My original post of DIOP was originally about Invariance of Plurigenera using adjoint ideals, local techniques, and minimal model techniques in Characteristic 0. Then I added a whole big section in mixed or positive. Now I’ve just split off the second part: Essentially the point of this preprint is to prove invariance of plurigenera for log pairs in mixed or positive characteristic, and also the existence of minimal models in this case.

Below I give the invariance of klt plurigenera with a result of the existence of minimal models in mixed characteristic / positive characteristic and relative dimension 2 over a DVR.

edit: (4/28) still in the process of improving the char p version. Here’s a rough recent draft:

SurfDIOPFp1.5_copy, Hash .

edit (5/7 ++) I’ve decided to start using the following:  code for hashing. Most recent updates below, I couldn’t  figure out the right vanishing for the effective version, so it’s just the asymptotic.

SurfDIOP2.4_copy , hash

Effective version seems to work again.. (and I seem to have gotten an effective version of Kawamata Viehweg vanishing for KLT log smooth surfaces).
SurfDIOP2.9_copy , hash

Well actually the effective (or at least computable) vanishing works for normal klt surface (over an algebraically closed field in positive characteristic).

SurfDIOP3.0 , hash

SurfDIOP3.1 , hash

SurfDIOP3.3_copy , hash : added non-general type case (needs a bit of work)

SurfDIOP3.4 , hash : added new alternative proof of main theorem via W2 DIOP.

(deleted 3.5-7, looks like a cited theorem was incorrectly stated)

SurfDIOP3.8_copy hash

SurfDIOP3.9 , hash :

SurfDIOP3.10 , hash : Although the nu = 1 case doesn’t seem to be working for DIOP, this gives the existence of minimal models in that case regardless (some details need to be expanded upon, but at least the sketch seems to be correct at this point).

SurfDIOP3.11hash : more details filled in…

SurfDIOP3.12_copy , hash