Loading Web-Font TeX/Math/Italic
| 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


      Add remark

      Problem 3.1.

      [Tucker] Is there Grauert-Riemenschneider vanishing for F-regular varieties which admit a resolution?

        • Global F-regularity and rational chain connectedness


          Add remark

          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


              Add remark

              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


                  Add remark

                  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.