For constructible sheaves $\mathcal F$ on real analytic manifolds $X$, there is a notion of the singular support $SS(\mathcal F)$ which is a radially invariant singular Lagrangian subset of the cotangent bundle $T^\ast X$.
For constructible $\ell$-adic (or $\mathbb F_\ell$ if you prefer) sheaves $\mathcal F$ on the etale site of a smooth variety $X$ in positive characteristic, Beilinson defined an analogous notion of singular support $SS(\mathcal F)\subseteq T^\ast X$. In positive characteristic, the singular support is always half-dimensional (every irreducible component is of the same dimension as $X$), but it need not be Lagrangian!
What is the simplest example of such a sheaf $\mathcal F$ in positive characteristic with non-Lagrangian singular support? I am led to believe that the sheaf $j_!\mathbb Q_\ell$ for $$j:\{(x,y,t)\in\mathbb A^3_{x,y,t}:t^2-t=xy\}\to\mathbb P^2_{x,y}$$ in characteristic $2$ is such an example (this example was communicated to me by Takeshi Saito as an example of something else). Is this example correct, and is there a simpler one?
