I'm a student interested in arithmetic geometry, and this implies I use étale cohomology a lot. Regarding its definition, étale cohomology is a purely algebro-geometric object. However, almost every material I found on étale cohomology focus on its number-theoretic applications, such as the Weil conjectures and Galois representations. So, this is my question:
Are there some applications of étale cohomology on pure algebro-geometric problems?
Here, "pure algebro-geometric problems" means some problems of algebraic geometry without number-theoretic flavors, such as birational geometry, the minimal model program, classifying algebraic varieties (curves, surfaces, etc..), especially over algebraically closed fields.
Since étale cohomology coincides with singular cohomology over $\mathbb{C}$, there must exist such problems over the complex numbers. Hence, I am looking for applications of étale cohomology which are also useful over algebraically closed fields which are not $\mathbb{C}$.
Best Answer
There are many, for example, Artin's proof in nonzero characteristic of Castelnuovo's criterion for the rationality of a surface, and a proof that the Neron-Severi group is finitely generated. Both of these are in Chapter V of Milne's book (3.25, 3.30).