## 5. Other questions

Analogs of some other questions from characteristic 0.-
### 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? -
### 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 $X$? -
### 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”? -
### 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.

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