Bidouble covers
of the quadric are parametrized by connected families depending on four positive integers
. In the special
case where we call
them –surfaces.
Such a Galois covering
admits a small perturbation yielding a general
–tuple covering
of with
branch curve ,
and a natural Lefschetz fibration obtained from a small perturbation of the composition
.
We prove a more general result implying that the braid monodromy factorization corresponding
to determines the
three integers in
the case of –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
–surfaces with the same values
of . This result hints at the
possibility that –surfaces
with fixed values of ,
although diffeomorphic but not deformation equivalent, might be not canonically
symplectomorphic.