Category Archives: Abelian Varieties

Understanding Integration III: Jacobians

Posted on November 19, 2012 by Charles Siegel

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 →

Posted in Abelian Varieties, AG From the Beginning, Algebraic Geometry, Complex Analysis, Curves | 2 Comments

Understanding Integration I: Riemann Surfaces

Posted on October 1, 2012 by Charles Siegel

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 →

Posted in Abelian Varieties, AG From the Beginning, Algebraic Geometry, Complex Analysis, Curves | 5 Comments

Prym Varieties

Posted on November 19, 2010 by Charles Siegel

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

Posted on November 3, 2010 by Charles Siegel

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.

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

Subvarieties of Jacobians

Posted on November 2, 2010 by Charles Siegel

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 →

Posted in Abelian Varieties, AG From the Beginning, Algebraic Geometry, Curves | Leave a comment

Jacobians of Curves

Posted on November 1, 2010 by Charles Siegel

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 →

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

The Schottky Problem (ICTP)

Posted on July 1, 2010 by Charles Siegel

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

Riemann’s Bilinear Relations

Posted on April 19, 2010 by Charles Siegel

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

Posted on September 10, 2009 by Charles Siegel

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

Posted on September 9, 2009 by Charles Siegel

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 | Leave a comment