| Register
\(\newcommand{\Cat}{{\rm Cat}} \) \(\newcommand{\A}{\mathcal A} \) \(\newcommand{\freestar}{ \framebox[7pt]{$\star$} }\)

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.