### About this blog

Rigorous trivialities is a web log about mathematics, but especially geometry, broadly construed. Contributors will be Charles Siegel, Jim Stankewicz and occasionally Matt Deland. Charles specializes in algebraic geometry, topology and mathematical physics. Jim specializes in arithmetic algebraic geometry. Matt has transitioned from algebraic geometry to work in industry.

Header is taken from the larger work by fdecomite under the creative commons license.

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# 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

## 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

## 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

## 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.

## 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

## 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

## The Schottky Problem (ICTP)

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

## Riemann’s Bilinear Relations

This post begins my series on some classical geometry of curves and abelian varieties. We’ll start with some talk of line bundles and polarizations on abelian varieties in general, and the first big theorem I’m really targeting is the Torelli … Continue reading

Posted in Abelian Varieties, Algebraic Geometry
2 Comments

## B-N-R Part 5: Spectral Curves

Today we’re back to some material from the first post in this series, and going to prove an actual theorem about vector bundles. Next time, we’ll be getting into the heart of the paper, and that may be my last … Continue reading

Posted in Abelian Varieties, Big Theorems, Curves, Vector Bundles
4 Comments

## B-N-R Part 4: Prym Varieties

The last post was on the generalities of Abelian varieties, and constructing a map. This time, we’re going to do it for a specific one, and the maps involved will all be useful later. We start out with a finite … Continue reading

Posted in Abelian Varieties, Big Theorems, Curves, Vector Bundles
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