By Eriko Hironaka

ISBN-10: 082182564X

ISBN-13: 9780821825648

This paintings reports abelian branched coverings of delicate advanced projective surfaces from the topological point of view. Geometric information regarding the coverings (such because the first Betti numbers of a gentle version or intersections of embedded curves) is said to topological and combinatorial information regarding the bottom area and department locus. targeted awareness is given to examples during which the bottom house is the complicated projective airplane and the department locus is a configuration of strains.

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**Example text**

Given an intersection graph / : T —* Y for C, a lifting map for / in X is a continuous map / ' : T -> X so that P(/'(T)) = /(T) for all 7 G I \ Note that given one lifting / ' there are others given by c o / ' where

Proof. We have a composition of coverings X - ^ X 'I F2, where p(x) = ~p(pj(x)) for all x £ X and lj is the Galois group for pj. Since ix,. is the inertia subgroup for Lj in the composition covering p, p is one to one over p~~1(Lj). Therefore, if V , ( C 1J C 2) is a n v element of G/Jj so that (1) ip(e\}e2)pj(f(ei)) and Pj(/'(e2)) lie one the same curve in p - 1 ( L j ) e (2) ^( i> ^2) is the image of ip(e\, e2) in G, then V , ( e i> e 2)/ / (ei) and /'(e2) lie on the same curve in p _ 1 (Lj).

9, there is a choice of lifting C" for each curve C in C in the covering p:X-+W2 so that (p,C)t->bipJc is lifting data for the C. ERIKO HIRONAKA 58 C. O r d e r i n g curves above C. To find the intersection matrix for the curves in p w l ( £ ) explicitly, we need to be able to order the curves in p~l(C) and find their intersection numbers. Recall that the curves in p~l(C) for any curve C in C are in one to one correspondence with cosets of the stabilizer subgroup He associated to C. Thus our goal now is to find the stabilizer subgroups explicitly.