### AG From the Beginning

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.

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.

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.

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.

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.

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.

Let ${M=\mathbb{P}^{nm-1}}$ be the projectivization of ${\mathrm{Mat}_{n\times m}(\mathbb{C})}$. Then for all ${k}$, we have a variety ${M_k}$ given as the set of matrices of rank at most ${k}$, which is given by the vanishing of the determinents of ${(k+1)\times (k+1)}$ minors. We call these the generic determinantal varieties.

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