By Eriko Hironaka

This paintings stories abelian branched coverings of gentle advanced projective surfaces from the topological standpoint. Geometric information regarding the coverings (such because the first Betti numbers of a tender version or intersections of embedded curves) is said to topological and combinatorial information regarding the bottom area and department locus. exact realization is given to examples within which the bottom area is the complicated projective airplane and the department locus is a configuration of strains.

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20 - 21). Each component of p*C counts with multiplicity \Ic\. 2, so we have |G|C2 = |/c|2 £ C'\ 28 ERIKO HIRONAKA The number of irreducible components in p~l{C) is the index oi He in G. Since the covering is Galois, all the components have the same self intersection. Therefore, \G\C* = for a given C" C p~l(C). C>\ Multiplying both sides of this equation by \Hc\ \IcV\G\ finishes the proof. • If a and /? (C") is nonempty only when they are the same curve. This only happens when aHc equals fiHc, or equivalently, when the intersection aHc H PHc is nonempty.

Our aim now is to show that given a lifting map / ' : T —• X for an intersection graph / : T —+ Y, we can find lifting data by a local study. 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.

2, Theorem 1 (see also [Ho], §2). 6 THEOREM. The first Betti number ofXu is given by w€Cln This completes S T E P 1 of the algorithm. 3 STEP 2 : Intersection Matrix for Curves Above Branch Locus. Summary. , a way to choose curves C" in the covering, one above each curve C in C, together with the information of which group action makes two curves meet above a specified point. , so that for some choice of lifting of the curves in C in the covering p : X —• P 2 , \P gives lifting data. 1 to find the intersection matrix / of the curves in p~~l(C).

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