Suppose that $X$ is a smooth projective algebraic variety over a perfect field $k$ of characteristic $p > 0$ (I'm also interested in the non-smooth case).

Suppose that $D$ is a divisor or possibly a $\mathbb{Q}/\mathbb{R}$-divisor on $X$ that is ample (I'd also be happy to consider weaker notions of positivity, big, semi-ample, nef, etc).

Now, consider the Frobenius map $$F : \mathcal{O}_X(-D) \to \mathcal{O}_X(-pD)$$ (or more generally to $\mathcal{O}_X(-p^eD)$ for $e \gg 0$). If $X$ is a $\mathbb{Q}$-divisor, then for this map, I mean $\mathcal{O}_X(\lfloor -D \rfloor) \to \mathcal{O}_X(\lfloor -p^eD \rfloor)$.

Take cohomology and consider $$H^i(X, \mathcal{O}_X(-D))\to H^i(X, \mathcal{O}(-p^eD)) $$ for $i \neq 0$. I'm particularly interested in the top cohomology, (notice that for $e \gg 0$, the lower cohomologies are uninteresting by Serre vanishing as Donu pointed out).

**Question:** Can I say anything about where $D$ lives in the nef/ample cone based on the dimension of the kernel of that map (ie, the rank of that map)? In particular, has this been written down somewhere? Even for surfaces?

**Question:** More generally, do the set of classes in the ample cone which have a kernel of dimension $\geq n$ representative, (ie a "bad" (low rank in the above sense)) live in a region with any nice (connectivity/convexity?) properties.

I'm somewhat aware of some of the various statements that can be made in this direction for curves (ie, Tango curves), but if someone has something interesting to say that also illuminates the higher dimensional picture, I'd be quite interested!

Maybe I'm asking the wrong questions too, so I'd be happy to hear what the right questions are.

**EDIT:** I should point out that for $D$ sufficiently ample Weil divisor, that the picture is very uninteresting (Serre vanishing applies).