Category Archives: MaBloWriMo

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

The Stanley-Reisner Ring

Posted on November 8, 2010 by Charles Siegel

Today, we’re going to do something completely different, but which most of my peers seem not to have seen, but is a very cool application of algebraic geometry.

Posted in AG From the Beginning, Algebraic Geometry, Cohomology, MaBloWriMo, Toric Geometry | 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

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

Everything is a Normal Cone

Posted on November 13, 2009 by Charles Siegel

Well, really, for intersection theory, it’s true.  We start with a closed subscheme, with normal cone .  We’re going to construct a family of embeddings that deforms to the zero section of .  Then, because intersections should vary nicely in … Continue reading →

Posted in Intersection Theory, MaBloWriMo | 3 Comments

Segre Classes of Subschemes and their Applications

Posted on November 11, 2009 by Charles Siegel

So, last time we talked about Segre classes and cones.  Now, we’re going to move ahead, and talk about a specific cone in detail, the Normal cone we defined on Monday.  Let be a subscheme, and let be its normal … Continue reading →

Posted in Intersection Theory, MaBloWriMo | 2 Comments