MMP for 3fold germs — September 13, 2015

MMP for 3fold germs

Minimal models exist for arithmetic 3fold germs in mixed characteristic (this should complete the proof started in surfdiop / syzsurfdvr). I will try to combine all three of these into one thing and clean it up shortly.

MMPThreefoldGerms0.1_copy , hash

MMPThreefoldGerms0.2 , hash

MMPThreefoldGerms0.3 , hash (working on combining things)

MMPThreefoldGerms0.4 , hash (slightly cleaned up)

MMPThreefoldGerms0.5_copy , hash (more cleaned up, and simplification of big case of termination)

MMPThreefoldGerms0.6 , hash (had forgotten to make the corresponding changes to the big case simplification in some preliminary lemmas)

MMPThreefoldGerms0.7 , hash (clarifies proof of effective KVV (actually I had the constant wrong in the last version), and also gives a weak version of a very recent conjecture of Fujino on effective bpf-ness, resulting in some effective finite generation theorems as well – this was going to be a new note, but since the finite generation is already there, I’ve just put it here).

MMPThreefoldGerms0.8_copy , hash : Took a closer look at the KVV, and found another hole in my proof – now I’ve ended up with a quite nasty constant if you actually compute it out, but which seems more reasonable. In any case, the main results don’t depend on this version of KVV (Tanaka’s original version suffices for all of the existence / minimal model type results) and I’ve tried to make that clearer – still it would be really nice to have some effective things, hence the inclusion in these past couple of drafts.

MMPThreefoldGerms0.9 , hash : removed unnecessary hypothesis (don’t have to go to canonical model for the effective KVV part)

MMPThreefoldGerms0.10 , hash : just cleaned up a few things –

MMPThreefoldGerms0.11 , hash : slight change of hypothesis on one of the finite generations

MMPThreefoldGerms0.12_copy, 10/2/2015 : Expanded proof of above slight change in hypothesis

MMPThreefoldGerms0.13 , 5 October 2015 Found that actually, that finite generation (for the special fiber) was recently proven by Hashizume, this makes my job a bit easier, since the proof is a bit nasty, and not my main result. Also, I’ve moved the effective bpf-ness to a separate note (becoming too long for advisor to read)

MMPThreefoldGerms0.14_copy , 6 October 2015 No significant changes, but removed unused references and clarified the big case of termination slightly (namely added the statement of a lemma that the the geometric valuations are linear after doing a small contraction)

MMPThreefoldGerms0.19_copy 10/29/2015 I think there was a tiny issue in the cone theorem of versions .15-.18 – which were all attempts to get around the log smoothness hypothesis possibly failing after a few steps of the MMP- in this one the argument is simplified by doing a local version that should hold after a finite number of such steps.

MMPThreefoldGerms1.0_copy 10/30/2015 Should be close to finished, versioned up to 1.0.

MMPThreefoldGerms1.02_copy 11/11/2015 fixed some more typos