On the point of the general relationship to Spec of a ring. I once had a conversation with an expert on Zariski's construction with valuations rings, where it seemed that a locale could immediately be written down as a presentation. There is also a well-known presentation of Spec as a locale, by divisibility basically. But the former point I hadn't seen mentioned anywhere (whether or not it is useful).

This is about inseparable coverings; there is a classic paper of John Tate I've never read (fortunately I've now found it is online) which refers to Emil Artin's concept of a "conservative function field", where the genus doesn't change under extension of constant field. Tate's paper seems to use the Cartier operator (before Cartier); I imagine this isse is now well understood. Could be another question, though.

I feel I should mention the phenomenon of "genus change in purely inseparable extension", given that this is something surprising in characteristic p; but I don't know whether there is a definitive treatment out there in what is a rather large literature by now.

Not as you have stated this.

Much more is in Waldschmidt's slides at modular.math.washington.edu/swc/aws/08/slides/… . The first result seems to have been by Siegel.

Added a reference where I think the whole business is sufficiently clarified.

Name familiar, face escapes me? That would be a long time ago.

That would be Hankel.

That's a negative result, as far as existence of common invariants is concerned. It removes one good reason why they might be there.

But it is possible that you have a group action (whatever your two involutions generate). If that group is finite, Galois theory might be more revealing. The formulation does suggest a slightly more algebraic approach to what is happening. It depends on the nature of F.