January 2008


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 X a topological space and take \mathscr{F},\mathscr{G} to be two presheaves on X. A morphism of presheaves is an abelian group homomorphism \phi(U):\mathscr{F}(U)\to\mathscr{G}(U) for each U\subset X open, that satisfies an extra condition.

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Second in our survey of some technical tools are sheaves. Sheaves have a reputation for being terrifying and technical and some people I know have trouble seeing their purpose at all. In my opinion, they’re very nice, convenient ways to organize information. Sheaves do require a bit of topology to use, and I recommend John Armstrong’s exposition as a basic reference. I talked a bit about the topology relevant in algebraic geometry (which is rather different than that in most point set topology settings, note that we make use of the concept of irreducibility quite often) but the basics and the notion of connectedness are both still rather important.

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Now that I’m settling into the semester, it’s time that I get back to rambling semi-coherently about algebraic geometry. I do, however, now have a somewhat more focused goal, and though it’s still quite a ways off, it has reminded me of important machinery that needs introduction. So for a bit, I’ll be talking more about algebra and some other tools of the trade and putting the geometry itself on the back burner for a bit (which isn’t to say that I’m abandoning it completely).

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The semester is starting up again now, so my schedule is a bit irregular. Sadly, I won’t be able to keep up the pace I set over break. However, I should still be putting out a couple of posts a week. So without further ado, tangent spaces. We’ve looked at tangent spaces before, and used them to define what it means for a point to be smooth versus singular. Today, we’re going to start back at the beginning of this stuff and justify the definitions a little.

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Back in the second post, we defined projective space to be the collection of lines through the origin in affine space. A natural generalization is to look at r-planes through the origin in affine space. At first glance, we might think that these spaces are legitimate generalizations of projective space, and that it might be valuable to do algebraic geometry on them.

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Today, we talk about the Segre Embedding. This will let us say a bit more about the products of varieties that were mentioned before.

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Now we’ve got some basic theory under our belts. Time to get a handle on specific objects. Nothing gives better intuition into algebraic geometry than working out a bunch of examples and seeing how they fit together. We’ll begin with the Veronese Embedding, which will give us a bunch of varieties that are isomorphic to projective space.

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We’ve now spoken about all sorts of invariants, the most important ones being derived from the Hilbert Polynomial, we’ve spoken about Bezout’s Theorem, and we’ve developed a nice bit of classical algebraic geometry. We’ve also mentioned algebraic groups, and it might be a good idea to refresh this stuff, because now we’ll return to it, and build up some more examples.

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Now we’re ready for our first big geometric theorem. We’ll prove Bezout’s Theorem as a special case of another, more general theorem.

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