# Minimal Model Program for surfaces over algebraically closed fields of characteristic p

Let $k$ be an algebraically closed field of characteristic $p>0$.

I have been trying to find out unsuccessfully if there is a mmp for algebraic surfaces over $k$. I know minimal surfaces are classified thanks to Zariski, Mumford, others in this setting and that is not my question. I want to 'run' LMMP of pairs not to find minimal model but as a tool to prove several stuff.

In particular I would like to know if there is a Cone Theorem (i.e. giving me that extremal rays have non-positive self-intersection, I suspect the answer is yes) and if there is a contraction theorem (I suspect not yet).

By contraction theorem I mean that if given $(X,D)$ where $X$ is a surface $D$ is an effective divisor and the pair is klt, if $K_X+D$ is not nef I can find a curve $E$ with negative self-intersection such that $E$ can be contracted.

I am aware I am being vague with the formulation but I do not want to constrain your imagination.

Now, if someone also knows if flips and termination of flips are possible, please share :)

• The fact that extremal rays have non-positive self-intersection is much easier than the Cone Theorem: it comes immediately from Riemann--Roch. See Koll'ar--Mori Lemma 1.20. – user5117 Jan 11 '12 at 18:27
• For surfaces, you have Lipman's theorem: any excellent, reduced, Noetherian scheme of dimension 2 has a desingularization. You can use this to prove the existence of a minimal resolution of singularities for a normal surface, and even to prove the existence of a minimal surface. See Theorem 8.3.44, Theorem 9.3.21, Proposition 3.32, and more in Liu's book "Algebraic geometry and arithmetic curves". I don't know anything about MMP so I might have misunderstood the question. – Ali Jan 11 '12 at 18:56