Category Archives: Examples

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

Rational Normal Scrolls

Posted on November 17, 2010 by Charles Siegel

Today, we’re going to talk about a very important class of rational varieties, that show up all the time, in quite a variety of different contexts, and at the end, we’ll talk about why.

Posted in AG From the Beginning, Algebraic Geometry, Examples, MaBloWriMo | Leave a comment

Determinantal Varieties

Posted on November 15, 2010 by Charles Siegel

Let be the projectivization of . Then for all , we have a variety given as the set of matrices of rank at most , which is given by the vanishing of the determinents of minors. We call these the … Continue reading →

Posted in AG From the Beginning, Algebraic Geometry, Examples, 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

Group Schemes and Moduli (IV)

Posted on September 9, 2008 by Matt DeLand

Unless there is some specific interest, I think this will be the final post in the series about taking quotients of varieties by actions of group schemes.  Recall that everything is being done over an algebraically closed field, .  I’ll … Continue reading →

Posted in AG From the Beginning, Algebraic Geometry, Examples | 3 Comments

Group Schemes and Moduli (II)

Posted on August 24, 2008 by Matt DeLand

We continue our quest to understand when quotients of schemes by actions of group schemes exist. Last time we defined group schemes, group actions, geometric quotients, and gave some examples. In this post, I’ll define what it means to be … Continue reading →

Posted in AG From the Beginning, Algebraic Geometry, Examples | 6 Comments

Constructing Nodal Curves

Posted on August 20, 2008 by Charles Siegel

So, I don’t really have a full length post in me at the moment. However, here’s a nice trick that I’ve learned recently, plus some motivation. With this, it’s easy to write down explicitly a curve of arbitrary degree in … Continue reading →

Posted in Algebraic Geometry, Curves, Examples | 11 Comments

Group Schemes and Moduli (I)

Posted on August 19, 2008 by Matt DeLand

Hi! My name is Matt DeLand, I’m a graduate student at Columbia and I’m responding to Charles’ call for cobloggers. I also study Algebraic Geometry, and have been enjoying Charles’ posts; hopefully I can help out and make some positive … Continue reading →

Posted in AG From the Beginning, Algebraic Geometry, Examples | 5 Comments

Canonical Linear Systems

Posted on August 6, 2008 by Charles Siegel

Given any curve , we have a natural linear system , consisting of all the effective canonical divisors. Now, sometimes this is unhelpful. After all, if , then the canonical system has negative degree, so . If , then the … Continue reading →

Posted in AG From the Beginning, Algebraic Geometry, Curves, Examples | 5 Comments