By Eriko Hironaka

This paintings stories abelian branched coverings of tender advanced projective surfaces from the topological perspective. Geometric information regarding the coverings (such because the first Betti numbers of a gentle version or intersections of embedded curves) is expounded to topological and combinatorial information regarding the bottom house and department locus. unique cognizance is given to examples during which the bottom area is the advanced projective aircraft and the department locus is a configuration of traces.

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P o i n t / c u r ve incidence matrix for £ . We begin by ordering the curves in C and the points S of intersection on £. , Lk of lines C and exceptional curves Eq for points q £ T. , ps through which more than two lines pass. Order the curves in £ as follows: L i , . . , Eqt. For each point pr E S with only two lines Ljx and Lj2 passing through p r , set dr = 1. There is a single corresponding point pr in Ljx H Lj7. For each point pr G 5 with pr = qu for some qu in T, there are dr distinct points p r > i , .

The projection Px sends the set of all intersections 5 on vCflC2 to distinct (necessarily real) points Q in C. PS. All points in S lie on the affine plane. P4- All slopes ma are nonzero. Add two more conditions. P5. For some jo, Pj € £* for all j > jo, and rotating the affine plane so that Lk becomes vertical doesn't change the ordering of the x-coordinates of points in 5 — £*. #)• P6. P*(p) > 0 for all peS. By shifting x by a constant xo greater than |Px(p*)| we can make sure property P6 holds without changing the previous conditions.

S ; j = 1 , . . ,k be the entries of M. Define the shift matrix Sh(£) with entries 6,-j inductively on i as follows. (1) Row 1: bltj = 0 for all j = 1 , . . , Jb. (2) Rowi: bi-ij if a$J- = 0 or j = i2 or j = i? i = * a n d •? 8, there is a choice of lifting Lj for each Lj in C so that tf : J ^ G (pi,Lj) *-+bitj are lifting data. Now we are ready to find the shift matrix for C. 3 Definition. For r = 1 , . . , s, let Sh r be the matrix defined as follows (1) If row r of M has only two columns j \ and J2 with entries equal to 1, then let Sh r be the 1 x (k + t) matrix with entries brjl in the j \ column, brtj2 in the J2 column and zeros elsewhere.