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On the topology of arrangements of a cubic and its inflectional tangents.

1. Introduction. In this article, we continue to study Zariski pairs for reducible plane curves based on the idea used in [3]. A pair ([B.sup.1], [B.sup.2]) of reduced plane curves in P2 is said to be a Zariski pair if (i) both [B.sup.1] and [B.sup.2] have the same combinatorics and (ii) ([P.sup.2], [B.sup.1]) is not homeomorphic to ([P.sup.2], [B.sup.2]) (see [2] for details about Zariski pairs). As we have seen in [2], the study of Zariski pairs, roughly speaking, consists of two steps:

(i) How to construct (or find) plane curves with the same combinatorics but having some different properties.

(ii) How to distinguish the topology of ([P.sup.2], [B.sup.1]) and ([P.sup.2], [B.sup.2]).

As for the second step, various tools such as fundamental groups, Alexander invariants, braid monodromies, existence/non-existence of Galois covers and so on have been used. In [3], the first and last authors considered another elementary method in order to study Zariski k-plets for arrangements of reduced plane curves and showed its effectiveness by giving some new examples. In this article, we study the topology of arrangements of a smooth cubic and its inflectional tangents along the same line.

1.1. Subarrangements. We here reformulate our idea in [3] more precisely. Let [B.sub.o] be a (possibly empty) reduced plane curve Bo. We define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to be the set of the reduced plane curves of the form [B.sub.o] + B, where B is a reduced curve with no common component with [B.sub.o].

Let B = [B.sub.1] + ... + [B.sub.r] denote the irreducible decomposition of B. For a subset I of the power set [2.sup.{1, ...,r}] of {1, ..., r}, which does not contain the empty set [empty set], we define the subset [[Sub.bar].sub.I] ([B.sub.o], B) of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] by:

[[Sub.bar].sub.I] ([B.sub.o], B) := {[B.sub.o] + [summation over (i [member of] I] Bi | I [member of] I}.

For I = [2.sup.{1, ..., r}] \ [empty set], we denote [Sub.bar]([B.sub.o], B) = [[Sub.bar].sub.I] ([B.sub.o], B).

Let A be a set and suppose that a map

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

with the following property is given: for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], if there exists a homeomorphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

We denote by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] the restriction of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to [Sub.bar]([B.sub.o], B). Note that if there exists a homeomorphism [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then we have the induced [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Remark 1.1. In [section] 2 we give four explicit examples for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] allowing to distinguish the k-Artal arrangements (see [section] 1.2 for the definition), using the Alexander polynomial, the existence of [D.sub.6]-covers, the splitting numbers and the linking set.

If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] have the same combinatorics, then any homeomorphism h : ([T.sup.1], [D.sub.o] + [D.sup.1]) [right arrow] ([T.sup.2], [D.sub.o] + [D.sup.2]) with h([D.sub.o]) = [D.sub.o] induces a map [h.sub.[sharp]] : [Sub.bar]([D.sub.o], [D.sup.1]) [right arrow] [Sub.bar]([D.sub.o], [D.sup.2]), where [T.sup.i] is a tubular neighborhood of [D.sub.o] + [D.sup.i] for i = 1,2. Let ([D.sub.o] + [D.sup.1], [D.sub.o] + [D.sup.2]) be a Zariski pair of curves in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that

* it is distinguished by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], i.e., any homeomorphism h : ([T.sup.1], [D.sub.o] + [D.sup.1]) [right arrow] ([T.sup.2], [D.sub.o] + [D.sup.2]) necessarily satisfies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and

* the combinatorial type of [D.sub.o] + [D.sup.1] and [D.sub.o] + [D.sup.2] is [C.bar].

