By Eriko Hironaka

This paintings reports 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 soft version or intersections of embedded curves) is said to topological and combinatorial information regarding the bottom house and department locus. particular consciousness is given to examples during which the bottom house is the complicated projective aircraft and the department locus is a configuration of traces.

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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.

2 LEMMA. For any curve C in C the curves in p~l(C) are disjoint. Proof. Take any pin C. 13. If there is such a curve D, then Ip is generated by the elements of Ic and ID, otherwise just the elements of Ic- In either case, Ip is contained in HeNow, suppose there are two curves C\ and Ci in p~l(C) intersecting at a point q in p"1(p). 1). Let Vq be the connected component of p~l{U) containing q. We will show C\ and Ci must be equal. For any point p' G CC\ [/, let (ji £ C\ and 02 € Ci be points lying in p~x(p') D Vq.

Thus, for all real points q G C there is a homeomorphism of any fiber Fq to Fqo, given by the local ordering, which is unique up to isotopy. Therefore, any path 7 in C — Q with real endpoints defines an element in Mod(Fqo). Explicitly, given any g G l - Q there is a unique isotopy class of maps [