We study the ring theory of the multi-parameter deformations of the quantum Schubert cell algebras obtained from 2-cocycle twists. This is a large family, which extends the Artin–Schelter–Tate algebras of twisted quantum matrices. We classify set theoretically the spectra of all such multi-parameter quantum Schubert cell algebras, construct each of their prime ideals by contracting from explicit normal localizations and prove formulas for the dimensions of their Goodearl–Letzter strata for base fields of arbitrary characteristic and all deformation parameters that are not roots of unity. Furthermore, we prove that the spectra of these algebras are normally separated and that all such algebras are catenary.

Semiclassical limits of generic multi-parameter quantized coordinate rings A=q(kn) of affine spaces are constructed and related to A, for k an algebraically closed field of characteristic zero and q a multiplicatively antisymmetric matrix whose entries generate a torsion-free subgroup of k×. A semiclassical limit of A is a Poisson algebra structure on the corresponding classical coordinate ring R=(kn), and results of Oh, Park, Shin and the authors are used to construct homeomorphisms from the Poisson-prime and Poisson-primitive spectra of R onto the prime and primitive spectra of~A. The Poisson-primitive spectrum of R is then identified with the space of symplectic cores in kn in the sense of Brown and Gordon, and an example is presented (over ℂ) for which the Poisson-primitive spectrum of R is not homeomorphic to the space of symplectic leaves in kn. Finally, these results are extended from quantum affine spaces to quantum affine toric varieties.