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

1. Fundamental MMP theorems

Analogs of the fundamentals theorems of the MMP in characteristic 0 whose positive characteristic forms remain open.
    1. Basepoint free theorem

      Problem 1.1.

      [Tanaka] Suppose that $k = \bar{k}$, $\text{char } k = p > 0$. Let $X/k$ be a terminal threefold, and $A$ an ample $\mathbb Q$-divisor on $X$. If $K_X+A$ is nef, must it be semiample?
          This is known in a couple cases: 1. $X$ smooth, $\nu(K_X+A) = 0$.

      2. $k = \bar{\mathbb F}_p$, $\dim X = 3$, $K_X+A$ big.
        • Termination in dimension 3

          Problem 1.2.

          Do klt flips terminate in dimension 3?
              This is probably OK for terminal varieties, with the proof as in characteristic $0$.
            • Connectedness

              Problem 1.3.

              Is there a positive-characteristic analog of the connectedness lemma?

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