Moduli spaces and braid monodromy types of bidouble covers of the quadric

Catanese, Fabrizio; Lönne, Michael; Wajnryb, Bronislaw

doi:10.2140/gt.2011.15.351

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Moduli spaces and braid monodromy types of bidouble covers of the quadric

Fabrizio Catanese, Michael Lönne and Bronislaw Wajnryb

Geometry & Topology 15 (2011) 351–396

DOI: 10.2140/gt.2011.15.351

Abstract

Bidouble covers $π : S \to Q : = ℙ^{1} \times ℙ^{1}$ of the quadric are parametrized by connected families depending on four positive integers $a, b, c, d$ . In the special case where $b = d$ we call them $a b c$ –surfaces.

Such a Galois covering $π$ admits a small perturbation yielding a general $4$ –tuple covering of $Q$ with branch curve $Δ$ , and a natural Lefschetz fibration obtained from a small perturbation of the composition $p_{1} \circ π$ .

We prove a more general result implying that the braid monodromy factorization corresponding to $Δ$ determines the three integers $a, b, c$ in the case of $a b c$ –surfaces. We introduce a new method in order to distinguish factorizations which are not stably equivalent.

This result is in sharp contrast with a previous result of the first and third author, showing that the mapping class group factorizations corresponding to the respective natural Lefschetz pencils are equivalent for $a b c$ –surfaces with the same values of $a + c, b$ . This result hints at the possibility that $a b c$ –surfaces with fixed values of $a + c, b$ , although diffeomorphic but not deformation equivalent, might be not canonically symplectomorphic.

Keywords

algebraic surface, moduli space, braid monodromy, equivalence of factorizations, symplectomorphism, bidouble cover, Lefschetz pencil

Mathematical Subject Classification 2000

Primary: 14J15

Secondary: 14J29, 14J80, 14D05, 53D05, 57R50

References

Publication

Received: 12 October 2009

Accepted: 8 November 2010

Published: 1 March 2011

Proposed: Ronald Fintushel

Seconded: Joan Birman, Ron Stern

Authors

Fabrizio Catanese
Mathematisches Institut Lehrstuhl Mathematik VIII Universität Bayreuth Universitätsstraße 30 D-95447 Bayreuth Germany
http://www.mathe8.uni-bayreuth.de/Personen/Catanese.html
Michael Lönne
Mathematisches Institut Lehrstuhl Mathematik VIII Universität Bayreuth Universitätsstraße 30 D-95447 Bayreuth Germany

Bronislaw Wajnryb
Department of Mathematics Rzeszów University of Technology ul W Pola 2 35-959 Rzeszów Poland

Volume 15, issue 1 (2011)