Rigorous Trivialities

Now, I highly doubt that I have any readers that don’t read Terry Tao’s blog, but regardless, please sign his petition (here). We need to fight threats to math education everywhere, and this is a big one. I’m behind Terry 100% on this one. So really, please head over and leave a comment expressing your support.

First off, I’d like to make a correction to the definition of an abstract variety: we’re going to need to assume both that is irreducible and that it is covered by finitely many affine varieties. I had both of these conditions in the back of my head when I wrote that, and fortunately the issues were brought up in the comments.

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WordPress is giving me the same troubles it gave John earlier, so if you spot tex that has lost its backslashes, please let me know and I’ll correct the post.

I won’t be posting much this week, unless I make an attempt at blogging about the Derived Categories conference at IAS in Princeton.  Hopefully I’ll understand something (the first few talks have encouraging titles…the later ones, not so much.)  Well, that’s all.

Last time, we defined locally ringed spaces and I linked back to an old post where I defined the notion of an abstract variety. We’re going to talk about these objects a bit more today, including, in particular, checking that our old quasi-projective varieties are all abstract varieties. We’re heading for another nice theorem over the next couple of posts, so these next few will be rather pointed.

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Sorry about February, got rather caught up in schoolwork. My posts should resume, though still not with the frequency they used to. So, anyway, we now have sheaves and their morphisms, and even some morphisms that we get when we have continuous maps of our spaces. Today, we’re going to put it all together and look at spaces with a special sheaf of rings.

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Ok, here’s my second talk. This one went a bit heavier on the technical stuff, and is mostly out of Geometric Invariant Theory by Mumford and Fogarty (I have access to the second edition). At some point in my Algebraic Geometry from the Beginning series, I’ll try to get to explaining all of the terms used here. Also, I’ll be getting back to that next week (hopefully) now that a big pile of my commitments have been resolved.  Also, this is technically the title of the talk, as you will shortly notice, I didn’t really stick to the topic I was intending to.

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Here are my lecture notes for a talk I gave yesterday on invariants of finite groups in the graduate student algebra seminar here. Next week I’m talking about quotients of varieties by finite groups, and I’ll post those here as well. As a side note, does anyone know how to get good commutative diagrams in the parts of Latex that are available for use on wordpress?

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Today, rather than posting in my Algebraic Geometry from the Beginning series, I’m just going to do a quick post on a few things that are going on mathematically. First off, by clicking through Kung Fu Physics’s Super Holiday Post because it linked here, I found the Academic Reader. It’s more aimed at physics than math, but it still seems nifty, and looks like it might be a bit more suited to my paper organizing and reading needs than Google Reader (which I do make massive use of).

On the topic of papers, I’m about to start reading a couple of papers about Non-Reductive Geometric Invariant Theory. One of them popped up on the Arxiv today, on Quotients by non-reductive algebraic group actions by Kirwan. Fortunately, the abstract has a link to an older paper, Towards non-reductive geometric invariant theory by Doran and Kirwan last year. I’m excited about these because I’m organizing the Graduate Student Algebra Seminar here at Penn this semester, and we decided that our topic was going to be algebraic groups and invariant theory. I’m giving a couple of the early talks focusing on invariants of finite groups, and I’ll probably post my talks here after I give them.

Additionally, I’m about to start reading Meet homological mirror symmetry by Ballard because I’ve been wanting to figure out what this stuff is, and I think now I’m getting to the place where I have a chance at understanding it, being in my second semester of Complex Algebraic Geometry, which is focusing on open problems (Geometric Langlands, Hodge Conjecture, and Mirror Symmetry seem to be the ones we’re focusing on), I’m studying Hartshorne and doing the problems in it whenever I have spare time to prep for my orals in the Fall, and I’m doing a reading course on Deformation Theory, starting with Manetti’s Lectures on Deformations of Complex Manifolds. Overall, I’ve got quite a semester in front of me, and I’ll keep updating here, well…whenever it crosses my mind or whenever I want to post but don’t feel like doing anything technical.

We’ve got sheaves now, so naturally we move on to morphisms between them. We begin by fixing a topological space and take to be two presheaves on . A morphism of presheaves is an abelian group homomorphism for each open, that satisfies an extra condition. (more…)