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

We describe the sixth worst singularity that a plane curve of degree d⩾5\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\geqslant 5$$\end{document} 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⩾3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d \geqslant 3$$\end{document} were computed. These six small log canonical thresholds, in order, are 2 / d, (2d-3)/(d-1)2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({2d-3})/{(d-1)^2}$$\end{document}, (2d-1)/(d2-d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({2d-1})/(d^2-d)$$\end{document}, (2d-5)/(d2-3d+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({2d-5})/({d^2-3d+1})$$\end{document}, (2d-3)/(d2-2d)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({2d-3})/(d^2-2d)$$\end{document} and (2d-7)/(d2-4d+1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$({2d-7})/({d^2-4d+1})$$\end{document}. We give examples of curves with these values as their log canonical thresholds using illustrations.


Introduction
Let C d ⊂ P 2 be a reduced plane curve of degree d over 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, mult P (C d ), Milnor number, μ(P), or log canonical threshold of the curve at P, lct P (P 2 , C d ). In this paper, we will use lct P (P 2 , C d ) to answer the above question. Recall that lct P (P 2 , C d ) = sup λ ∈ Q | the log pair (P 2 , λC d ) is log canonical at P . This implies that the smaller the value of lct P (P 2 , C d ), the worse the singularity of the curve C d at P.
In order to answer Question 1.1 for d 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 lct P (P 2 , C d ) = 1.
All the three parameters mentioned earlier give the same answer to Question 1.1, since mult P (C d ) d, μ(P) (d − 1) 2 , with mult P (C d ) = d, μ(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. To present this answer, we introduce certain types of singularities in Sect. 2 and we call these types of singularities K n , T n , K n , T n , M n , M n , M n , where n = mult P (C d ). In [1], the following result was obtained.
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 Fig. 4 C with a K n singularity at P and its blow-up at P Theorem 1.8 Suppose d 5 and Then the curve C d has singularity of type 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.

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 T n , K n , T n , K n , M n , M n and M n , where n = 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 → 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.

Singularities of type K n (cusps)
A curve C having singularity of type K n can be defined with the help of its geometric properties as given below. These singularities are also called cusps.
Recall from [5, Theorem 1.1] that the log canonical threshold of a cuspidal curve is 5 C with a T n singularity at P and its blow-up at P Remark 2.1 Suppose S = P 2 . Let C be a curve of degree d 3 having a K n singularity at P. Then n 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 K d−1 at the point P = [0 : 0 : 1].

Singularities of type T n
A curve C having singularity of type T n at P can be defined using the following geometric properties: • the point P 1 is an ordinary double point of C 1 (Fig. 5).

Remark 2.2
Suppose S = P 2 and C is a curve of degree d. Let L be a line in 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 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 which is absurd. Thus, C = L + Z and L ∩ Z = P where Z has singularity of type K d−2 at the point P.

Singularities of type T n
A curve C having singularity of type T n at P can be defined using the following geometric properties: • 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).

Remark 2.3
Suppose S = P 2 and C is a curve of degree d. Let L be a line in P 2 passing through the point P, whose proper transform L 1 in S 1 passes through the point P 1 . Fig. 6 C with a T n singularity at P and its blow-up at P C with a K n singularity at P and its blow-up at P Similar computations as in Remark 2.2 imply C = Z + L so that L ∩ 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 K d−2 at the point P, which is introduced in the next subsection.

Singularities of type K n
A curve C with singularity of type K n can be defined using the following geometric properties: • 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).

Remark 2.4
Suppose S = P 2 and C is a curve of degree d. Then C with a K d−1 singularity at P exists. Such a curve can be reducible, for example, which does not contain L as an irreducible component and has singularity of type K d−2 at the point P.

Singularities of type M n
A curve C with singularity of type M n at P can be defined using the following geometric properties: Fig. 8 C with an M n singularity at P and its blow-up at P • 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).

Remark 2.5 Suppose
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.

Singularities of type M n
A curve C with singularity of type M n at P can be defined using the following geometric properties: • mult P (C) = n 5, • C 1 intersects E 1 tangentially at the point Q 1 with (C 1 .E 1 ) Q 1 = 2 and is smooth at this point ( Fig. 9).

