Quantum Schubert cells via representation theory and ring theory

We resolve two questions of Cauchon and Meriaux on the spectra of the quantum Schubert cell algebras U^-[w]. The treatment of the first one unifies two very different approaches to Spec U^-[w], a ring theoretic one via deleting derivations and a representation theoretic one via Demazure modules. The outcome is that now one can combine the strengths of both methods. As an application we solve the containment problem for the Cauchon-Meriaux classification of torus invariant prime ideals of U^-[w]. Furthermore, we construct explicit models in terms of quantum minors for the Cauchon quantum affine space algebras constructed via the procedure of deleting derivations from all quantum Schubert cell algebras U^-[w]. Finally, our methods also give a new, independent proof of the Cauchon-Meriaux classification.