I'm trying to develop some intuition for the (local) Picard-Lefschetz formula (which I'm encountering for the first time in Deligne's paper "La Conjecture de Weil, I").

To summarize the setup, we have a proper family of varieties $\pi : X \rightarrow D$, where $D$ is thought of as a small complex disk (or really its algebro-geometric analogue). All the fibers $\pi^{-1}(t)$ are smooth except the central fiber, which has a double point.

There is a monodromy action of the fundamental group of the punctured disk on the cohomology of the "generic fiber", which is actually only non-trivial in middle dimension, i.e. if $\dim X = n$ then the monodromy action is only non-trivial on $H^n$. See the linked page for the precise formula.

Anyway, my question is if there is some nice intuition for why the monodromy action takes this shape. Most sources that I can find online jump straight into very abstract formulations in the language of derived functors and perverse sheaves. For concreteness, here are some specific questions:

- Is there some "easy" way to see that the monodromy should act as a transvection?
- Is there some "easy" way to see that the action should be trivial except in middle dimension?
- Is there some "easy" reason that one would expect the monodromy formula to depend on the parity of the dimension?

I thought that I could see 2 by applying the Lefschetz Hyperplane Theorem to the family. The point is that I get a family $Y \subset X$ over $D$, such that the cohomology of $Y$ maps isomorphically to that of $X$ except in middle degree. If I choose $Y$ generally, then it will miss the singular point of $X$ and thus be a *smooth* family, so the monodromy action should be trivial. But this argument had nothing to do with the critical point of $\pi$ being nondegenerate, so maybe I'm missing something?

I would also appreciate reference recommendations that treat this subject and how it naturally evolves into the modern derived category formalism.