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

Bidouble covers pi: S -> Q:= P-1 x 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 abc-surfaces. Such a Galois covering pi admits a small perturbation yielding a general 4-tuple covering of Q with branch curve Delta, and a natural Lefschetz fibration obtained from a small perturbation of the composition p(1) omicron pi. We prove a more general result implying that the braid monodromy factorization corresponding to Delta 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.