A survey on Zariski pairs
A survey on Zariski pairs
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DOI：
10.2969/aspm/05010001
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
Enrique Artal Bartolo;J. Agustín;H. Tokunaga
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文献类型：
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作者：
Enrique Artal Bartolo;J. Agustín;H. Tokunaga
正如Zebraki在[130]的导言中指出的那样，这个问题首先由Enriques考虑，并且问题被简化为找到给定曲线的补的基本群（补这个词被理解并且经常被省略）。Zebriki认为一些明确的情况下，并证明了重要成果。这里我们详细介绍一些最相关的：（Z1）如果两条曲线位于一个连通的等奇异曲线族中，那么它们有同构的基本群。(Z2)若连续族{Ct}t∈[0，1]对t ∈（0，1]是等奇异的，且C 0是约化的，则存在自然满射π1（P\C0，p0）π1（P\Ct，pt），其中基点pt（t ∈ [0，1]）连续依赖于t. (Z3)一条n阶不可约曲线的基本群，如果只具有普通的二重点，则它是n阶循环的（[130，定理7]），见注1。(Z4)考虑从P中的一般三次曲面到P的投影，中心在曲面外部的一般点。它的分支轨迹是一个有六个尖点的六次C6，其基本群同构于Z/2 Z <$Z/3 Z。(Z5)他指出，（Z4）中描述的任何六次曲线的六个尖点都满足位于二次曲线上的额外条件-而不会减少其族的维数。此外，如果C6是一个有六个尖点的六次曲线，并且它的基本群在三字母对称群上有一个表示，则C6是一个三次曲面的分支曲线，并且它的六个尖点位于一个二次曲线上。特别地，如果存在一个六次C′ 6，其六个尖点不在二次曲线上，则π1（P \ C6，po）6 <$= π1（P \ C′ 6，po）。
As Zariski pointed out in the Introduction of [130], this question was first considered by Enriques and the problem is reduced to finding the fundamental group of the complement of the given curve (the word complement is understood and often omitted for short). Zariski considered some explicit cases and proved important results. Here we detail some of the most relevant: (Z1) If two curves lie in a connected family of equisingular curves, then they have isomorphic fundamental groups. (Z2) If a continuous family {Ct}t∈[0,1] is equisingular for t ∈ (0, 1] and C0 is reduced, then there is a natural epimorphism π1(P\C0, p0) π1(P\Ct, pt), where the base point pt (t ∈ [0, 1]) depends on t continuously. (Z3) The fundamental group of an irreducible curve of order n, possessing ordinary double points only, is cyclic of order n ([130, Theorem 7]), see Remark 1. (Z4) Consider the projection from the general cubic surface in P onto P, centered at a general point outside the surface. Its branch locus is a sextic C6 with six cusps whose fundamental group is isomorphic to Z/2Z ∗ Z/3Z. (Z5) He noted that the six cusps of any sextic described in (Z4) satisfy the extra condition of lying on a conic –without decreasing the dimension of their family. Moreover, if C6 is a sextic with six cusps and its fundamental group has a representation onto the symmetric group of three letters, then C6 is the branch curve of a cubic surface and its six cusps lie on a conic. In particular if a sextic C′ 6 with six cusps not on a conic exists, then π1(P \ C6, po) 6∼= π1(P \ C′ 6, po).