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4. Sections and section rings

Questions about the existence of sections, and properties of various section rings.

    1. Fano and MDS


      Add remark

      Problem 4.1.

      [Cascini] Is every Fano variety a Mori Dream Space?
          This result is known in characteristic $0$. It’s also true that if $X$ is Fano then $X$ is strongly $F$-regular, if $p \gg 0$.

        • Invariance of plurigenera


          Add remark

          Problem 4.2.

          [Cascini] What is the status of invariance of plurigenera?
              It’s known that $h^0(mK_X)$ need not be constant in flat families if $m=1$. For $m \geq 2$ it remains a possibility.


            • Add remark

              Problem 4.3.

              [Hacon] What if we look at $\text{dim } S^0(mK_X+A)$ instead?

                • General type in families


                  Add remark

                  Problem 4.4.

                  [McKernan] Suppose that $\mathcal X \to \Delta$ is a family, and a special fiber is of general type. Does it follow that a general fiber is of general type?

                    • Embedding by $S^0(mK_X)$


                      Add remark

                      Problem 4.5.

                      [McKernan] Suppose that $X$ is of general type. Is there an effective constant $m = m(n)$ such that $S^0(mK_X)$ (the canonical linear system of Schwede) defines a birational map? What about the usual linear system $|mK_X|$?

                        • Effective Fujita vanishing


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                          Problem 4.6.

                          [McKernan] Is there an effective Fujita vanishing result in flat families? Suppose that $f : \mathcal X \to S$ is flat, $\mathcal F$ is a coherent sheaf on $\mathcal X$, and $\mathcal L$ is $f$-ample. Does there exist a constant $m_0 = m_0(\mathcal X,\mathcal F,\mathcal L)$ such that $H^i(X_s,(\mathcal F \otimes \mathcal L^m \mathcal )\vert_{X_s} ) = 0$ if $s$ is any point of $S$, $m \geq m_0$ and $\mathcal M$ is any $f$-nef line bundle on $X$?
                              This is probably OK in the equal characteristic case, adapting the usual proof.

                            • Sections of nef $K_X+A$


                              Add remark

                              Problem 4.7.

                              [Cascini] Suppose that $A$ is Cartier and ample, and $K_X+A$ is nef. Must $H^0(X,K_X+A)$ be nonzero?

                                • $F$-regularity and finite generation


                                  Add remark

                                  Problem 4.8.

                                  [Schwede] Suppose that $X$ is an $F$-regular variety (maybe not $\mathbb Q$-Gorenstein). If $D$ is a Weil divisor, must $\bigoplus_{m \geq 0} \mathcal O_X(mD)$ be finitely generated?

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