Rigorous Trivialities

Legalities Question

Ok, this might be a silly question, but I’m not sure if something I want to do is legal or illegal.So, I’ve got some books in pdf format that are obtained legally (ie, EGA as downloaded from Numdam) which are rather out of print. I’d li...

~ published: Tuesday at 13:45 ~ permalink

Group Schemes and Moduli (II)

We continue our quest to understand when quotients of schemes by actions of group schemes exist. Last time we defined group schemes, group actions, geometric quotients, and gave some examples. In this post, I’ll define what it means to be a reductiv...

~ published: 08/24 at 09:34 ~ permalink

Constructing Nodal Curves

So, I don’t really have a full length post in me at the moment. However, here’s a nice trick that I’ve learned recently, plus some motivation. With this, it’s easy to write down explicitly a curve of arbitrary degree in the plane...

~ published: 08/20 at 10:45 ~ permalink

New Theme

So, we’ve got a new theme. This is for one simple reason: the old one didn’t list the author of posts. This wasn’t an issue before, but today, just before this, is the first post from one of the new cobloggers, Matt DeLand, from Columb...

~ published: 08/19 at 15:52 ~ permalink

Group Schemes and Moduli (I)

Hi! My name is Matt DeLand, I’m a graduate student at Columbia and I’m responding to Charles’ call for cobloggers. I also study Algebraic Geometry, and have been enjoying Charles’ posts; hopefully I can help out and make some posit...

~ published: 08/19 at 15:37 ~ permalink

Request: Projective Elimination Theory

We talked before about elimination theory, doing it entirely in the affine case. The question was asked about how to do it projectively. There are a couple of subtleties to it, but the idea is simple: we eliminate in each affine chart and then glue toge...

~ published: 08/14 at 11:13 ~ permalink

Less posting for a bit

Ok, well, I’m going to be posting less, and hopefully settling into a regular schedule at some point, but the rest of the month is going to be erratic at best, with my orals being (in theory) the first week of September. I had been intending to wr...

~ published: 08/11 at 09:38 ~ permalink

Phylogenetics and Algebraic Geometry

Ok, this post is going a little bit out of my field. If anyone can fill in some of the gaps in my understanding of phylogenetics and in how to get from there to the math, please do so in the comments. So, anyway, phylogenetics is the study of how variou...

~ published: 08/07 at 10:03 ~ permalink

Geometric Form of Riemann-Roch

Now, the way that the Riemann-Roch theorem was phrased before, the geometry wasn’t obvious. We had to extract it in terms of rational functions with poles given by a divisor. Now that we’ve talked about canonical curves, we can use them to g...

~ published: 08/07 at 07:00 ~ permalink

Canonical Linear Systems

Given any curve , we have a natural linear system , consisting of all the effective canonical divisors. Now, sometimes this is unhelpful. After all, if , then the canonical system has negative degree, so . If , then the canonical system is trivial, and...

~ published: 08/06 at 07:00 ~ permalink

Plücker Formulas

Let be a curve and be the dual curve. We say that has traditional singularities if every point of and is smooth, a node or a cusp. That is, if we take affine coordinates around and get equation for , and look in the ring of [...]...

~ published: 08/05 at 07:00 ~ permalink

Book Club?

I’ve decided that I need a bit more fiction or non-technical nonfiction in my life, to counteract the intensive studying and such. Would any of my half dozen or so readers be interested in some sort of book club, likely nerdy, organized via this b...

~ published: 08/04 at 16:59 ~ permalink

Dual Curves

Today we think about plane curves. But first a bit on projective space. Recall that the Grassmannian is the parameter space of -planes in -dimensional space, and so . We noted at the time that . So that means that we also get a projective space . So ...

~ published: 08/04 at 09:39 ~ permalink

Hurwitz’s Theorem on Automorphisms

This is this blog’s 100th post. Now, not quite my hundredth, nor nearly the hundredth with actual math content, but still, it’s a number which, when expressed in base ten, happens to have some zeros. More importantly, however, tomorrow marks...

~ published: 08/01 at 09:03 ~ permalink

Hurwitz’s Theorem

Ok, back to curves. We’d wandered a bit in the direction of this topic before, having discussed Bezout’s Theorem and the Riemann-Roch Theorem. Today we’ll talk about the Hurwitz formula, also called the Riemann-Hurwitz formula. ItR...

~ published: 07/31 at 08:52 ~ permalink

Computing Hilbert Functions

Today, we’ll link the computational thread back to the thread involving Hilbert schemes, by working out how to compute the Hilbert function (and thus polynomial) for any ideal in the ring . The trick involves Groebner bases and flat families, and r...

~ published: 07/30 at 08:25 ~ permalink

Resultants

Let’s say that you have two polynomials, and you REALLY need to know if they have a common root. Now, if they’re quadratic, you’re in luck, because we can solve them both completely and just check. In fact, if you’re patient, you...

~ published: 07/29 at 06:00 ~ permalink

Elimination and Extension Theorems

Last time we talked about Groebner bases and Buchberger’s algorithm, so today we’ll do an application of them. In fact, a few, because the Elimination Theorem and the Extension Theorem are extremely useful results, and we’ll talk a bit ...

~ published: 07/28 at 09:10 ~ permalink

Carnival of Math 37

Well, I’m taking the day off, but instead of looking for my stuff, you can all go over to the Logic Nest and read the 37th Carnival of Mathematics. Hmm…I had been intending to write something for it, but it slipped my mind. Oh well....

~ published: 07/25 at 08:19 ~ permalink

Groebner Bases and Buchberger’s Algorithm

After all that technical work and careful but rather abstract technique developed for the construction of a bunch of moduli spaces, I’ve decided to take a break from that line of reasoning and to look to one that’s rather dear to me. It’...

