Usage of étale cohomology in algebraic geometry 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 $\mathbb{C}$.
 A: 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).
A: The Betti numbers of many (complex) moduli spaces have been computed by counting points over finite fields, using the Weil conjectures, as proved by Deligne, and comparison theorems for étale and singular cohomology. The first example of which I am aware is Lothar Göttsche's calculation of the Betti numbers of Hilbert schemes of points on a smooth projective surface.
A: Another interesting (at least to me) application of étale cohomology is the following. The étale cohomological dimension of the complement of a variety $V$ gives a lower bound on the arithmetical rank of $V$, i.e., the minimum number of equations needed to define $V$ set-theoretically. For example, Bruns and Schwänzl [BS90] used this to prove that over an algebraically closed field the determinantal variety defined by $t$-minors of a generic $m\times n$ matrix has arithmetical rank $mn-t^2+1$.
[BS90] Bruns, Winfried, Schwänzl, Roland - The number of equations defining a determinantal variety. Bull. London Math. Soc. 22 (1990), no. 5, 439–445.
A: Borel has a nice \'etale cohomological proof of Matsushima's criterion: A homogeneous space of a connected reductive group is affine if and only if the isotropic group is reductive.
(The paper is https://doi.org/10.1007/BF01194008)
