Some results related to adjoint rings, syzygies, and minimal models on a surface of relative dimension 2 over a DVR. I will be working on this for next two weeks as I have a short break from teaching, I hope to actually get something done.

SyzygiesSurfDVR0.1 , hash

SyzygiesSurfDVR0.2 , hash

SyzygiesSurfDVR0.3 , hash

SyzygiesSurfDVR0.4 , hash

SyzygiesSurfDVR0.5 , hash

SyzygiesSurfDVR0.6 , hash

SyzygiesSurfDVR0.8 , hash

SyzygiesSurfDVR0.9_copy hash :the proof in this one seems to hold for nu = 0,2, or 1 if serre duality holds. The nu = 1 general proof, is now mainly in SurfDIOP3.10+, which is receiving more updates now.