Proving that a variety is not (isomorphic to) a toric variety

Question

Is there an algorithmic (or other) way to prove that a (projective) variety is not isomorphic to a toric variety?

I'd be happy with an algebraic answer (for affine or projective varieties), using the fact that toric ideals are binomial prime ideals. There ne could use that the coordinate rings are characterized as those admitting a fine grading by an affine semigroup , i.e. presented by a binomial prime ideal (Prop. 1.11 in Eisenbud/Sturmfels "Binomial ideals").

This question resulted from an Example that I discussed with Mateusz Michalek. The example is: let $V$ be the Zariski closure of the image of the parameterization: $$(p_1,p_2,a_1,a_2,a_3,b_1,b_2,b_3) \to \begin{pmatrix} p_1a_1a_2a_3+p_2b_1b_2b_3 \\ p_1a_1a_2b_3+p_2b_1b_2a_3 \\ p_1a_1b_2a_3+p_2b_1a_2b_3 \\ p_1a_1b_2b_3+p_2b_1a_2a_3 \\ p_1b_1a_2a_3+p_2a_1b_2b_3 \\ p_1b_1a_2b_3+p_2a_1b_2a_3 \\ p_1b_1b_2a_3+p_2a_1a_2b_3 \\ p_1b_1b_2b_3+p_2a_1a_2a_3 \\ \end{pmatrix}$$

Implicitization using Macaulay2 is quick and yields a complete intersection: $$\langle et-ry-qu+wo, wt-qy-ru+eo, we-qr-yu+to \rangle \subset k[q,w,e,r,t,y,u,o]$$

How to prove that $V$ is not toric?

Answer 1

The question of algorithmically deciding if an ideal is binomial after a (suitable, e.g. linear) automorphism of affine space is decidable and various algorithms are discussed in "When is a polynomial ideal binomial after an ambient automorphism?" by Lukas Katthän, Mateusz Michałek, and Ezra Miller.

I discussed the methods in this paper with the authors and tried them on the example in the question, but the complexity was too high. Together with Lukas Katthän, however, we found a different way to prove that this ideal is in fact toric. It is a complete intersection and given some standard invariants and tables of toric complete intersections it was easy to identify a single candidate toric variety that would be isomorphic to this one.

The ideal in question can be rewritten as: $$ I = \langle x_1x_2 + x_3x_4 - y_1y_2 - y_4y_3, x_1x_4 + x_2x_3 - y_1y_4 - y_2y_3, x_1x_3 + x_2x_4 - y_1y_3 - y_2y_4\rangle $$ This makes the structure a bit more visible. The candidate isomorphic toric ideal is $$J = \langle x_1y_1 - x_2y_2, x_1y_1 - x_3y_3, x_1y_1 - x_4y_4\rangle$$ We examined the singular locus and in particular its minimal primes which are all linear. This revealed what the linear automorphism was: $$ \begin{align} x_1 & \mapsto x_1+x_2+x_3+x_4 + (y_1+y_2+y_3+y_4) \\ x_2 & \mapsto x_1+x_2-x_3-x_4 + (y_1+y_2-y_3-y_4)\\ x_3 & \mapsto x_1-x_2+x_3-x_4 + (y_1-y_2+y_3-y_4)\\ x_4 & \mapsto x_1-x_2-x_3+x_4 + (y_1-y_2-y_3+y_4)\\ y_1 & \mapsto x_1+x_2+x_3+x_4 - (y_1+y_2+y_3+y_4)\\ y_2 & \mapsto x_1+x_2-x_3-x_4 - (y_1+y_2-y_3-y_4)\\ y_3 & \mapsto x_1-x_2+x_3-x_4 - (y_1-y_2+y_3-y_4)\\ y_4 & \mapsto x_1-x_2-x_3+x_4 - (y_1-y_2-y_3+y_4). \end{align}$$ The way it is written here it maps the toric ideal to the original ideal.