$, Theorem 3.7 implies $\mu (m_0-m_1) =\mu C_d^2.E_1^2 > 2(3-\mu (m_0+m_1))$ or $\lambda m_1=\lambda C_d^2.E_2 > 2(2-\lambda m_0)$. That is, $\mu (3m_0+m_1) > 6$ or $\mu (2m_0+m_1) > 4$. We have

$$\begin{aligned} 3m_0+m_1&=2m_0+m_0+m_1\leqslant 2(d-2)+d=3d-4, \end{aligned}$$

(5)

$$\begin{aligned} 2m_0+m_1&\leqslant d-2+d=2d-2. \end{aligned}$$

(6)

Using the above equations (5) and (6), both of the above mentioned inequalities obtained from using Theorem 3.7 result in contradiction, hence proving our claim that $P_2 \notin E_1^2$. $\blacksquare$

Claim 3

$P_2 \notin L^2$.

Proof

Suppose $P_2 \in L^2$. Since L is not an irreducible component of $C_d$, we have and this implies that $m_0+m_1+m_2 \leqslant d$. Also, applying Lemma 3.3, we get $\mu d \geqslant \mu (m_0+m_1+m_2)>3$ which results in a contradiction since $\mu d < 3$. Thus, $P_2 \ne L^2 \cap E_2$. $\blacksquare$

Thus, we have that $(S_2, \mu C_d^2+(\mu (m_0+m_1)-2)E_2)$ is not log canonical at the point $P_2$. Then from Remark 3.2, $(S_3, \mu C_d^3+(\mu (m_0+m_1)-2)E_2^3+(\mu (m_0+m_1+m_2)-3)E_3)$ is not log canonical at some point in $E_3$, say $P_3$.

We have $2m_1 \leqslant m_0+m_1 \leqslant d,$ and

$$\begin{aligned} \mu (m_0+m_1+m_2)\leqslant \mu (m_0+2m_1)\leqslant \mu (d-2+d)=4. \end{aligned}$$

(7)

Therefore $\mu (m_0+m_1+m_2)-3 \leqslant 1$.

$P_3 \in C_d^3$, since if not, then this would imply that $(S_3,(\mu (m_0+m_1)-2)E_2^3+(\mu (m_0+m_1+m_2)-3)E_3)$ is not log canonical at the point $P_3$. But since the coefficients of $E_i\leqslant 1$ and $E_2^3,E_3$ are SNC divisors over the point $P_3$, this is not possible.

Claim 4

$P_3 \notin E_2^3$.

Proof

Suppose $P_3 \in E_2^3$. Observe that

$$\begin{aligned} C_d^3\sim _{{\mathbb {Q}}}f_3^*(C_d^2)-m_2E_3. \end{aligned}$$

We thus have

$$\begin{aligned} C_d^3.E_2^3=f_3^*(C_d^2.E_2)-m_2(E_3.E_2^3)=m_1-m_2. \end{aligned}$$

Therefore, Theorem 3.4 implies

$$\begin{aligned} \mu (m_1-m_2)+(\mu (m_0+m_1+m_2)-3)=(\mu C_d^3+(\mu (m_0+m_1+m_2)-3)E_3).E_2^3 > 1, \end{aligned}$$

which implies $\mu (m_0+2m_1)>4$. From (7) we know that $\mu (m_0+2m_1)\leqslant 4$. This contradiction proves our claim. $\blacksquare$

Therefore, the log pair $(S_3, \mu C_d^3+(\mu (m_0+m_1+m_2)-3)E_3)$ is not log canonical, at the point $P_3$. Thus from Remark 3.2, $(S_4,\mu C_d^4+ (\mu (m_0+m_1+m_2)-3)E_3^4+(\mu (m_0+m_1+m_2+m_3)-4)E_4)$ is not log canonical at a point $P_4 \in E_4$. We have

$$\begin{aligned} m_2+m_3 \leqslant 2m_2 \leqslant m_0+m_1 \leqslant d. \end{aligned}$$

Thus,

$$\begin{aligned} \mu (m_0+m_1+m_2+m_3) \leqslant \mu (2(m_0+m_1)) \leqslant \frac{2}{d-1}\, 2d < 5. \end{aligned}$$

(8)

Claim 5

$P_4 \notin E_3^4$.

