LECTURES ON ZARISKI VAN-KAMPEN THEOREM 1. Introduction ...

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32 ICHIRO SHIMADA ã ˜b ˜d ˜c Figure 7.9. The new lassos after the monodromy action of β Combining all of these, we see that 〈 π 1 (P 2 \ C ∨ ,a)= a, b, c, d We can reduce the relations to ∣ 〉 dcba =1, cdc = dcd, ada = dad, . a = c, db = ad, dc = bd Putting α = cd, β = cdc, wehave π 1 (P 2 \ C ∨ ,a)=〈 c, d | cdc = dcd, c 2 d 2 =1〉. π 1 (P 2 \ C ∨ ,a)=〈 α, β | α 3 = β 2 =(βα) 2 〉. This group is the binary 3-dihedral group. Thus we obtain the following theorem due to Zariski: Theorem 7.2.1. The fundamental group of the complement to a three cuspidal quartic curve is isomorphic to the binary 3-dihedral group. Remark 7.2.2. Let us see the structure of the group G := 〈 a, b | aba = bab, a 2 b 2 =1〉. From a 2 b 2 = 1, we have ab 2 a = ba 2 b = 1. Hence (aba) 2 = a(ba 2 b)a = a 2 , (bab) 2 = b(ab 2 a)b = b 2 . From aba = bab, we have a 2 = b 2 . We put c := a 2 = b 2 . Then c is of order 2 and in the center of G. Since the group G/〈c〉 = 〈 a, b | aba = bab, a 2 = b 2 =1〉 is isomorphic t S 3 , we see that G is a central extension of S 3 by Z/(2). In particular, G is a non-abelian finite group of order 12.
LECTURES ON ZARISKI VAN-KAMPEN THEOREM 33 7.3. Zariski conjecture and Zariski pairs. If a reduced plane curve C ⊂ P 2 consists of irreducible components of degree d 1 ,...,d k , then H 1 (P 2 \ C, Z) is isomorphic to Z k /(d 1 ,...,d k )Z. Suppose that π 1 (P 2 \ C) is abelian. Then it is isomorphic to H 1 (P 2 \ C, Z), and hence it is determined by the degrees of the irreducible components. When is π 1 (P 2 \ C) abelian We have the following theorem, which had been known as Zariski conjecture since the publication of the paper [11], and was proved by Fulton and Deligne around 1970 in [4] and [3]. Theorem 7.3.1. If C is nodal, then π 1 (P 2 \ C) is abelian. This theorem was proved, not by Zariski-van kampen’s theorem, but by Fulton- Hansen’s connectedness theorem [6]. See [7] for the proof. Several improvements of this theorem are known. One of them is the following theorem, due to Nori [9]. Theorem 7.3.2. Let C be an irreducible curve of degree d with n nodes and k cusps. If 2n +6k
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LECTURES ON ZARISKI VAN-KAMPEN THEOREM 1. Introduction ...

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