Suppose that $f\colon X\to \mathbb P^N$ is a finite morphism, where $X$ is a smooth projective variety over $\mathbb C$. Then one may consider monodromy of the (singular) cohomology of the subvariety $f^{-1}(H)$, where $H\subset\mathbb P^N$ is a general hyperplane (monodromy as $H$ varies). I strongly suspect that properties of this monodromy are to a great extent parallel to those of monodromy of the hyperplane section (i.e., to the Picard-Lefschetz theory). In particular, I believe that the monodromy group is generated by reflexions in "vanishing cycles" admitting a more or less explicit description.

My question is whether there exists a text in which this theory is written up (may be somebody's Ph.D. thesis?).

Thanks in advance, Serge