By Eriko Hironaka

This paintings experiences abelian branched coverings of soft complicated projective surfaces from the topological perspective. Geometric information regarding the coverings (such because the first Betti numbers of a gentle version or intersections of embedded curves) is expounded to topological and combinatorial information regarding the bottom area and department locus. specific consciousness is given to examples during which the bottom area is the complicated projective aircraft and the department locus is a configuration of strains.

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**Abelian Coverings of the Complex Projective Plane Branched Along Configurations of Real Lines**

This paintings reviews abelian branched coverings of tender complicated projective surfaces from the topological standpoint. Geometric information regarding the coverings (such because the first Betti numbers of a soft version or intersections of embedded curves) is said to topological and combinatorial information regarding the bottom area and department locus.

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**Example text**

S. Note that the endpoints of all paths defined above lie in M. — Q. As can be seen by the previous diagram, the fundamental group TTI(C — Q, qo) is generated by T i , . . , T,, where each Tj is defined by 7V7+)-1. , <7jk-i, where ERIKO HIRONAKA 38 each (T{ is the braid 1 t-1 /+i X /+2 k and has relations for \i — j \ > 2 and for i = 1 , . . ,s — 2. Recall that Fg0 equals C minus k ordered points lying on the real line. The braid <7; corresponds to the element of Mod(F^0) which can be represented by a homeomorphism which rotates a disk D, containing only the ith and i -f 1st point and centered between them, by 180 degrees and fixes all points outside of a disk D containing D.

4 Definition. Let / ' : T —• X be a lifting map for an intersection graph / : T —* Y for C. Let J be the set of pairs of edges of T labelled by the same curve C c C , meeting at a common vertex. Let j>:l-+G be a map so that for each (ei,e2) G I , there is a curve C" C p - 1 ( C ) such that V , (ei,e 2 )/ / (ei) and / , ( e 2 ) lie on C". We call tp the jAt/Unjr rfata for / ' : T - • Y. The similarity of the notation with the lifting data associated to a C lifting will be explained in the next lemma.

If q G M — Q, then the points t i , . . 5 Monodromy. Let go € R be a point so that qo > q for all q E Q. The monodromy of the fibration is the image of the natural map (*) *1(C-Q,flo)->Mod(Ffo) where Mod(Fqo) is the mapping class group, or group of isotopy classes of homeomorphisms of Fqo to itself which fix everything outside of a large disk in Fqo containing Tqo. ERIKO HIRONAKA 36 There is a canonical homomorphism (**) Bk - Mod(F f0 ), where Bk is the braid group on k strands [Mo]. Let £:xi(C-S,«o)->B* be the map E which takes a loop 7 : [0,1] —• C — S based at go to the braid obtained by following Ty^ as 0 ranges between 0 and 1.