open questions

Here I will store some questions which have either come up in research, or which I am idly curious about; my criterion for posting such a question here is that it should be easy to state and have interesting consequences.  Please let me know if you have an answer!  If you would like to know why I think these questions are interesting, feel free to send me an email.

1. Let $C, D$ be smooth projective (connected) curves over $\overline{\mathbb{F}_q}$ of genus at least $2$.  Do they necessarily have a finite etale cover in common?  See e.g. this paper of Bogomolov and Tschinkel for an answer to a related question.
2.  Let $X$ be a smooth projective (connected) variety over $\mathbb{C}$.  Suppose there exists an ample vector bundle $\mathscr{E}$ on $X$ such that $$\text{Hom}(\mathscr{E}, T_X)\neq 0.$$ Is it the case that $X\simeq \mathbb{P}^n$?  See e.g. this paper of mine for some remarks on this question, as well as this paper of Paltin Ionescu.  UPDATE:  Now proven!  See this paper by Jie Liu.
3. Let $C$ be a curve over a finite field $k$.  Let $x_0, x_1\in C(k)$ be rational points.  Let $$\cdots C_i\to C_{i-1}\to \cdots C_1\to C$$ be a tower of $\ell$-power finite etale covers of $C$, and let $y_0^i, y_1^i\in C_i(\bar k)$ be lifts of $x_0, x_1$ to $C_i$.  How rapidly does the $\ell$-part of the order of $$[y_0^i]-[y_1^i]\in \text{Jac}(C_i)(\bar k)$$ grow in $i$?  A good understanding of this question would be useful for bounding the "integral $\ell$-adic periods" that show up in this paper of mine.
4. In this note, I define a natural action of $\mathfrak{sl}_2\times S_n$ on the free vector space spanned by labelled graphs with $n$ vertices.  Does this vector space arise naturally as the cohomology of some polarized variety with an $S_n$-action on it, where the $\mathfrak{sl}_2$-action comes from the Hard Lefschetz theorem?  UPDATE: Will Sawin points out that one may take $X=(\mathbb{P}^1)^{n\choose 2}$, with $S_n$ acting via the natural action on $2$-element subsets of $\{1, \cdots, n\}$.  The quotient by this action gives a pretty interesting variety.
5. Let $k$ be a field of characteristic $p>0$ and $X$ a projective $k$-variety.  Let $\mathscr{E}$ be an ample vector bundle on $X$ and $\mathscr{F}$ be a coherent sheaf on $X$.  Is it true that for $n\gg 0$ and $i\geq \text{rk}(\mathscr{E})$, $$H^i(X, \mathscr{E}^{(p^n)}\otimes \mathscr{F})=0?$$  In this paper I show that the answer is "yes" if $X$ admits a Frobenius lift and $\mathscr{E}$ lifts.  This result would be very useful for proving vanishing theorems for ample vector bundles; the partial results I prove already imply a generalization of the Bott-Danilov-Steenbrink vanishing theorem.
6. Is there an example of a smooth projective variety $X/\mathbb{C}$ and a birational morphism $f: Y\to X$ (with $Y$ also smooth and projective) such that
• $f$ exhibits $Y$ as an iterated blowup of $X$ along smooth subvarieties of codimension at least $3$, and
• $X$ has finite automorphism group, but $Y$ has infinite automorphism group?

If one replaces the number $3$ above with the number $2$, one may find examples of rational surfaces with this property.