what you call "the two-sheeted hyperboloid" has as real part a sphere rather than the union of two projective planes. For double covers with branch locus of higher degree I don't see why it should be the union of two projective planes. if it's true, can you give an argument?

Fair enough, I have edited the question.

The support of $\mathcal{F}$ is a closed subscheme of $\mathbb{P}^n$ and I assume that this is in fact a variety (thus reduced). Why does the formula not make sense?

this should follow from Bertini's theorem: the intersection with a general hyperplane does not add any new singularities.

if not in general, perhaps if we further assume that $M$ is invertible in $\mathbb{Z}_{(2)}$?

thank you for that detailed answer! the thing you say about $\mathbb{Z}[\frac{1}{2}]$, is it somehow hidden in O'Meara's book? when I went through it, I couldn't find it, or have I just overlooked it? Can one say something similar about $\mathbb{Z}_{(2)}$?

okay, many thanks! I'll have to read these works.

Thanks, that sounds great. Can you give me some references for the two statements that you use?