Algebraic Geometry


These are interesting times to look over the algebraic geometry arxiv postings. Just over a week ago, there was a posting by Tanaka which claimed the minimal model program was false in characteristic two. Then yesterday at the top of the page was a paper by Castravet and Tevelev claiming that the Mori Dream Space conjecture for \overline{M_{0,n}} was false. Then today, there is a paper by Fontanari claiming instead that the Mori Dream Space conjecture is TRUE for the same space, but modded out by the finite group S_n.

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Posting is slowing down a bit, I’ve got a paper I’m trying to get out, and a couple of projects that are hitting some preliminary results, plus, I’m getting ready for holiday travel, and then two months at Humboldt.  Trying out an experiment with more rigid personal scheduling, and hopefully I’ll post more often.  Also, I’m reviewing Atiyah-Macdonald, Eisenbud, and Schenck so that perhaps in March I can begin a “Commutative Algebra from the Beginning” series, or perhaps just a series on geometric interpretation of commutative algebra theorems.

However, for today, we’re going to take something most of us first saw in differential geometry (I first met this map in do Carmo‘s book) and translate it into algebraic geometry.

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Now, we’re going to talk a bit about the geometry of the periods, which were completely analytic in nature.  As we mentioned, for a compact Riemann surface X, we have a period matrix \Omega that encodes the complex integration theory on the surface.

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Last time, we discussed integration theory of functions along paths on Riemann surfaces, and then we decided that we wanted to talk about compact Riemann surfaces.  Unfortunately, there aren’t any holomorphic functions on them, and meromorphic functions are the wrong choice about what to integrate along curves.  Today, we’ll talk about the correct things to integrate, and some of their properties.

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I’m back! And now, posting from Kavli IPMU in Japan.  Now, I’m going to start a series on theta functions, Jacobians, Pryms, and abelian varieties more generally, hopefully with some applications, with my goal being at least one post a week, and eventually establishing a regular posting schedule again.  But today, we’ll start with basics, something that should be completely understandable to graduate students and advanced undergraduates.

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Let {\pi:\tilde{C}\rightarrow C} be an unramified double cover, where {C} is geneus {g}. Then {\tilde{C}} has genus {2g-1} by the Riemann-Hurwitz formula. Now, {J(C)} encodes lots of information about the geometry of {C}, especially with the additional data of the theta divisor. It turns out that for double covers, there’s an abelian variety that contains a lot of this data.

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Today, we’re going to talk about a very important class of rational varieties, that show up all the time, in quite a variety of different contexts, and at the end, we’ll talk about why.

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