# What is known about lower etale cohomology of unirational varieties?

Which information is currently known about $$H^1_{et}(X,\mathbb{Z}_l)$$ and $$H^2_{et}(X,\mathbb{Z}_l)$$, where $$X$$ is a smooth unirational variety over an algebraically closed field of finite characterstic $$p\neq l$$? Actually, I am interested in the case where $$X$$ is a quotient of a rational variety $$Y$$ by a free action of a finite group (whose order is prime to $$p$$). Does the situation differ much from the complex variety case; are there any examples where it does?

Certainly, my question is closely related to (the torsion) of $$\operatorname{Pic}(X)$$ and to the Brauer group of $$X$$.

• If you want your varieties to behave like in characteristic $0$, it's probably better to say separably unirational. In positive characteristic there are all sorts of 'strange' unirational varieties; for example there exist unirational K3 surfaces. Maybe this doesn't matter for your question, but it seems that you're thinking about the separable case anyway. Feb 18, 2020 at 2:10

Concerning the $$H^2$$, for $$X$$ a smooth projective rationally chain connected variety over an algebraically closed field $$k$$ and $$\ell \in k^\ast$$, it follows from Theorem 1.2 in https://arxiv.org/abs/1703.05735 that
$$\mathrm{NS}(X)\otimes \mathbb{Z}_{\ell} = \mathrm{H}^2_{et}(X,\mathbb{Z}_{\ell}(1))$$