Regularizing rational morphisms.

Question

Given a rational morphism $f:X\rightarrow Y$ between varieties $X$ and $Y$, is it possible to find a variety $X'$ that is birational to $X$ and a regular morphism $f':X'\rightarrow Y$ that coincides with $f$ generically?

Answer 1

If by "rational morphism" you mean what is usually called a rational map, and if you are taking all varieties to live over an algebraically closed field of characteristic $0$: yes, this is part of the Resolution of Singularities package proved initially by Hironaka, and goes under the name "resolution of indeterminacy." Furthermore, Hironoka shows that you can do this via a sequence of blowups along smooth subvarieties of the singular locus.

If your base field has characteristic $p > 0$, then this is still an open question. I'm not sure if this applies directly to resolving indeterminacies, but many applications of ordinary resolution of singularities can actually still be obtained in characteristic $p$ using de Jong's theory of alterations, which says that a singular variety is always dominated by a smooth variety of the same dimension (i.e. the "resolution morphism" is generically finite, not birational).