By Eriko Hironaka

This paintings experiences abelian branched coverings of soft advanced projective surfaces from the topological perspective. 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 house and department locus. detailed realization is given to examples within which the bottom area is the complicated projective aircraft and the department locus is a configuration of strains.

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**Example text**

F d — 1. Let R be the first (global) index so that the line corresponding to IR has positive slope and the line corresponding to IR+\ has negative slope. If R is between £ and £ + d — 1 then the center of rotation of the disk occurs somewhere between R and R+l. Thus the fibers fibers Fp(7+(^)) and Fp^-(0\\ vary as in the following pictures. ) 0=1/3 0 = 2/3 If R is not between £ and £+d— 1 then the center of rotation is either somewhere above £ (if R < £) or somewhere below £+ d— 1 (if R > £+ d— 1).

K, 40 ERIKO HIRONAKA where cr = (6 r — 6/) + (mr — mi)qj. For some d > 2 c r = 0, for all r = £ , . . , £ + c f - 1. On F^+^y we have T 7 +^^ = {mie T ^ + 6i + miqj,... j}. Similarly, on F - , ^ , we have T 7+(*) = { ~ m i e ^ + 6i + m i g ; , . . i—rnke*1 +bk + rrikqj}. Thus, the element of Mod(Fqo) corresponding to j + and j rotates a disk containing */,.. ,^+

1. IV. 2 Definition. Define E i , . . , E, in Bk as follows. (1) Look at the first row of M. 11. Let Ei equal E*^. (2) Given the previous E r , let o7 be the element of the symmetric group on k elements in the image of E r under the natural map Bk —•Syrn^-. Define Er = E 7 ( E M ) 2 E r \ where £ equals