Category Archives: Curves

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 II: 1-Forms and Periods

Posted on October 15, 2012 by Charles Siegel

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 … Continue reading →

Posted in AG From the Beginning, Algebraic Geometry, Complex Analysis, Curves | Leave a comment

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

Low Genus Moduli of Curves

Posted on November 9, 2010 by Charles Siegel

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 … Continue reading →

Posted in AG From the Beginning, Algebraic Geometry, Curves, Examples, MaBloWriMo, Moduli of Curves | 4 Comments

A short post on bitangents

Posted on November 4, 2010 by Charles Siegel

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.

Posted in AG From the Beginning, Algebraic Geometry, Curves, Enumerative Geometry, Examples, MaBloWriMo | 1 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 →

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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