Proof

Suppose $P_4 \in E_3^4$. From inequality (8), $m_0+m_1+2m_2 < {5}/{\mu }$. Also, observe that

$$\begin{aligned} C_d^4\sim _{{\mathbb {Q}}}f_4^*(C_d^3)-m_3E_4 \end{aligned}$$

so that $C_d^4.E_3^4=m_2-m_3$. Then from Theorem 3.4 we have

$$\begin{aligned} (\mu C_d^4+(\mu (m_0+m_1+m_2+m_3)-4)E_4).E_3^4>1, \end{aligned}$$

which implies

This contradicts inequality in (8). Thus $P_4 \notin E_3^4$. $\blacksquare$

Thus, ($S_4, \mu C_d^4+(\mu (m_0+m_1+m_2+m_3)-4)E_4)$ is not log canonical at the point $P_4$. From Lemma 3.6, we have

$$\begin{aligned} \mu \mathrm {mult}_{P_4}(C_d^4)+ (\mu (m_0+m_1+m_2+m_3)-4)>1, \end{aligned}$$

which implies

(9)

Now using (9) and a geometric construction of a special curve in $S_4$ we will try to arrive at a contradiction. We may assume that the line L is given by $x=0$ and . Let ${\mathscr {C}}$ be the conic in ${\mathbb {P}}^2$ that is given by

$$\begin{aligned} xz+Axy+By^2=0, \end{aligned}$$

where $A,B \in {\mathbb {C}}$ and $B \ne 0$. Then ${\mathscr {C}}$ is smooth and is tangent to the line L. Denote the proper transform of ${\mathscr {C}}$ in $S^i$ by ${\mathscr {C}}^i$. It follows from Claims 2, 3, 4 and 5, that there exist A and $B\ne 0$ such that ${\mathscr {C}}^i$ on $S^i$ contain $P_i$ for $i=1,2,3$. So we can assume that A, B are chosen this way. Then we have

$$\begin{aligned} {\mathscr {C}}^4 \sim 2L^4+E_1^4+2E_2^4+E_3^4. \end{aligned}$$

Thus the pencil $\vert {\mathscr {C}}^4 \vert$ does not have base points.

Also, let ${\mathscr {L}}$ be a pencil of conics in ${\mathbb {P}}^2$ given by

$$\begin{aligned} s x^2 + t (x+Axy+By^2)=0, \end{aligned}$$

where $s,t\in {\mathbb {C}}$. It is generated by 2L and ${\mathscr {C}}$.

Let $\phi _{\vert {\mathscr {C}}^4 \vert }:S^4 \rightarrow {\mathbb {P}}^1$ be the morphism defined by the pencil $\vert {\mathscr {C}}^4 \vert$. Similarly, let $\phi _{{\mathscr {L}}}:{\mathbb {P}}^2 \dashrightarrow {\mathbb {P}}^1$ be the rational map defined by the pencil ${\mathscr {L}}$. These make the following diagram commutative:

Choose a curve $Z^4$ in $\vert {\mathscr {C}}^4 \vert$ that passes through the point $P_4$. Then $Z^4$ is a smooth irreducible curve. Let the proper transform of $Z^4$ in ${\mathbb {P}}^2$ be denoted by Z. Thus Z is a smooth conic in the pencil ${\mathscr {L}}$. Suppose Z is not an irreducible component of the curve $C_d$, then we have

$$\begin{aligned} 2d-(m_0+m_1+m_2+m_3)=C_d^4.Z^4 \geqslant \mathrm {mult}_{P_4}(C_d^4) . \end{aligned}$$

(10)

Equations (9) and (10) result in a contradiction since $\mu < {5}/({2d})$.

Thus, $C_d=Z+C_{d-2}$ where $C_{d-2}$ is an irreducible curve of degree $d-2$ which does not contain the conic Z as an irreducible component.

Let $C_{d-2}^1,C_{d-2}^2,C_{d-2}^3,C_{d-2}^4$ be the proper transforms of the curve $C_{d-2}$ on the surfaces $S_1, S_2, S_3$ and $S_4$, respectively. Denote by $n_0=\mathrm {mult}_P(C_{d-2})$, $n_1=\mathrm {mult}_{P_1}(C_{d-2}^1)$, $n_2=\mathrm {mult}_{P_2}(C_{d-2}^2)$, $n_3=\mathrm {mult}_{P_3}(C_{d-2}^3)$, and $n_4=\mathrm {mult}_{P_4}(C_{d-2}^4)$. Thus $(S_4, \mu C_{d-2}^4 +\mu Z^4 +((\mu (n_0+n_1+n_2+n_3+4)-4)E_4$) is not log canonical at $P_4$.

Applying Theorem 3.4 to the above gives $\mu (2(d-2)-n_0-n_1-n_2-n_3)+\mu (n_0+n_1+n_2+n_3+4)-4>1$, which implies $\mu > {5}/({2d})$. But $\mu < {5}/({2d})$ and thus this contradiction proves the proposition.$\square$

This in turn proves that $m_0\leqslant d-2$ is not possible for the chosen value of $\lambda$, hence completing the Proof of Theorem 1.8.