By Eriko Hironaka
This paintings reviews abelian branched coverings of soft advanced projective surfaces from the topological perspective. Geometric information regarding the coverings (such because the first Betti numbers of a soft version or intersections of embedded curves) is expounded to topological and combinatorial information regarding the bottom house and department locus. distinct cognizance is given to examples during which the bottom area is the complicated projective airplane and the department locus is a configuration of strains.
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Additional info for Abelian Coverings of the Complex Projective Plane Branched Along Configurations of Real Lines
B . P r e s e n t a t i o n of TTI(F - £ ) . _ We now use M to find a presentation for 7Ti(C2 — C). Let E(Fj), j = 1 , . . 13, let M be the homomorphism 2 M : Bk — Aut FT ERIKO HIRONAKA 54 where Fk is the free group on generators / i i , . . 1, and let for i = 1 , . . , k and j = 1 , . . ,jo — 1. 2, < j i i , . . ,i, > is a presentation for the fundamental group of P 2 — £. To compute Rij explicitly we use the following definition. 1. IV. 2 Definition. Define E i , . . , E, in Bk as follows.
2) If T is empty and k > 2, let P 2 equal P 2 and let a be the identity map. (3) If T is nonempty, let a : P 2 -* P 2 be the blowup of P 2 at the points in T. 7) of p : X —» P 2 . Then we call X the Hirzebruch covering associated to £ and n. One particularly useful property of Hirzebruch coverings is the following. 2 LEMMA. ([Hirz], p. 122) Hirzebruch coverings X are smooth. We give a proof in Remark III. 6 using the language developed in Chapter I. In the process we show how to find the generators of the stabilizer and inertia subgroups of the branch locus of p and p.
If p lies on only one curve C in C U 5 , let E'p be any curve in X mapping to p' which intersects $(p, C)C'. Let S be the set of intersections on C and define $: J-+G so that $(tf,C) equals $((7(2), C) for any q G S C\ C and W(q,Ep) equals the identity for all q G S 0 Ep. 4 PROPOSITION. 25 The map ¥ is hfting data for the C lifting. Proof. Take any q G S. If