Remark 2.6
Suppose S = P 2 and C is a curve of degree d. Then n = d − 1 is possible and a curve with singularity of type M d−1 exists. For example, 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.

Singularities of type M n
A curve C with singularity of type M n at P can be defined using the following geometric properties:

Fig. 9
C with an M n singularity at P and its blow-up at P Fig. 10 C with an M n singularity at P and its blow-up at P Fig. 10). 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.

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: 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.

Preliminaries
Let S be a smooth surface and P be a point in S. Let D be an effective non-zero Q-divisor on the surface S. Then, where each C i is an irreducible curve on S and a i ∈ Q 0 . Let π : S → S be a birational morphism such that S is smooth. One can then conclude that π is a composition of n blow-ups of points. For each C i , we denote its proper transform by C i and the exceptional curves of the blow-up by F 1 , F 2 , . . . , F n . Then, Similarly, the log pair (S, D) is said to be Kawamata log terminal at P if Let π 1 : S 1 → 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 This is called the log pull-back of the log pair (S, D). Observe that This implies that the log pair (S, D) is not log canonical at P if mult P (D) > 2, and is not Kawamata log terminal if mult P (D) 2.  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 Supp(D). Let μ be a non-negative rational number.  D) is not log canonical at P and mult P (D) 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. Proof Suppose mult P (D) 1. Let us seek for a contradiction. Since (S, D) is not Kawamata log terminal at P, we have that (S 1 , D 1 + (1 − mult P (D))E 1 ) is not Kawamata log terminal at some point P 1 ∈ E 1 . From Lemma 3.5 we have that this point P 1 is unique. This implies that mult P (D) > 1, by Lemma 3.3, which in turn contradicts our assumption.
Let Z 1 and Z 2 be irreducible curves on the surface S such that Z 1 and Z 2 are not contained in Supp(D) and P ∈ Z 1 ∩ Z 2 . Also, suppose that Z 1 and Z 2 are smooth at P and intersect transversally at P. Let μ 1 and μ 2 be non-negative rational numbers.

