Lowest log canonical thresholds of a reduced plane curve of degree d
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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, \(({2d3})/{(d1)^2}\), \(({2d1})/(d^2d)\), \(({2d5})/({d^23d+1})\), \(({2d3})/(d^22d)\) and \(({2d7})/({d^24d+1})\). We give examples of curves with these values as their log canonical thresholds using illustrations.
Keywords
Log canonical threshold Plane curves \(\alpha \)invariant of Tian singularityMathematics Subject Classification
14H20 14H50 14J70 14E051 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?
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 Open image in new window .
Example 1.3
Example 1.4
All the three parameters mentioned earlier give the same answer to Question 1.1, since mult\(_P(C_d) \leqslant d\), \(\mu (P) \leqslant (d1)^2\), with \(\mathrm{mult}_P(C_d) = d\), \(\mu (P)=(d1)^2\) 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 Open image in new window 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 Open image in new window . Then the curve \(C_d\) has singularities of type \({\mathbb {T}}_{d1},{\mathbb {K}}_{d1},\widetilde{{\mathbb {T}}}_{d1},\widetilde{{\mathbb {K}}}_{d1}\) at P and the values of their log canonical thresholds at P are \(({2d3})/{(d1)^2}< ({2d1})/(d^2d)< ({2d5})/({d^23d+1}) < ({2d3})/(d^22d)\), 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
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.
2.1 Singularities of type \({\mathbb {K}}_n\) (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).
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 \(n \leqslant d1\). If \(n=d1\), then the curve C is irreducible. Such curves do exist. For example, the curve given by \(zx^{d1}+y^d=0\) has singularity of type \({\mathbb {K}}_{d1}\) at the point Open image in new window .
2.2 Singularities of type \({\mathbb {T}}_{n}\)

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

the point \(P_1\) is an ordinary double point of \(C^1\) (Fig. 5).
Remark 2.2
2.3 Singularities of type \(\widetilde{{\mathbb {T}}}_{n}\)

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

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).
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 \(d1\) that does not contain L as an irreducible component and Z has singularity of type \(\widetilde{{\mathbb {K}}}_{d2}\) at the point P, which is introduced in the next subsection.
2.4 Singularities of type \(\widetilde{{\mathbb {K}}}_{n}\)

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

\(C^1\) 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).
Remark 2.4
Suppose \(S={\mathbb {P}}^2\) and C is a curve of degree d. Then C with a \(\widetilde{{\mathbb {K}}}_{d1}\) singularity at P exists. Such a curve can be reducible, for example, \(y(x^{d2}y^{d1})=0\) or can be irreducible, for example, \(x^{d2}y+y^d+x^d=0\). If C is reducible, then \(C=L+Z\) where Z is a curve of degree \(d1\) which does not contain L as an irreducible component and has singularity of type \({\mathbb {K}}_{d2}\) at the point P.
2.5 Singularities of type \({\mathbb {M}}_{n}\)

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

\(C^1\) 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).
Remark 2.5
Suppose \(S={\mathbb {P}}^2\) and C is a curve of degree d. A curve having singularity of type \({\mathbb {M}}_{d1}\) at P exists, for example, \(x(x^2y^2)(x^{d4}y^{d3})=0\). 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 \(d1\) which does not contain L as an irreducible component.
2.6 Singularities of type \(\widetilde{{\mathbb {M}}}_n\)

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

\(P_1\) is an ordinary double point of \(C^1\) with Open image in new window ,

\(C^1\) intersects \(E_1\) tangentially at the point \(Q_1\) with Open image in new window and is smooth at this point (Fig. 9).
Remark 2.6
Suppose \(S={\mathbb {P}}^2\) and C is a curve of degree d. Then \(n=d1\) is possible and a curve with singularity of type \(\widetilde{{\mathbb {M}}}_{d1}\) exists. For example, \(y(zx^2+y^3)(zy^{d4}+x^{d3})=0\) has an \(\widetilde{{\mathbb {M}}}_{d1}\) singularity at the point Open image in new window . 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 \(d1\) degree irreducible curve that does not contain L as an irreducible component.
2.7 Singularities of type \(\widehat{{\mathbb {M}}}_n\)

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

\(P_1\) is an ordinary double point of \(C^1\) with \((C_1.E_1)_{P_1}=n2\),

\(Q_1\) is an ordinary double point of \(C^1\) with \((C_1.E_1)_{Q_1}=2\) (Fig. 10).
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 \(n=d1\) is possible, for example, C given by \(x(zx^{d4}+y^{d3})(z^2y^2+x^4)=0\) has an \(\widehat{{\mathbb {M}}}_{d1}\) singularity at the point Open image in new window . 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 \(d1\) which does not contain L as an irreducible component.
2.8 Defining equations

