Algebraic Invariance of Plurigenera via minimal models in large, non-general type Kodaira dimensions. — April 9, 2015

Algebraic Invariance of Plurigenera via minimal models in large, non-general type Kodaira dimensions.

(This is just rewriting my recent edit as it’s own post). Basically I’ve combined some proofs from the past year and a half into one write-up. (algebraic) KLT Deformation Invariance of Plurigenera for Kodaira dimension >= dim(X) – 3: First I remark that klt abundance should already hold in that case using some previous results (Hacon / Lai ?), then I use either adjoint ideals or Hodge Theory to derive the desired theorem on plurigenera which so far is only known to hold using the techniques of complex analysis. Needs some editing / proofreading (April 6, 2015)

DIOPKLT3_copy0.2_copy , hash

edit: The above link somehow went down for a bit, but seems to be working again!

(July 9 ) Below is my original attempt for all non-negative kodaira dimensions (from last summer, although recently cleaned up) which my advisor doesn’t like but has so far failed to convince me is incorrect:

DIOPKLT3_copy0.7 , hash

DIOPKLT3_copy0.8 , hash

DIOPKLT3_copy0.9 , hash

random update — June 19, 2014

random update

Just a random update. I’ve been working on some problems in birational geometry this past year. There were a few that were too small to publish and now I’m working on another that might be? enough to get a paper out of. In any case, that’s my goal right now, to get a paper haha. In any case, it seems like I’ve been improving my research methods and learning some new techniques in algebraic geometry, so that’s good I guess. In other news, as part of my second year of RTG fellowship, I’m helping the first year grad students prepare for qualifying exams which has been fun, going over those types of nice, cut-and-dry problems they usually put on those exams. (June 19?)

edit: (Nov 10) Well that problem I was working on has I was able to find a solution about a month ago. Unfortunately, the intended application that someone else was working on didn’t quite pan out. However, since the proof seems somewhat interesting, maybe it will be written up as a preprint sometime.

edit2: (Nov 16) and here are a couple of special cases to an algebraic proof of Siu’s Deformation Theorem: (DIOPKLTProofOfExistence and an earlier one: TheoremCBetter5.1. For fun, I hashed these into the bitcoin blockchain…

edit3: (April 6, 2015) I’ve just combined the above. Needs a big check for mistakes. Proof.

Hartshorne Solutions — September 16, 2013

Hartshorne Solutions

October 3 Version

Well, I’ve been up all weekend typing, yet here they are: (I’ve taken then down for now in case, if you need it, please e-mail me).
all the solutions to Hartshorne.

edit (2/2/2015): I was looking through (related to a problem I’m working on) and found an error on the hodge inequality for surfaces -> here’s the real way, based off an exercise in another book: you can find m,n in Z such that D1 (mD1+nD2)=0 so that both (mD1+nD2)^{2} <= 0 by hodge index, and also mD1^{2}=-nD1D2. Now expand the square <= 0 equation, make substitutions, and cancel.

goal date for hartshorne problems — July 24, 2013

goal date for hartshorne problems

So I’ve been working on learning theorems for the past few weeks and probably the next few weeks. I think around September 15th seems like a good goal date to try and finish typing those Hartshorne problems from earlier this summer.

Some very rough notes on extension theorems — July 5, 2013
After many hours of pain — May 29, 2013
Interesting — May 20, 2013
Script for Math drawing — April 29, 2013

Script for Math drawing

Here is a quick and dirty script for drawing math pictures I use when solving problems. An example picture:

Usually I use this script attached to a hotkey in conjunction with pinta (basically MS paint) on linux. It gives a prompt where you can enter latex. Then it puts the latex into your clipboard as a png file so you can paste it into paint. The end result? paint with latex. The reason I wrote this is because the inkscape / latex plugin I used to use was too resource heavy and slow for when solving problems.

a = io.popen(“zenity –entry”):read(“*a”)
b = string.gsub(a, “\n”, “”)
os.execute(“/home/andrew/mathtex.cgi \””..b..”\” -o textmp”)
os.execute(“python /home/andrew/”)

Also you will need (I found the bulk of this script somewhere else on the web a long time ago and have forgotten the source):
#! /usr/bin/python

import pygtk
import gtk
import os
import sys

def copy_image(f):
assert os.path.exists(f), “file does not exist”
image = gtk.gdk.pixbuf_new_from_file(f)

clipboard = gtk.clipboard_get()


Chern Classes and Hodge Bundles — February 19, 2013
Hurwitz schemes, Hilbert Numbers, and Severi varieties — January 16, 2013