about — July 9, 2015


This is my old math related stuff blog divisibility.wordpress.com. I will soon be moving my blog to a new service (tried wordpress and medium and decided they aren’t customizable enough, so will update here after a while with the new location).

I frequently update older posts (especially the ones containing math preprints) since I like to keep similar topics together in one post. Much of the work here is hashed into the bitcoin blockchain using code from here. This allows me to keep track of when I created things.

thesis — February 22, 2016
Merry christmas! — December 21, 2015
MMP Arithmetic Threefolds (not germs) — November 20, 2015
A Big Result On Graph Isomorphism — November 6, 2015

A Big Result On Graph Isomorphism

I thought GI was supposed to be harder than factoring! yikes

Gödel's Lost Letter and P=NP


Lszl Babai is one of the world experts on complexity theory, especially related to groups and graphs. He also recently won the 2015 ACM Knuth Prize, for which we congratulate him.

Today we wish to discuss a new result that he has announced that will place graph isomorphism almost in polynomial time.

More exactly Lszl shows that Graph Isomorphism is in Quasipolynomial Time: that is time of the form

$latex displaystyle 2^{O(log(n))^{c}}, &fg=000000$

for some constant $latex {c}&fg=000000$. Polynomial time is the case when $latex {c=1}&fg=000000$, but any $latex {c}&fg=000000$ is a huge improvement over the previous best result.

Luca Trevisan already has made a post on this result, and Scott Aaronson likewise. Luca further promises to be in Chicago next Tuesday when Lszl gives his talk on the result—here is the abstract of the talk:

We outline an algorithm that solves the Graph Isomorphism (GI) problem and the…

View original post 555 more words

Degrees / basepoint freeness / etc. — October 5, 2015

Degrees / basepoint freeness / etc.

In this note, I have taken the basepoint-freeness stuff out of the MMP for arithmetic threefold germs note, (mainly because it was becoming too long for my advisor to read). I’ll probably polish it up in the future, but for now mainly focus on that paper.

Degrees0.1 , 5 October 2015

Degrees0.3_copy , 10/27/2015

Nov 3. -looks like someone has posted some similar things  http://arxiv.org/abs/1510.08885 also using Tanaka’s techniques (maybe slightly weaker since they don’t work with the pair and requires an ample)

Degrees0.4_copy , 11/13/2015 in this version, I use what is shown in degrees 0.3 to show that the MMP for 3f germs paper holds more generally over a Dedekind Domain.

Degrees0.5_copy , 11/15/2015 just a very slight clarification + a few typo fixes

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

Adjoint Rings and Syzygies in mixed or positive characteristic — August 7, 2015

Adjoint Rings and Syzygies in mixed or positive characteristic

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.

Mixed Characteristic Invariance of Plurigenera and Finite Generation for Relative Dimension 2 — June 12, 2015

Mixed Characteristic Invariance of Plurigenera and Finite Generation for Relative Dimension 2

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

Positive Characteristic Invariance of Plurigenera of Smooth Sections Under Certain W2 Lifting Assumptions — May 12, 2015

Positive Characteristic Invariance of Plurigenera of Smooth Sections Under Certain W2 Lifting Assumptions

I’ve split this off from the bottom of one of the previous surface DIOP papers. I’m hopeful, haven’t managed to convince advisor yet. I’ll probably be expanding upon this in the next few weeks. The point is to attempt to get a positive characteristic analogue of Levine’s theorem in the case we have a lifting of a KLT pair (X, \Delta)

W2DIOP0.1_copy , hash

W2DIOP0.8 , hash

W2DIOP0.9 , hash

W2DIOP1.0 , hash : added an application of the main Theorem (an alternate proof to SurfDIOP main theorem).