Download Abelian Coverings of the Complex Projective Plane Branched by Eriko Hironaka PDF

By Eriko Hironaka

This paintings experiences abelian branched coverings of delicate complicated 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 said to topological and combinatorial information regarding the bottom area and department locus. designated recognition is given to examples during which the bottom area is the advanced projective airplane and the department locus is a configuration of traces.

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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. 3. If £ is any finite union ofk lines in P 2 , then Hi(P 2 - £ ; Z ) is generated by m for all L C £ and has the only relation LEMMA LCC Proof.

For configurations of real lines, the monodromy is easier to describe explicitly than in the general situation. 3. To find lifting data for the branch locus, we also study the local topology of real line configurations in F 2 . 2) for the line configuration. 5 shows how to convert this to lifting data. l Hirzebruch covering surfaces. The covering surfaces that we will deal with throughout the rest of this paper were defined in [Hirz]. Here is an alternative definition using the language developed in section II.

The image of r is contained in ft([o, 1] x r ) , THE COMPLEX PROJECTIVE PLANE 47 so any lift of r with initial point in pj(f'(ei)) has endpoint on />j(/'(e 2 )). On the other hand, since 7 doesn't pass through any points in 5, the image of 7 is contained in Lj, so any lift of 7 " 1 with initial point on Pj{f'{e2)) has endpoint on the same curve in p~l(Lj) as the one containing pj(f'(e2)). Therefore, the action of <£(T7~ 1 ) takes points on pj(f'(ei)) curve in p~ 1 (L ; ) containing Pj(f'(e2))> D to points on the We now have left to find <£(r7~1).

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