Do Neron-Severi groups of smooth projective unirational varieties contain $\ell$-torsion? 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 $H^2_{et}(X,\mathbb{Z}_l)$.
 A: 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 $NS(Y_\ell) = Pic(Y_\ell)/Pic^0(Y_\ell)$.
