Is there any (general) theory on singularities of higher dimensional toric sections. I mean singularities of an intersection of an n-tori with a linear space. So for example singularities of the variety defined by \begin{align*} x_1^2 + x_2^2 - 1 \\ x_3^2 + x_4^2 - 1 \\ x_5^2 + x_6^2 - 1 \\ 2\cdot x_1 - x_3 \\ x_4 + x_5 \end{align*} I'm mostly interested in a way to find them. I tried approaches with the principal minors of the jacobian. But there are soon (if you go to higher dimensions of the tori and/or linear spaces) too many polynomials, to compute a groebner basis.
I would very much appreciate a pointer to a resource of some kind, if there are any.
kind greetings, Marc