Category Archives: Abelian Varieties

Understanding Integration III: Jacobians

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 , we have a period matrix that encodes the complex integration theory on … Continue reading

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Understanding Integration I: Riemann Surfaces

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 … Continue reading

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Prym Varieties

Let be an unramified double cover, where is geneus . Then has genus by the Riemann-Hurwitz formula. Now, encodes lots of information about the geometry of , especially with the additional data of the theta divisor. It turns out that … Continue reading

Posted in Abelian Varieties, AG From the Beginning, Algebraic Geometry, Cohomology, Curves, Examples, Hodge Theory, MaBloWriMo | Leave a comment

Theta Characteristics and Quadrics in Characteristic Two

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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Subvarieties of Jacobians

For this whole post, we’ll take to be a curve and the Jacobian of the curve.  We’re going to construct several special subvarieties (not special in any technical sense, though) of , which encode a great deal of geometric information … Continue reading

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Jacobians of Curves

As promised in the last post, I’m making another go at MaBloWriMo…maybe others will as well.  I don’t know if I’m going to have a coherent topic over the course of this month, but I’ll be starting with abelian varieties … Continue reading

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The Schottky Problem (ICTP)

These are my notes, and are only a rough approximation of the actual talk:

Posted in Abelian Varieties, Conferences, Curves, Hodge Theory, ICTP Summer School, Moduli of Curves | 1 Comment