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

\(\mathbb {T}_n\): \(\displaystyle x\biggl (x^{n1}y^{n}+\sum _{{i=2}}^{n+1}\, a_i x^{i1}y^{n+1i}+\mathrm {H.O.T}\biggr )=0\),

\(\widetilde{T}_n\): \(\displaystyle x\biggl (y(y^{n1}x^{n2})+\sum _{{i=2}}^{n+1}\,a_i x^{i1} y^{n+1i}+\mathrm {H.O.T}\biggr )=0\),

\(\widetilde{K}_n\): \(\displaystyle y(x^{n1}y^{n})+\sum _{{i=1}}^{n+1}\,a_ix^{i1}iy^{n+1i}+\mathrm {H.O.T}=0\),

\(\mathbb {M}_n\):\(\displaystyle x\biggl ((x^2y^2)(x^{n3}y^{n2})+\!\!\!\sum _{{i=2, i \ne 3}}^{n} \!\!\! a_i x^i y^{n+1i}+\mathrm {H.O.T}\biggr )=0\),

\(\widetilde{\mathbb {M}}_n\): \(\displaystyle y\biggl ((x^2+y^3)(y^{n3}+x^{n2})+\sum _{{i=1}}^{n1}\, a_i x^i y^{ni}+\!\!\!\! \sum _{{i=0 , i \ne n2}}^{n+1}\!\!\!\!\!\! b_i x^i y^{n+1i}+\mathrm {H.O.T}\biggr )=0\),

\(\widehat{\mathbb {M}}_n\): \(\displaystyle x\biggl ((x^{n3}+y^{n2})(y^2+x^4)+\!\!\!\sum _{{i=0 , i\ne 1}}^{n} \!\!\! a_i x^{i1} y^{n+1i}+ \sum _{{i=0}}^{n}\, b_i x^{i1} y^{n+2i}+\!\!\! \sum _{{i=0 , i \ne 5}}^{n+2}\!\!\!c_i x^{i1} y^{n+3i}+\mathrm {H.O.T}\biggr )=0\).
3 Preliminaries
Definition 3.1

\(a_i \leqslant 1\) for every \(C_i\) such that \(P \in C_i\),

Open image in new window for every Open image in new window such that Open image in new window .

