Loading Web-Font TeX/Math/Italic
| Register
\newcommand{\Cat}{{\rm Cat}} \newcommand{\A}{\mathcal A} \newcommand{\freestar}{ \framebox[7pt]{$\star$} }

5. Other questions

Analogs of some other questions from characteristic 0.

    1. Pulling back forms to a resolution


      Add remark

      Problem 5.1.

      [Xu] Suppose that X is log canonical, and that there exists a log resolution f : \tilde{X} \to X which is an isomorphism over the smooth locus. Let \omega be an (n-1)-form on X. Does f^\ast \omega\vert_{X_{\text{smooth}}} extend to an (n-1)-form on \tilde{X} with log poles along the exceptional locus?
          This is true in characteristic 0 by work of Greb-Kebekus-Kovács and Greb-Kebekus-Kovács-Peternell. In fact it works for k-forms for any k, but the case k = n-1 should be the easiest.

        • Universal lower bounds for Seshadri constants


          Add remark

          Problem 5.2.

          [Mustaţă] Suppose that X is a smooth variety over an uncountable field k of positive characteristic. Does there exist a constant c = c(n) such that \epsilon(L,x) \geq c for any ample divisor L and very general point x of X?
              The analogous result is true in characteristic 0: we have \epsilon(L,x) \geq 1 if n =2 and \epsilon(L,x) \geq 1/n if n \geq 3. The proof in that case relies on generic smoothness.

            • Nefness under mod p reduction


              Add remark

              Problem 5.3.

              [Kollár, Cascini] Suppose that X is a variety over k, with \text{char } k = 0, and L is a nef divisor on X. Must L_p be nef on X_p for infinitely many p? What about in the case L= K_X? What if “nef” is replaced by “semiample”?
                  It’s not the case that L_p is nef for almost all p; Shepherd-Barron gives an example where this fails on Shimura surfaces. If the abundance conjecture is true, then in the L = K_X case semiampleness should hold for almost all p.

                • Singularities and point-counting


                  Add remark

                  Problem 5.4.

                  [Takagi] Suppose that X is a variety over a finite field. Is there any relation between the singularities of X and the number of points over \mathbb F_q? For example, suppose that X is Fano and F-regular.
                      The motivation is a result of Esnault: suppose that F is smooth, geometrically connected, and rationally chain connected. Then \# X(\mathbb F_q) \equiv 1 \mod q.

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