Sections of elliptic surfaces and Zariski pairs for conic-line arrangements via dihedral covers

In this article, we make use of geometry of sections of elliptic surfaces and elementary arithmetic on the Mordell-Weil group in order to study existence problem of dihedral covers with given reduced curves as the branch loci. As an application, we give some examples of Zariski pairs (B1, B2) for “conic-line arrangements” satisfying the following conditions: ( i ) deg B1 = deg B2 = 7. ( ii ) Irreducible components of Bi (i = 1, 2) are lines and conics. ( iii ) Singularities of Bi (i = 1, 2) are nodes, tacnodes and ordinary triple points. Introduction. Let φ : S → P be a relatively minimal elliptic surface over P with a distinguished section O. Let MW(S) be the set of sections of S. It is well-known that one can define a structure of an abelian group on MW(S) with identity element O and that MW(S) is called the Mordell-Weil group of φ : S → P. We denote the group law by +̇ and the multiplication-by-m map (m ∈ Z) on MW(S) by [m]s for s ∈ MW(S). Also we identify a section with its image on S. Take s1, . . . , sk ∈ MW(S). Then ∑ i[ai]si gives another element of MW(S) and its image on S gives rise to a new curve on S. In this article, we consider p-divisibility (p: odd prime) of ∑ i[ai]si in MW(S) and a reduced divisor on S given by the union of [ai]si (i = 1, . . . , k) in order to study dihedral covers of the Hirzebruch surface Σd of degree d (d : even) or its blowing-ups Σ̂d. As an application, we give examples of Zariski pairs of degree 7 for conic-line arrangements. This can be considered as a continuation of the author’s previous articles ([22], [23], [24], [25]). Before we go on to explain our results in detail, let us first recall the definition of a Zariski pair. Definition 1. A pair (B1, B2) of reduced plane curves Bi (i = 1, 2) of degree n in P = P(C) (the base field of this article is always the field of complex numbers C) is called a Zariski pair of degree n if it satisfies the following condition: ( i ) Bi (i = 1, 2) are curves of degree n such that the combinatorial type (see Definition 2 below) of B1 is the same as that of B2. ( ii ) (P, B1) is not homeomorphic to (P, B2). 2010 Mathematics Subject Classification. Primary 14E20; Secondary 14J27.

[1]  H. Tokunaga.  Some sections on rational elliptic surfaces and certain special conic-quartic configurations , 2012 .

[2]  M. Friedman,et al.  On the structure of conjugation-free fundamental groups of conic-line arrangements , 2011, 1111.5291.

[3]  Jin-Gen Yang,et al.  Discrminantal groups and Zariski pairs of sextic curves , 2009, 0903.2058.

[4]  M. Teicher,et al.  On the fundamental group of the complement of two tangent conics and an arbitrary number of tangent lines , 2006, math/0612346.

[5]  H. Tokunaga.  TWO-DIMENSIONAL VERSAL G-COVERS AND CREMONA EMBEDDINGS OF FINITE GROUPS , 2006 .

[6]  M. Oka.  Zariski Pairs on sextics II , 2005, math/0507052.

[7]  M. Oka.  Zariski pairs on sextics I , 2005, math/0507051.

[8]  J. I. Cogolludo,et al.  Invariants of Combinatorial Line Arrangements and Rybnikov's Example , 2004, math/0403543.

[9]  Hiroyasu Tsuchihashi.  GALOIS COVERINGS OF PROJECTIVE VARIETIES FOR DIHEDRAL AND SYMMETRIC GROUPS , 2003 .

[10]  Enrique Artal Bartolo,et al.  Topology and combinatorics of real line arrangements , 2003, Compositio Mathematica.

[11]  M. Teicher,et al.  Fundamental groups of tangent conic-line arrangements with singularities up to order 6 , 2003, math/0305418.

[12]  M. Teicher,et al.  Fundamental groups of some quadric-line arrangements , 2002, math/0208248.

[13]  H. Tokunaga.  Some examples of Zariski pairs arising from certain elliptic K3 surfaces, II: Degtyarev's conjecture , 1999 .

[14]  Enrique Artal Bartolo,et al.  Zariski pairs, fundamental groups and Alexander polynomials. , 1998 .

[15]  G. Rybnikov.  On the fundamental group of the complement of a complex hyperplane arrangement , 1998, math/9805056.

[16]  H. Tokunaga.  Some examples of Zariski pairs arising from certain elliptic K3 surfaces , 1998 .

[17]  H. Tokunaga.  Dihedral coverings branched along maximizing sextics , 1997 .

[18]  H. Tokunaga.  On Dihedral Galois Coverings , 1994, Canadian Journal of Mathematics.

[19]  T. Shioda,et al.  The Mordell-Weil Lattice of a Rational Elliptic Surface , 1991 .

[20]  T. Shioda.  Mordell–Weil Lattices , 1990, Mordell–Weil Lattices.

[21]  U. Persson.  Configurations of Kodaira fibers on rational elliptic surfaces , 1990 .

[22]  R. Miranda,et al.  On extremal rational elliptic surfaces , 1986 .

[23]  R. Miranda.  The moduli of Weierstrass fibrations over ℙ1 , 1981 .

[24]  E. Horikawa.  On deformations of quintic surfaces , 1975 .

[25]  K. Kodaira.  On compact analytic surfaces II , 1963 .

[26]  O. Zariski,et al.  ON THE PURITY OF THE BRANCH LOCUS OF ALGEBRAIC FUNCTIONS. , 1958, Proceedings of the National Academy of Sciences of the United States of America.

[27]  O. Zariski.  The Topological Discriminant Group of a Riemann Surface of Genus p , 1937 .

[28]  Oscar Zariski,et al.  On the Problem of Existence of Algebraic Functions of Two Variables Possessing a Given Branch Curve , 1929 .

[29]  Enrique Artal Bartolo,et al.  A survey on Zariski pairs , 2006 .

[30]  M. Namba,et al.  On the Fundamental Groups of Galois Covering Spaces of the Projective Plane , 2004 .

[31]  H. Tokunaga.  Dihedral covers and an elementary arithmetic on elliptic surfaces , 2004 .

[32]  R. Miranda.  The basic theory of elliptic surfaces : notes of lectures , 1989 .

[33]  堀川 穎二.  On deformation of quintic surfaces , 1974 .