Assuming the existence of such a Zariski pair for the combinatorial type [C.bar], we construct Zariski pairs with glued combinatorics. We first note that the following proposition is immediate:

Proposition 1.2. Choose [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] with the same combinatorial type. Let [[Sub.bar].sub.[C.bar]]([B.sub.o], [B.sup.j]) (j = 1,2) be the sets of subarrangements of [B.sub.o] + [B.sup.j] having the combinatorial type [C.bar] (j = 1,2), respectively. If

(i) any homeomorphism h : ([T.sup.1], [B.sub.o] + [B.sup.1] [right arrow] ([T.sup.2], [B.sub.o] + [B.sup.2]) necessarily satisfies h([B.sub.o]) = [B.sub.o], where [T.sup.i] is a tubular neighborhood of [B.sub.o] + [B.sup.i] for i = 1, 2, and

(ii) for some elements [a.sub.1] [member of] A,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

then ([B.sub.o] + [B.sup.1], [B.sub.o] + [B.sup.2]) is a Zariski pair.

Remark 1.3. If for all automorphism [sigma] of the combinatorics of [B.sub.o] + [B.sup.j], [sigma]([B.sub.o]) = [B.sub.o] then hypothesis (i) of Proposition 1.2 is always verified. In particular, it is the case if deg([B.sub.o]) [not equal to] deg([B.sub.i]), for i = 1, ..., r.

1.2. Artal arrangements. In this article, we apply Proposition 1.2 to distinguish Zariski pairs formed by Artal arrangements. These curves are defined as follows:

Let E be a smooth cubic, let [P.sub.i] (1 [less than or equal to] i [less than or equal to] 9) be its 9 inflection points and let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the tangent lines at [P.sub.i] (1 [less than or equal to] i [less than or equal to] 9), respectively.

Definition 1.4. Choose a subset I [subset] {1, ..., 9}. We call an arrangement C = E + [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] an Artal arrangement for I. In particular, if k = [sharp] (I), we call C a k-Artal arrangement.

The idea is to apply Proposition 1.2 to the case when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let [C.sup.1] and [C.sup.2] be two k-Artal arrangements. Note that if there exists a homeomorphism h : ([P.sup.2], [C.sup.1]) [right arrow] ([P.sup.2], [C.sup.2]), h(E) = E always holds. In [1], E. Artal gave an example of a Zariski pair for 3-Artal arrangements. Based on this example, we make use of our method to find other examples of Zariski pairs of k-Artal arrangements and obtain the following

Theorem 1.5. There exist Zariski pairs for k-Artal arrangements for k = 4, 5, 6.

Remark 1.6. Note that the case of k = 5 is considered in [4], where it is shown that there exists a Zariski pair for 5-Artal arrangements.

2. Some explicit examples for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. We here introduce four examples for [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The last two were recently considered by the second author and J.-B. Meilhan [4] and the third author [6], respectively.

2.1. [D.sub.2p]-covers. For terminologies and notation, we use those introduced in [2, [section] 3], freely.

Let [D.sub.2p] be the dihedral group of order 2p. Let [Cov.sub.b] ([P.sup.2], 2B, [D.sub.2p]) be the set of isomorphism classes of [D.sub.2p]-covers branched at 2B.

We now define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Note that satisfies the required condition described in the Introduction. Thus, we define the map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as the restriction of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [Sub.bar]([B.sub.o], B).

2.2. Alexander polynomials. For the

Alexander polynomials of reduced plane curves, see [2, [section] 2]. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the map assigning to a curve of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] its Alexander polynomial. We define the map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] {0,1} by: [degrees]

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

As previously, we define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as the restriction of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to Sub([B.sub.o], B).

2.3. Splitting numbers. Let [B.sub.o] + B be a reduced curves such that [B.sub.o] is smooth, and let m be the degree of B. Let [[pi].sub.B] : X [right arrow] [P.sup.2] be the unique cover branched over B, corresponding to the surjection of [[pi].sub.1] ([P.sup.2] \ B) [right arrow] Z/mZ sending all meridians of the [B.sub.i] to 1. The splitting number of [B.sub.o] for [[pi].sub.B], denoted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the number of irreducible components of the pull-back [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (see [6] for the general definition). By [6, Proposition 1.3], the application:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

verify the condition of Proposition 1.2, where [N.sup.*] is the set of integers more than 0. We can then define the map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as the restriction of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to [Sub.bar]([B.sub.o], B).

