the main problem in this area is to find a nonsingular algebraic variety u, that is birationally equivalent to an irreducible algebraic variety v, such that the mapping f: u v is regular but not biregular.

This is all lower case. Can you fix that?
The main problem in this area is to find a nonsingular algebraic variety U, that is birationally equivalent to an irreducible algebraic variety V, such that the mapping f: U V is regular but not biregular.