# Abelian Coverings of the Complex Projective Plane Branched by Eriko Hironaka By Eriko Hironaka

This paintings experiences abelian branched coverings of soft 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 expounded to topological and combinatorial information regarding the bottom area and department locus. precise cognizance is given to examples within which the bottom area is the complicated projective aircraft and the department locus is a configuration of traces.

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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. f3(C') = j^\aHcCM3Hc\C2, and we have proven (*).

HI. 10 Examples. The corresponding braid for d = 2 is the generator element cr*. The corresponding braid for d = 3 is i

E r L _ i be the edges in T labelled L so that Px{^i) is the interval between Px(pi) and Px(pi+i)- Let { be any element ofG mapping to (rj~l) in G/IL as defined above for e,-_i and e,-. For each pj G 5 f l L , let f Define °0 V :J ifj = 1 otherwise. ->G so that for each L and p G S C\ L V»(p,L) = Vy Then there exists a lifting V of L for each L C C so that ty is lifting data for V'. Proof. Define L' to be the lift of L containing the edge f'(e\). 6. • The rest follows We are now ready to find lifting data for a C lifting in p : X —• Y.