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Bidouble covers $\pi : S \mapsto Q$ of the quadric Q are parametrized byconnected families depending on four positive integers a,b,c,d. In the specialcase where b=d we call them abc-surfaces.  Such a Galois covering $\pi$ admits a small perturbation yielding a general4-tuple covering of Q with branch curve $\De$, and a natural Lefschetzfibration obtained from a small perturbation of the composition of $ \pi$ withthe first projection.  We prove a more general result implying that the braid monodromyfactorization corresponding to $\De$ determines the three integers a,b,c in thecase of abc-surfaces. We introduce a new method in order to distinguishfactorizations which are not stably equivalent. This result is in sharpcontrast with a previous result of the first and third author, showing that themapping class group factorizations corresponding to the respective naturalLefschetz 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 ofa+c, b, although diffeomorphic but not deformation equivalent, might be notcanonically symplectomorphic.

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