\(a_i < 1\) for every \(C_i\) such that \(P \in C_i\),

Open image in new window for every Open image in new window such that Open image in new window .
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 Open image in new window . Let \(\mu \) be a nonnegative 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
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 Open image in new window 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 Open image in new window 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 nonnegative 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 \(({2d3})/(d^22d)<\mathrm {lct}_P(S,C_d) < ({2d7})/({d^24d+1})\). This means that there exists \(\lambda < ({2d7})/({d^24d+1})\) 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 \({\mathbb {M}}_{d1},\widetilde{{\mathbb {M}}}_{d1}\) or \(\widehat{{\mathbb {M}}}_{d1}\) 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
 (i)
\(\lambda < {2}/({d1})\),
 (ii)
\(\lambda < ({2k+1})/{kd}\) , for \(k\in {\mathbb {Z}}_{>0}\) such that \(k \leqslant d3\),
 (iii)
\(\lambda < ({2k+1})/({kd+1})\) for \(k\in {\mathbb {Z}}_{>0}\) such that \(k\leqslant d5\),
 (iv)
\(\lambda <{3}/{d}\),
 (v)
\(\lambda <{2}/({d2})\),
 (vi)
\(\lambda <{6}/({3d4})\),
 (vii)
\(\lambda < {5}/{2d}\).
The proof is straightforward.
Since \((S,\lambda C_d\)) is not Kawamata log terminal at the point \(P\in C_d\), by Remark 3.2 one has that \((S_1,\lambda C_d^1+(\lambda m_01)E_1)\) 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 \(m_0 \leqslant d1\), we have \(\lambda m_0 \leqslant \lambda (d1)\). 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, \((S_1,\lambda C_d^1+(\lambda m_01)E_1)\) 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 \(m_0=d1\). Then L is an irreducible component of \(C_d\).
Proof
Lemma 4.4
Suppose \(m_0=d1\). Then \(C_d\) has singularity of type \({\mathbb {M}}_{d1}\), \(\widetilde{{\mathbb {M}}}_{d1}\) or \(\widehat{{\mathbb {M}}}_{d1}\) 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 \(C_d=C_{d1}+L\) where \(C_{d1}\) is an irreducible curve of degree \(d1\) which does not contain L as an irreducible component. Let \(n_0=\mathrm {mult}_{P}(C_{d1})\). Since \(m_0=\mathrm {mult}_P(C_d)=d1\), we have \(n_0=m_01=d2\) .
Let \(f_1:S_1 \rightarrow S\) be the blowup at the point P and \(n_1=\mathrm {mult}_{P_1}(C_{d1}^1)\). We have \(n_1=m_11\). We also have \(P_1 \in C_{d1}^1\) since if not, it would mean that \((S_1, \lambda L^1+(\lambda (d1)1)E_1)\) is not log canonical at the point \(P_1\) which is a contradiction since \(\lambda <1 \) and \(\lambda (d1)1<1\) and Open image in new window are SNC divisors over \(P_1\). Thus, \(n_1 \geqslant 1\).
Observe that Lemmas 4.3 and 4.4 complete the proof of the main result if \(m_0=d1\). In the remaining part of the section, we will prove that \(m_0 \leqslant d2\) is not possible. In particular, we prove the following proposition.
Proposition 4.5
If \(m_0 \leqslant d2\), then \(\mathrm {lct}_P(S, C_d) \geqslant {2}/({d1})\).
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, \(m_0 \leqslant d2\) is not possible, since \(\lambda <{2}/({d1})\). Let us prove this proposition by the method of contradiction.
Proof
Suppose \(m_0 \leqslant d2\) and Open image in new window . Let \(\mu ={2}/({d1})\). 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 \(C_d=L+C_{d1}\), where \(C_{d1}\) is an irreducible curve of degree \(d1\) in \({\mathbb {P}}^2\) and does not contain L as an irreducible component. Let \(f_1:S_1 \rightarrow S\) be the blowup at the point P in \(C_d\). Let \(n_0=\mathrm {mult}_P(C_{d1})\).
The two inequalities in (3) and (4) imply that \(\mu (d1) > 2\) 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 \(m_0+m_1 \leqslant d\).
We know that \(P_2 \in C_d^2\), since if not, it would imply \((S_2,(\mu m_01)E_1^2+(\mu (m_0+m_1)2)E_2)\) is not log canonical at the point \(P_2\). This is not possible since \(\mu m_01<1\), \(\mu (m_0 +m_1)2 \leqslant \mu d 2 <1\), and \(E_1^2,E_2\) are SNC divisors at \(P_2\).
Claim 2
\(P_2 \notin E_1^2\).
Proof
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 Open image in new window 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 \(P_3\).
\(P_3 \in C_d^3\), 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
Claim 5
\(P_4 \notin E_3^4\).
Proof
Thus, \(C_d=Z+C_{d2}\) where \(C_{d2}\) is an irreducible curve of degree \(d2\) which does not contain the conic Z as an irreducible component.
Let \(C_{d2}^1,C_{d2}^2,C_{d2}^3,C_{d2}^4\) be the proper transforms of the curve \(C_{d2}\) on the surfaces \(S_1, S_2, S_3\) and \(S_4\), respectively. Denote by \(n_0=\mathrm {mult}_P(C_{d2})\), \(n_1=\mathrm {mult}_{P_1}(C_{d2}^1)\), \(n_2=\mathrm {mult}_{P_2}(C_{d2}^2)\), \(n_3=\mathrm {mult}_{P_3}(C_{d2}^3)\), and \(n_4=\mathrm {mult}_{P_4}(C_{d2}^4)\). Thus \((S_4, \mu C_{d2}^4 +\mu Z^4 +((\mu (n_0+n_1+n_2+n_3+4)4)E_4\)) is not log canonical at \(P_4\).
Applying Theorem 3.4 to the above gives \(\mu (2(d2)n_0n_1n_2n_3)+\mu (n_0+n_1+n_2+n_3+4)4>1\), which implies \(\mu > {5}/({2d})\). But \(\mu < {5}/({2d})\) and thus this contradiction proves the proposition.\(\square \)
This in turn proves that \(m_0\leqslant d2\) is not possible for the chosen value of \(\lambda \), hence completing the Proof of Theorem 1.8.
Notes
Acknowledgements
The author would like to thank Erik Paemurru for his valuable comments and suggestions.
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