A Zariski Pair in Affine Complex Plane

We present a Zariski pair in affine complex plane consisting of two line arrangements, each of which has six lines. In the seminal paper [3], Zariski started the study of the fundamental groups of the complements of plane algebraic curves. Among other things, he constructed a pair of plane curves with the same degree and the same local singularities, but non-isomorphic fundamental groups, which is called in the literature a Zariski pair. The two curves in Zariski’s example are sextics. Several families of Zariski pairs in projective complex plane have been found. Recently, K.-M. Fan [1] has given a Zariski pair consisting of two arrangements of lines in projective complex plane CP . Each arrangement consists of seven real lines. Although there is some relationship between Zariski pairs in affine and projective planes, this relationship is not direct. Therefore, the study of Zariski pairs in affine complex plane is of independent interests. In this note, we present a Zariski pair consisting of six lines in affine complex plane C. We need the following special version of Oka-Sakamoto theorem. Theorem 1. (Oka-Sakamoto [2]) Let be given m + n lines in C: K1, . . . ,Km and L1, . . . , Ln such that K1 ∪ · · · ∪Km intersects L1 ∪ · · · ∪Ln in mn distinct points. Then π(C2\(K1∪· · ·∪Km∪L1∪· · ·∪Ln)) ∼= π(C 2\(K1∪· · ·∪Km))×π(C 2\(L1∪· · ·∪Ln)). Corollary 2. If L1, L2, . . . , Ln can be divided into m groups K11, . . . ,K1p1 ; . . . ; Km1, . . . ,Kmpm , (p1 + · · ·+ pm = n), such that (Ki1 ∪ · · · ∪Kipi) ∩ (Kj1 ∪ · · · ∪Kjpj ) consists of pipj distinct points, then π(C \ (L1 ∪ · · · ∪ Ln)) ∼= m ∏ i=1 π(C \ (Ki1 ∪ · · · ∪Kipi)). Corollary 3. Let L1 intersects L2 ∪ · · · ∪ Ln in n− 1 distinct points, then π(C \ (L1 ∪ · · · ∪ Ln)) ∼= π(C 2 \ (L2 ∪ · · · ∪ Ln))× Z. A Zariski pair. Let A be the arrangement in C consisting of three pairs of parallel lines with 12 normal crossings (see figure 1). Let B be an arrangement in C consisting of a triangle and three parallel lines which do not pass the vertices of the triangle, and are not parallel to any edge of the triangle (see figure 2). The following proposition says that these two arrangements form a Zariski pair in C. Received June 4, 2003; accepted for publication February 6, 2004. Institute of Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, 100080 P. R. China ([email protected]). The first author was supported by NSF undewr the grant 10071087. Department of Mathematics and Information Science, Mailbox 66, Faculty of Science, Beijing University of Chemical Technology, Beisanhuandonglu 15, Beijing 100029, P. R. China ([email protected]). The second author was supported by NSF under the grant 102710023, and SRF for ROCS, SEM. 473 474 J. YU AND G. JIANG