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



An analog of Picard-Lefschetz theory for finite coverings in lieu of embeddings
If you take a divisor of bidegree $(1,1)$ in $\mathbb P^n \times \mathbb P^n$ and take the fiber product over the first $\mathbb P^n$ with $X$, you get a family of varieties over the second $\mathbb P^n$, the fibers of which are the $f^{-1}(H)$. Essentially, you are asking what the monodromy of this family is. The singular fibers will be a codimension $1$ subset. I think the fundamental group of the complement of the singular locus will always be generated by small loops around it. For those small loops, simply choose a $\mathbb P^1$ containing them and apply regular Picard-Lefschetz theory.


