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## 3. Geometry of F-singularities

Questions dealing with the properties of F-singularities
1. ### Grauert-Riemenschneider for F-regular varieties

#### Problem 3.1.

[Tucker] Is there Grauert-Riemenschneider vanishing for $F$-regular varieties which admit a resolution?
• ### Global $F$-regularity and rational chain connectedness

#### Problem 3.2.

[Tucker] Does globally $F$-regular imply rationally chain connected?
It’s known that if $F$ is of globally $F$-regular type (i.e. $X$ is defined over a field of characteristic $0$, and $X_p$ is globally $F$-regular for $p \gg 0$), and $\mathbb Q$-Gorenstein, then $F$ is rationally connected.
• ### Global $F$-regularity and Fano type

#### Problem 3.3.

[Schwede] If $X$ is of globally $F$-regular type, does it follow that it is log Fano (in the sense that there exists a boundary divisor $\Delta$ with $(X,\Delta)$ klt and $-(K_X+\Delta)$ ample)? Even weaker, does it follow that $-K_X$ is big?
This is known if $\dim X = 2$, or if $X$ is a Mori dream space. If $X$ is in fact globally $F$-regular, then it is log Fano for all $p$. It’s also true that if $X$ is of globally $F$-regular type, then $-K_{X_p}$ is big for $p \gg 0$.
• ### Construction of $F$-pure centers

#### Problem 3.4.

[Schwede] Suppose that $(X,\Delta)$ is a strongly $F$-regular pair. Can we find a boundary $D \geq 0$ such that $(X,\Delta+D)$ is $F$-pure, with $p \nmid \text{index}(K_X+\Delta+D)$?

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