2.4. Linking set. Let [B.sub.o] be a non-empty curve, with smooth irreducible components. A cycle of [B.sub.o] is a [S.sup.1] embedded in [B.sub.o]. For [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we define the linking set of [B.sub.o], denoted by [lks.sub.B] ([B.sub.o]), as the set of classes in [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] of the cycles of [B.sub.o] which do not intersect B, where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is the subgroup of [H.sub.1] ([P.sup.2] \ B) generated by the meridians in [B.sub.o] around the points of [B.sub.o] [intersection] B. This definition is weaker than [4, Definition 3.9]. By [4, Theorem 3.13], the map defined by:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

verify the condition of Proposition 1.2. We can thus define the map [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] as the restriction of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to [Sub.bar]([B.sub.o], B). [degrees]

3. The geometry of inflection points of a smooth cubic. Let E [subset] [P.sup.2] be a smooth cubic curve and let O [member of] E be an inflection point of E. In this section we consider the elliptic curve (E, O). The following facts are well-known:

(a) The set of inflection points of E can be identified with [(Z/3Z).sup.[direct sum]2] [subset] E, the subgroup of three torsion points of E. This identification comes from the cubic group law with neutral element, a flex.

(b) Let P, Q, R be distinct inflection points of E. Then P, Q, R are collinear if and only if P + Q + R = O [member of] [(Z/3Z).sup.[direct sum]2].

From the above facts we can study the geometry of inflection points and the following proposition follows:

Proposition 3.1. Let E be a cubic curve and {[P.sub.1], ..., [P.sub.k]} [member of] E be a set of distinct inflection points of E. Let n be the number of triples [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] such that they are collinear. Then the possible values of n for k = 3, ..., 9 are as in the following table:
Table

k      3       4      5      6     7    8    9

n    0, 1    0, 1    1, 2   2, 3   5    8    12


4. Proof of the main theorem.

4.1. The case of 3-Artal arrangements.

Using the four invariants introduced in [section] 2, we can prove the original result of E. Artal. Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Theorem 4.1. For a 3-Artal arrangement [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we have:

(a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(c) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(d) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof. (a) This is the result of the last author ([7]).

(b) This is the result of E. Artal ([1]).

(c) By [6, Theorem 2.7], we obtain [[PHI].sup.split.sub.E] (C) = 3 if the three tangent points are collinear, and [[PHI].sup.split.sub.E] (C) = 1 otherwise.

(d) Using the same arguments as in [5], we can prove that, in the case of 3-Artal arrangements, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Using the previous point we obtain the result.

Remark 4.2. It is also possible to consider [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (E). But in this case, we have no method to compute it in the general case. But, if E is the cubic defined by [x.sup.3] - x[z.sup.2] - [y.sup.2] z = 0, the computation done in [4] implies the result.

Corollary 4.3. Choose {[i.sub.1], [i.sub.2], [i.sub.3], [i.sub.4]} [subset] {1, ..., 9} such that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are collinear, while [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are not collinear. Put [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Then ([C.sub.1], [C.sub.2]) is a Zariski pair.

4.2. The other cases. Choose a subset J of {1, ..., 9} such that 4 [less than or equal to] [sharp] J [less than or equal to] 6 and let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

be a k-Artal arrangement. To distinguish these arrangements in a geometric way (as the collinearity in the case of 3-Artal arrangements), let us introduce the type of a k-Artal arrangement.

Definition 4.4. For k = 4, 5, 6, we say an arrangement of the form [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] to be of Type I if the number n of collinear triples in {[P.sub.1], ..., [P.sub.k]} is n = k - 3, while we say C to be of Type II if the number n of collinear triples in {[P.sub.1], ..., [P.sub.k]} is n = k - 4.

