Do there exists some language $\mathcal{L}$ of rational polyhedral cones in rational vector spaces and a theory $T$ over $\mathcal{L}$ whose models $\mathcal{M}$ are resolutions of toric singularities?
The motivation of this question is that if such theory exists, a constructive resolution of toric singularities may be a quantifier-free theory equivalent to $T$.
