Category Archives: Examples

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

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

Rational Normal Scrolls

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.

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Determinantal Varieties

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

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Low Genus Moduli of Curves

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 | 3 Comments

A short post on bitangents

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)

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)

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