By Eriko Hironaka

This paintings stories abelian branched coverings of tender 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. distinctive awareness is given to examples within which the bottom area is the advanced projective airplane and the department locus is a configuration of strains.

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Order the curves above C (using generators for the stabilizer subgroups). D. Make an intersection matrix for curves above C. E. Find the nullity of the intersection matrix. 3, the difference b\ — Null(J) equals b\(X). l INPUT. The format of the input is important for the later calculations. 1 Conditions on coordinates x,y. 4. 4. 50 THE COMPLEX PROJECTIVE PLANE 51 PI. Each LQ in C is given by an equation of the form y = max -f bQi where mQ and ba are real. P2. The projection Px sends the set of all intersections 5 on vCflC2 to distinct (necessarily real) points Q in C.

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.

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.

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