## 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 :)

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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. – Artie Prendergast-Smith Jan 11 2012 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 2012 at 18:56

In the surface case, MMP in char p is known. See Koll'ar-Kov'ac's preprint on Koll'ar's webpage.

In dimensional 3, the existence of divisorial contractions and flipping contractions is known as EWM (so the target is only known as a algebraic space). See Keel's paper BASEPOINT FREENESS FOR NEF AND BIG LINE BUNDLES. I'm not sure about the termination of flips. The existence of flips is certainly not known (at this moment).

In higher dimensions, I think almost nothing is known.

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Dear CX -- Is "everything" about MMP for surfaces known in characteristic p? For instance, does the work of Keel-McKernan, Chenyang Xu, etc., on rational curves on log Del Pezzo surfaces also work in characteristic p? (I should know the answer to this myself, but at the moment I do not.) – Jason Starr Jan 11 2012 at 18:23
Dear Jason: By 'MMP', I mean precisely the things we need for running the minimal model program (in the surface case, it just means the contraction exists.). With further digging on the literature, it might be also true that the abundance is also known for the surfaces pairs in char p. On the other hand, in the paper "Strong Rational Connectedness of Surfaces", there is an example due to Koll'ar giving a del Pezzo surface with only A_1 singularities, whose smooth locus doesn't contain free curves. Of course it still may be rationally connected, which I didn't really check. – CX Jan 11 2012 at 18:55
Dear CX -- Thanks a lot. – Jason Starr Jan 11 2012 at 19:03
This is very useful actually. What about LMMP, i.e. what if I take a pair (X,D) which is KLT where X is a surface over an algebraically closed field of characteristic p and D a divisor on that surface? – Jesus Martinez Garcia Jan 16 2012 at 17:38
To Garcia: What I said is all for the general log pair case. I reedit my answer because I realize that Keel only show in dimension 3, flipping and divisorial contractions exist as EWM. In other words, so far we only know the target space for these contracts exists as an algebraic space. – CX Jan 17 2012 at 22:19