Let $X$ be a smooth projective unirational variety over an algebraically closed field of characteristic $p>0$, and $\ell\neq p$ a prime. My question: can the Neron-Severi group of $X$ contain (non-zero) $\ell$-torsion? This appears to be closely related to the presense of torsion in the etale cohomology group $H^2\_{et}(X,\mathbb{Z}\_l)$.



Do Neron-Severi groups of smooth projective unirational varieties contain $\ell$-torsion?
It seems that existence of $\ell$-torsion is possible. Here's one example.


Let $k$ be an algebraically closed field of characteristic $p$. In the paper "An Example of Unirational Surfaces in Characteristic p." (<https://eudml.org/doc/162649>) Shioda proves that a hypersurface $X\_n = \{x\_1^n + x\_2^n + x\_3^n + x\_4^n = 0\} \subset \mathbb{P}^3\_k$ is unirational if $p+1$ is divisible by $n$. For $n = \ell$, where $\ell \neq p$ is a prime number, such unirational hypersurface admits a free action of the group $\mathbb{Z}/\ell$ given by the formula:
$$
k\in \mathbb{Z}/\ell \quad k \cdot (x\_1,\ldots,x\_4) = (\xi^{ka\_1} x\_1, \xi^{ka\_2} x\_2, \xi^{ka\_3} x\_3, \xi^{ka\_4} x\_4),
$$ 
for $\xi$ a primitive $\ell$-th root of unity and $\{a\_i\}$ pairwise distinct integers not divisible by $\ell$. 
The quotient $Y\_\ell = X\_\ell / \mathbb{Z}/\ell$ then has non-trivial $H^1\_{et}(Y\_\ell,\mathbb{Z}/\ell)$ and hence non-trivial $\ell$-torsion in the Picard group. Since $H^1(X\_\ell,\mathcal{O}\_{X\_\ell}) = 0$ and $p$ is coprime to $\ell$, we see that $H^1(Y\_\ell,\mathcal{O}\_{Y\_\ell}) = 0$ and hence $Pic^0(Y\_\ell) = 0$. Consequently our $\ell$-torsion in $Pic(Y\_\ell)$ yields $\ell$-torsion in $NS(Y\_\ell) = Pic(Y\_\ell)/Pic^0(Y\_\ell)$.