~ published: 07/24 at 08:50 ~ permalink

Wordle

I’ve seen a lot of people playing with Wordle on their blogs lately, so here’s a wordle picture to represent the Algebraic Geometry from the Beginning series. I’ve removed some English words, removed plurals on a lot of words, and chang...

~ published: 07/23 at 20:17 ~ permalink

Examples of Moduli Spaces

Now we’re done constructing , so it’s time to get the general Hilbert scheme done, and then to construct some other moduli spaces. Now, as , we can see that is a subfunctor of , so we want to try to find a subscheme of to represent it.Now, ...

~ published: 07/23 at 09:41 ~ permalink

Nakayama’s Lemma

I promised a minipost on Nakayama when I talked about Flattening Stratifications, and I’ve got a moment now, so I’ll do it quickly. This post is all commutative algebra. So we’ll quickly state Nakyama:Nakayama’s Lemma: Let be an...

~ published: 07/22 at 11:30 ~ permalink

Dr. Horrible goes to Broadway?

This may just be me getting hopeful, but after reading this at the LA Times blog, and catching the following quote, I think I have reason:“We’re too busy talking about the giant Broadway adaptation, the much longer film version and the musical comment...

~ published: 07/22 at 11:02 ~ permalink

Flattening Stratifications

Now we move on to the next ingredient in the construction: flattening stratifications. We’ll start with just stating the theorem that Kollár used:Theorem: Let be a projective scheme and ample. Let be a coherent sheaf on . For every polynomial...

~ published: 07/22 at 10:22 ~ permalink

Constructing the Hilbert Scheme II

Last time we did a quick run through of how to put together the Hilbert Scheme. A few questions came up in the comments, the first being: how can we guarantee the existence of the we used, which works uniformly for ideals with a given Hilbert Polynomial...

~ published: 07/21 at 09:24 ~ permalink

The Hilbert Scheme

Now that we have a notion of moduli space, we’re going to look at several concrete examples. I mentioned that the Grassmannian is a fine moduli space for -planes in . We’ll make use of this to construct several more. First up on the list: t...

~ published: 07/18 at 08:58 ~ permalink

Moduli Spaces and Base Change

Last time we spoke of representable functors and talked about how to check if a functor is representable. The whole idea being that if we can first construct a functor that SHOULD be the functor of points for the scheme we want, and then we check represe...

~ published: 07/17 at 06:00 ~ permalink

Representable Functors

Before we can talk about moduli theory, we need to shift our methods of thinking a bit. So far we’ve thought only of geometric objects like varieties and schemes, and we’ve worked with “points.” However, the points of these object...

~ published: 07/16 at 07:34 ~ permalink

The Giant’s Shoulders 1

There’s a new Carnival around, and I’m in the first incarnation of it. Check out A Blog Around the Clock to see the first The Giant’s Shoulders. It’s all about classic papers in the sciences (and mathematics…and let’...

~ published: 07/16 at 05:46 ~ permalink

Supervillain Musical

By Joss Whedon. It’s fun. Enjoy....

~ published: 07/15 at 19:14 ~ permalink

New page!

I’ve created a new page where I’m listing all of the problems that my committee gives me to work on to study for my orals, and I’m striking out any problems as I finish them to their satisfaction. Now, I know some of you can solve them...

~ published: 07/15 at 15:33 ~ permalink

Flat Modules and Morphisms

Especially after yesterday, today’s going to feel very algebraic. That’s because we’re gearing up and pulling out the tools we’ll need to discuss one of the major success stories of algebraic geometry - moduli space theory. The p...

~ published: 07/15 at 09:10 ~ permalink

Fairly new blog

I just came across a fairly new mathblog, Mathematics Prelims, in which a student is studying for preliminary exams for his university. He’s doing analysis stuff right now. Go on and check it out....

~ published: 07/14 at 22:35 ~ permalink

Bertini’s Theorem

Today we’re going to prove a theorem of Bertini’s. Next time, we’ll focus on using the theorem to do something that’s even more geometric. For now, we’re only going to prove the theorem in characteristic zero. In fact, we&...

~ published: 07/14 at 08:52 ~ permalink

Carnival of Mathematics

Welcome to the 36th Carnival of Mathematics. I think I’m supposed to have something cute to say about the number 36, so we’ll start with something from my personal background. Though I don’t claim to be one of them (or believe in it in...

~ published: 07/11 at 07:44 ~ permalink

Begging for Submissions

Ok, I know I’m harping on this, but I really need submissions to the Carnival, which I’m posting Friday morning. That means I need them Thursday night. Please either go to the automated submission page on Blog Carnivals, or email submissio...

~ published: 07/09 at 19:17 ~ permalink

Blowing things up

In my post on Resolution of Singularities, I talked about three means of resolving singularities of varieties. One, normalization, which resolves codimension one singularities, has been covered. Now, we’re moving on to the real heavy guns. Hirona...

~ published: 07/09 at 10:01 ~ permalink

Social Networking Site

I’ve just joined a social networking site for grad students and postdocs, called the Graduate Junction. Looks like it could be really good, probably more useful to me than that facebook thing. I also just created a group for “Algebraic Geo...

~ published: 07/08 at 10:56 ~ permalink

Normalization and Normal Varieties

I got four people that seemed more-or-less to want me to keep going with these, so I have decided to do it, and not just because it means I have less work to do regarding finding a topic for each post. Anyway, on my post which was an overview of resoluti...

~ published: 07/08 at 09:58 ~ permalink