@user44191: Fixed, thanks.

@BrendanMcKay: Fixed, thanks.

Isn't a "theory," by definition, a particular generating set, though? That seems to be what Qfwfq is bringing up.

Oh, are you just saying that you want the equation for the surface swept out by dragging a conic section along a path that is itself a conic section?

By "toric surface" I initially thought you meant a two-dimensional toric variety, especially given the algebraic geometry tag, but now I suspect you mean something like the illustration for the article "toric lens" on Wikipedia. Is that what you're talking about?

Yeah, once people get used to relativizing things it's clear enough to them what things ought to be so they can get a little sloppy with their definitions. I agree it can make things infuriating to read for the uninitiated, though.

At the level of sets, $G \times_X P$ is the disjoint unions of the sets $G_x \times P_x$. I don't know if we explicitly said it above but we need to insist that each $G_x \times P_x$ maps into $P_x$. Then you have a bunch of individual actions of $G_x$ on $P_x$.