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 :)



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


