Category Archives: Cohomology

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

The Stanley-Reisner Ring

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

Monodromy Representations

A departure from directly working with varieties, we’re going to do something that’s strictly topological (at first glance) but which really has deep and important connections with Hodge theory. We’re going to talk about monodromy and monodromy representations. Let be … Continue reading

Posted in Algebraic Geometry, Algebraic Topology, Cohomology, Hodge Theory, Vector Bundles | 7 Comments

The Hodge Theorem

Previously, we talked a bit about the category of Hodge structures, and did some basic constructions.  However, I’d claimed that this was algebraic geometry (at least, in the categories on the post) so today, we’ll talk about a LOT of … Continue reading

Posted in Algebraic Geometry, Cohomology, Hodge Theory | 2 Comments

Hodge Structures

Back to blogging for a bit, though likely infrequently.  Doing a new series that might count as AG from the beginning, so I’ll put it up there once I’ve got a couple done.  We’re going to start doing some Hodge … Continue reading

Posted in AG From the Beginning, Algebraic Geometry, Cohomology, Hodge Theory | 2 Comments

The Grothendieck-Riemann-Roch Theorem, a proof-sketch

By this time I’m sure everyone whose curiousity was piqued by the statement of the Grothendieck-Riemann-Roch theorem has read it themselves. Nevertheless, in case you haven’t, I will proceed to outline the steps of the surprisingly “easy” proof.  It is … Continue reading

Posted in Algebraic Geometry, Big Theorems, Cohomology | 4 Comments

Applications of the Schubert Calculus

Ok, this is going to be my last post in enumerative geometry for a while, as I’m kind of drifting away from the subject.  However, this one will be fun.  We’ve already established the structure of the cohomology ring for … Continue reading

Posted in Algebraic Geometry, Algebraic Topology, Cohomology, Combinatorics, Computational Methods, Enumerative Geometry, Intersection Theory, Uncategorized | Leave a comment