
## 5. Other questions

Analogs of some other questions from characteristic 0.
1. ### Pulling back forms to a resolution

#### Problem 5.1.

[Xu] Suppose that $X$ is log canonical, and that there exists a log resolution $f : \tilde{X} \to X$ which is an isomorphism over the smooth locus. Let $\omega$ be an $(n-1)$-form on $X$. Does $f^\ast \omega\vert_{X_{\text{smooth}}}$ extend to an $(n-1)$-form on $\tilde{X}$ with log poles along the exceptional locus?
This is true in characteristic $0$ by work of Greb-Kebekus-Kovács and Greb-Kebekus-Kovács-Peternell. In fact it works for $k$-forms for any $k$, but the case $k = n-1$ should be the easiest.
• ### Universal lower bounds for Seshadri constants

#### Problem 5.2.

[Mustaţă] Suppose that $X$ is a smooth variety over an uncountable field $k$ of positive characteristic. Does there exist a constant $c = c(n)$ such that $\epsilon(L,x) \geq c$ for any ample divisor $L$ and very general point $x$ of $X$?
The analogous result is true in characteristic $0$: we have $\epsilon(L,x) \geq 1$ if $n =2$ and $\epsilon(L,x) \geq 1/n$ if $n \geq 3$. The proof in that case relies on generic smoothness.
• ### Nefness under mod p reduction

#### Problem 5.3.

[Kollár, Cascini] Suppose that $X$ is a variety over $k$, with $\text{char } k = 0$, and $L$ is a nef divisor on $X$. Must $L_p$ be nef on $X_p$ for infinitely many $p$? What about in the case $L= K_X$? What if “nef” is replaced by “semiample”?
It’s not the case that $L_p$ is nef for almost all $p$; Shepherd-Barron gives an example where this fails on Shimura surfaces. If the abundance conjecture is true, then in the $L = K_X$ case semiampleness should hold for almost all $p$.
• ### Singularities and point-counting

#### Problem 5.4.

[Takagi] Suppose that $X$ is a variety over a finite field. Is there any relation between the singularities of $X$ and the number of points over $\mathbb F_q$? For example, suppose that $X$ is Fano and $F$-regular.
The motivation is a result of Esnault: suppose that $F$ is smooth, geometrically connected, and rationally chain connected. Then $\# X(\mathbb F_q) \equiv 1 \mod q$.

Cite this as: AimPL: The minimal model program in characteristic p, available at http://aimpl.org/minimalmodcharp.