By Eriko Hironaka

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 delicate version or intersections of embedded curves) is expounded to topological and combinatorial information regarding the bottom house and department locus. distinctive consciousness is given to examples within which the bottom area is the advanced projective airplane and the department locus is a configuration of traces.

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**Example text**

L-£ are L{ : y = mix -f 6,-, for i = 1 , . . , fc, where mi > m 2 > • • • > m-r and (*) 6i < b2 < • • • < £p By property P5, the ordering of the x-coordinates of points in S is also perserved. Clearly, if we follow with C2, Lk goes to the line at infinity. , Lj become Li :y= - 6 , x +m,-. ) Thus, by (*), the ordering of the slopes remains the same. Furthermore, if £ i , . . , x8 are the ordered x-coordinates for points in S under the coordinate system obtained after Step (1), the new x coordinates _ 1 __L X\ ' X2 ' _1 Xs ' so the ordering remains the same for points in 5 as well.

Suppose f{e\) and /(e2) lie on the line L = Lj and f{e\) and /(e2) are joined at the point p G 5. Locally near p, £ fl M2 looks like the following picture. 4 and assume, by a suitable change of coordinates if necessary, that Px(p) = 0 and PJJ"1([—1, l])OS = Define 7:[0,1]-F2 so that j(0) equals the point P~1(eir,e) D L and define r : [0,1] — P2 ERIKO HIRONAKA 46 so that T(6) equals (sin(jr0)i, 0) + P~ 1 (COS(TT^)) n L. Note that 7(0) = r(0), 7(1) = r ( l ) , the x coordinates of 7(0) and r($) are equal, 7(0) C L and r(0) C A forallO<0

3D'= J^ pecnD \<*Cnpiyclp-l{p)\. 8, we have Summing over all p E C C\ D gives the formula (**). are, for C H A P T E R III. HIRZEBRUCH COVERING SURFACES In this chapter we apply the previous results specifically to Hirzebruch surfaces and describe techniques that lead to an effective algorithm for computing the first Betti number 61 and lower bound for the Picard number p of Hirzebruch surfaces X associated to configurations of real lines. The algorithm is given in Chapter IV. We define Hirzebruch covering surfaces and give some properties, following [Hirz], in III.