Theorem 4.5. Let [C.sub.1] be an arrangement of Type I and [C.sub.2] be an arrangement of Type II. Then ([P.sup.2], [C.sub.1]) and ([P.sup.2], [C.sub.2]) are not homeomorphic as pairs.

Furthermore if [C.sub.1] and [C.sub.2] have the same combinatorics, [C.sub.1], [C.sub.2] form a Zariski pair.

Proof. Let C be a k-Artal arrangement (k = 4, 5, 6). We denote by [Sub.bar][(E, [[laplace].sub.J]).sub.3] the set of 3-Artal arrangements contained in [Sub.bar](E, [[laplace].sub.J]). Let [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] be the restrictions of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], respectively. Then by Theorem 4.1, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If a homeomorhism h : ([P.sup.2], [C.sub.1]) [right arrow] ([P.sup.2], [C.sub.2]) exists, it follows that [h.sub.[sharp]] ([Sub.bar][([C.sub.1]).sub.3]) = [Sub.bar][([C.sub.2]).sub.3]. This contradicts the above values. Hence our statements follow.

Remark 4.6. Here are two remarks:

(i) If the j-invariant of the cubic E is not equal to 0, no three inflectional tangent lines are concurrent. Hence [C.sub.1] and [C.sub.2] have only double points as their singularities and their combinatorics are the same.

(ii) We note that for k = 1,2, 7, 8, 9 it can be proved that there do not exist Zariski pairs consisting of k-Artal arrangements with only double points as their singularities.

doi: 10.3792/pjaa.93.50

Acknowledgement. The second author is supported by a JSPS post-doctoral grant.

References

[1] E. Artal Bartolo, Sur les couples de Zariski, J. Algebraic Geom. 3 (1994), no. 2, 223-247.

[2] E. Artal Bartolo, J. I. Cogolludo and H.

Tokunaga, A survey on Zariski pairs, in Algebraic geometry in East Asia Hanoi 2005, 1-100, Adv. Stud. Pure Math., 50, Math. Soc. Japan, Tokyo, 2008.

[3] S. Bannai and H. Tokunaga, Geometry of bisections of elliptic surfaces and Zariski N-plets for conic arrangements, Geom. Dedicata 178 (2015), 219-237.

[4] B. Guerville-Balle and J.-B. Meilhan, A linking invariant for algebraic curves, arXiv:1602.04916.

[5] B. Guerville-Balle and T. Shirane, Non-homotopicity of the linking set of algebraic plane curves, arXiv:1607.04951.

[6] T. Shirane, A note on splitting numbers for Galois covers and [[pi].sub.1]-equivalent Zariski k-plets, Proc. Amer. Math. Soc. 145 (2017), no. 3, 1009-1017.

[7] H. Tokunaga, A remark on E. Artal's paper, Kodai Math. J. 19 (1996), no. 2, 207-217.

By Shinzo BANNAI, *) Benoit GuERVILLE-BALLE, **) Taketo SHIRANE ***) and Hiro-o TOKuNAGA ****)

(Communicated by Heisuke Hironaka, M.J.A., May 12, 2017)

2010 Mathematics Subject Classification. Primary 14H50, 14H45, 14F45, 51H30.

*) Department of Natural Sciences, National Institute of Technology, Ibaraki College, 866 Nakane, Hitachinaka, Ibaraki 312-8508, Japan.

**) Department of Mathematics, Tokyo Gakugei University, 4-1-1 Nukuikita-machi, Koganei, Tokyo 184-8501, Japan.

***) National Institute of Technology, Ube College, 2-14-1 Tokiwadai, Ube, Yamaguchi 755-8555, Japan.

****) Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1 Minami-Osawa, Hachioji, Tokyo 192-0397, Japan.
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Author:Bannai, Shinzo; Guerville-Balle, Benoit; Shirane, Taketo; Tokunaga, Hiro-o
Publication:Japan Academy Proceedings Series A: Mathematical Sciences
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Date:Jun 1, 2017
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