# New PDF release: Abelian Coverings of the Complex Projective Plane Branched By Eriko Hironaka

ISBN-10: 082182564X

ISBN-13: 9780821825648

This paintings reviews abelian branched coverings of soft 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. certain awareness is given to examples within which the bottom house is the complicated projective aircraft and the department locus is a configuration of strains.

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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. 9 PROPOSITION. For each proper transform L G C of a line L in C, let V be the curve in p w l (L) corresponding to V under the birational map ? : X —• X. For each point p ETy let E'p be the curve in p~l(Ep) mapping to f'(p) under ?. Let \$ : J-+G be defined by V(q,L) = ^(cr(g),L) for all lines L in C and let \$ ( £ p ) be the identity element. Then ^ is lifting data for the liftings.

Jv > be a presentation for 7Ti(P2 — C). Let A be the matrix of Fox derivatives 2(dRi\ dfij The matrix A is called the Alexander matrix for the presentation. 5), p. 5 PROPOSITION. The Alexander matrix A is a presentation matrix for the first homology group Ui(Xu1F;Z) considered as a Z[G]-module, for F any fiber. D, Computing b\. Let Clk be the set of ^-tuples of nth roots of unity and, for each element w = (u>i,... ,o>p) in Q,-£, let rw :Z[G]-*Z[fi n ] be the Z-module homomorphism defined by Tu,(0,-)=Wt, where Qi = <^(/i»).

Hirz], p. 122) Hirzebruch coverings X are smooth. We give a proof in Remark III. 6 using the language developed in Chapter I. In the process we show how to find the generators of the stabilizer and inertia subgroups of the branch locus of p and p. 4 that, if : Hi(P 2 - £ ; Z ) -+ H X (P 2 - £ ; Z / n Z) is the defining map of the covering, to each line L in £ there is a canonically associated element HL € Hi(P 2 — £ ; Z ) which can be realized as a positively oriented meridianal loop around L.