Of course it has, it's the stabilizer of a point (or a line) in its $GL_3(2)$ incarnation. What I am asking is how one can go about computing an explicit equation of the cover.

My question is whether there are any similar problems solved in the literature. Perhaps I could use some of their approaches, if such exist.

Are you sure that you don't just have really bad numerical behavior when $z$ is close to $\alpha$?

If you have a list of hyperplanes and your points tend to be close to each other, then I would argue that you may want to (occasionally?) order the hyperplanes according to the values at the points and then first look at the ones that were close to flipping to the outside.

Yes, because at the core of the argument is $p=d \sin(\beta)$ where $d$ is the diameter of the circle.

I am glad it helps. Chances are, this already exists in some form elsewhere in the literature, maybe better suited to the birational map case (as opposed to generically finite case in the paper). So maybe someone else will post a better answer/reference.

Then it is a question of whether one can find a hypersurface $f=0$ whose distance from each of the points in $X$ is smaller than some number. It feels like a sensible question of minimization of maximum distance, although some $L^2$ norm may be more tractable.

Actually if $Y\supseteq X$, then $Y$ could be generated in lower degree than $X$. E.g. $X$ could be cut out by quadrics and cubics but if one considers quadrics only, then one can get some $Y$. So maybe an easier question is not "generated" but simply having a hypersurface of small degree passing through it.

The problem is that if there are no polynomials of small degree that vanish on $X$, then there there would not be any that vanish on a larger set $Y$. Perhaps you want $Y$ to be "close to $X$ but not containing it"?

Perhaps, Theorem 4.8 of arxiv.org/pdf/math/0206241.pdf could be useful.

It would be more natural to consider the stratification of $Y$ induced by both $E_i$ and the proper preimages of $D_j$. Anyway, the whole set up is basically smooth toroidal geometry. Whatever formulas you get under the assumption that everything is toric should also hold in the toroidal setting.

What do you want as far as dimensions of $X$ and $Y$ go? I am assuming you are not interested in $Y $ being the union of $X$ and a line, right?

I do mean the minus, see Alekseyev's calculation. But what is the reason to look at the monomial expansion in $x_i$, if it would be better after a shift?

I know nothing about this, but wouldn't it make sense to look switch from $x_i$ to $y_i=x_i - \frac 12$ and look for the monomial expansion in $y_i$?

What's known for $n=2$ or $n=3$?

@Meow Yes, this looks like what I was asking about. Now, if I could only remember why ... :)