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

Abstract

Bidouble covers $\pi :S\to Q:={\mathbb{P}}^1\times {\mathbb{P}}^1$ 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 $p_1\circ \pi$.

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.