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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


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          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


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                  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)


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                      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


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                              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


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                                  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.