3
$\begingroup$

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

enter image description here

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?

Improve this question
$\endgroup$
2
  • 3
    $\begingroup$ Presumably resolution for 3-folds... $\endgroup$
    – Daniel Litt
    Commented Nov 11, 2016 at 22:19
  • 1
    $\begingroup$ Daniel is correct. It is far too long for anyone to sketch the proof here, but take a look at Lipman's "Introduction to resolution of singularities", in the Arcata proceedings for an exposition of some of the ideas. $\endgroup$
    – Donu Arapura
    Commented Nov 12, 2016 at 0:37

0

Reset to default

You must log in to answer this question.