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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

Bidouble covers π: S Q := 1 × 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 abc–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 p1 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 abc–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 abc–surfaces with the same values of a + c,b. This result hints at the possibility that abc–surfaces with fixed values of a + c,b, although diffeomorphic but not deformation equivalent, might be not canonically symplectomorphic.

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
Received: 12 October 2009
Accepted: 8 November 2010
Published: 1 March 2011
Proposed: Ronald Fintushel
Seconded: Joan Birman, Ron Stern
Fabrizio Catanese
Mathematisches Institut
Lehrstuhl Mathematik VIII
Universität Bayreuth
Universitätsstraße 30
D-95447 Bayreuth
Michael Lönne
Mathematisches Institut
Lehrstuhl Mathematik VIII
Universität Bayreuth
Universitätsstraße 30
D-95447 Bayreuth
Bronislaw Wajnryb
Department of Mathematics
Rzeszów University of Technology
ul W Pola 2
35-959 Rzeszów