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 5 on a smooth surface S such that P ∈ C d and let . This means that there exists λ < (2d − 7)/(d 2 − 4d + 1) such that (S, λC d ) is not Kawamata log terminal at P. Let us also assume that m 0 = 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 M d−1 , M d−1 or 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:
The proof is straightforward.
We will now introduce some notations. Let S = S 0 = P 2 and D = (P 2 , λC d ).
Consider a sequence of blow-ups f i : S i → 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 ∈ S i−1 . We have Also, let f : S k+1 → S be the composition of the blow-ups, i.e., f = f 1 • f 2 •· · ·• f k+1 . The f i -exceptional divisor during each blow-up is denoted by E i . The proper transform of the exceptional divisors E j in S i is denoted by E i j for all j < i. Also, after the f i blow-up, the curve C d is denoted by C i d 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, λC d ) is not Kawamata log terminal at the point P ∈ C d , by Remark 3.2 one has that (S 1 , λC 1 d + (λm 0 − 1)E 1 ) is not Kawamata log terminal at some point in E 1 . Let this point be P 1 .
From Lemma 3.5 this implies that the point P 1 is a unique point on E 1 at which is not Kawamata log terminal. Let L be the line in 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 If L is not an irreducible component of the curve C d , then we have where m 1 = mult P 1 (C 1 d ). Thus we have m 0 + m 1 d. P 1 ∈ C 1 d , since if not, then (S 1 , (λm 0 − 1)E 1 ) is not log canonical at the point P 1 , which is not possible since λm 0 − 1 < 1 from Lemma 4.2. Since m 0 = d − 1, m 1 = 1. Therefore, C 1 d is smooth at P 1 . Let 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 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 ∈ C 1 ∩ E 1 , f 3 is the blow-up of P 2 ∈ C 2 ∩ 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 ∩ E i and using the notations described earlier, we have (2) Let the coefficients of E k+1 i in (2) be denoted by b i . Then since 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 b j < b i for all j < i and we have This contradiction implies that k > d − 3. We also know that Therefore, these inequalities imply k = d − 1 or d − 2. Thus, when • k = d − 1, then C d has singularity of type K d−1 at P (see Sect. 2.1).
• k = d − 2, then C d has singularity of type 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 lct P (P 2 , , respectively. Since we assume lct P (P 2 , C d ) > (2d − 3)/(d 2 − 2d), neither of these values for k are possible. Therefore, this contradiction implies that L is an irreducible component of the curve C d .
Let f 1 : S 1 → S be the blow-up at the point P and n 1 = mult P 1 (C 1 d−1 ). We have is not log canonical at the point P 1 which is a contradiction since λ < 1 and λ(d − 1) − 1 < 1 and L 1 , E 1 are SNC divisors over P 1 . Thus, n 1 1. Consider that is, n 0 + n 1 d − 1 and since n 0 = d − 2, we have n 1 = 1. Thus the curve C 1 . We claim k > d − 5. Instead, suppose k 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 k+1 i in the above equation. Since the curves in the divisor 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 b j < b i for all j < i. In particular, b j < b k+1 < 1 for all j < k + 1. This contradiction implies k > d − 5.
We also know that Thus, these inequalities imply that k = d − 2 or d − 3 or d − 4. Thus, when If C d has either one of the above singularities at the point P, then lct P (P 2 , In the remaining part of the section, we will prove that m 0 d − 2 is not possible. In particular, we prove the following proposition. Since (S, μC d−1 +μL) is not log canonical at P, we have that (S 1 , μC 1 d−1 +μL 1 + (μ(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 which implies that n 0 + n 1 d − 1. But n 0 = m 0 − 1 d − 3, using our assumption. Also, 2n 1 n 0 + n 1 which implies 2n 1 d − 1. We can then conclude that μn 1 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 which implies that μ(d − 1) > 2 or which implies that μ(n 0 + 2) > 2. The two inequalities in (3) and (4) imply that μ(d − 1) > 2 which is absurd. Thus, L is not an irreducible component of C d .
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 d.
Since (S, μC d ) is not log canonical at the point P and since μ < 1, we have that is not log canonical at some point in E 1 , say P 1 . We also have 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 / ∈ E 2 1 .
Thus, we have that (S 2 , μC 2 d +(μ(m 0 +m 1 )−2)E 2 ) is not log canonical at the point P 2 . Then from Remark 3.2, is not log canonical at some point in E 3 , say P 3 .
is not log canonical at the point P 3 . But since the coefficients of E i 1 and E 3 2 , E 3 are SNC divisors over the point P 3 , this is not possible.
Therefore, the log pair (S 3 , μC 3 is not log canonical, at the point P 3  Thus, 3 . From inequality (8), m 0 + m 1 + 2m 2 < 5/μ. Also, observe that Then from Theorem 3.4 we have which implies This contradicts inequality in (8). Thus 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 P = [0 : 0 : 1]. Let C be the conic in P 2 that is given by xz + Ax y + By 2 = 0, where A, B ∈ C and B = 0. Then C is smooth and is tangent to the line L. Denote the proper transform of C in S i by C i . It follows from Claims 2, 3, 4 and 5, that there exist A and B = 0 such that 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 Thus the pencil |C 4 | does not have base points. Also, let L be a pencil of conics in P 2 given by sx 2 + t(x + Ax y + By 2 ) = 0, where s, t ∈ C. It is generated by 2L and C. Let φ |C 4 | : S 4 → P 1 be the morphism defined by the pencil |C 4 |. Similarly, let φ L : P 2 P 1 be the rational map defined by the pencil L. These make the following diagram commutative: Choose a curve Z 4 in |C 4 | that passes through the point P 4 . Then Z 4 is a smooth irreducible curve. Let the proper transform of Z 4 in P 2 be denoted by Z . Thus Z is a smooth conic in the pencil L. Suppose Z is not an irreducible component of the curve C d , then we have Equations (9) and (10) result in a contradiction since μ < 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.
This in turn proves that m 0 d − 2 is not possible for the chosen value of λ, hence completing the Proof of Theorem 1.8.