Consider the following from this paper "Correspondences, Motifs and Monoidal Transformations" of Manin here.

Theorem. Nonsingular three-dimensional projective unirational varieties $V$ over a finite field satisfy the Riemann-Weil hypothesis.

 

Sketch of the Proof. We consider a rational map $f: \mathbb{P}^3 \to V$ of finite degree. By virtue of Abhyankar's results there exists a commutative diagram of the form

 

in which $g$ is a birational morphism which splits into a sequence of monoidal transformations with nonsingular centers, and $h$ is a morphism of finite degree (at a generic point).

Question. What are Abhyankar's results here, and could anybody give a sketch of a proof of them?

Consider the following from this paper "Correspondences, Motifs and Monoidal Transformations" of Manin here.

Theorem. Nonsingular three-dimensional projective unirational varieties $V$ over a finite field satisfy the Riemann-Weil hypothesis.

Sketch of the Proof. We consider a rational map $f: \mathbb{P}^3 \to V$ of finite degree. By virtue of Abhyankar's results there exists a commutative diagram of the form

in which $g$ is a birational morphism which splits into a sequence of monoidal transformations with nonsingular centers, and $h$ is a morphism of finite degree (at a generic point).

Question. What are Abhyankar's results here, and could anybody give a sketch of a proof of them?