Non-regularizable birational automorphisms

A birational automorphism $f$ of an algebraic variety $X$ is an algebraic isomorphism of one Zariski open subset of $X$ to another open subset. A natural question arises: is there a birational model of $X$ where $f$ induces a regular automorphism. This question is related to the invariants of the inverse image action of $f$ on the singular cohomology groups of $X$. I am going to talk about this relation with an example of a birational automorphism of a three dimensional projective space which preserves a cubic surface pointwise.