### Curves

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.

Pretty much everything in this post is in Mumford’s “Curves and their Jacobians,” but I do a couple of things slightly differently, and I intend to supply a bit more detail in some places.  The goal here is to construct the moduli of curves of genus $g$ for small $g$.

Today’s post will be, as the title says, a bit short.  It will, more-or-less finish our current discussion of theta characteristics, and then we’ll get back to something else.  But we’ll derive a nice case of a classical formula.

Last time we defined theta characteristics as square roots of the canonical bundle.  Today, we’re going to analyze the notion a bit, and relate them to quadrics in characteristic two.

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