Category Archives: Cohomology

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

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

Monodromy Representations

Posted on February 25, 2010 by Charles Siegel

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 | 9 Comments

The Hodge Theorem

Posted on February 22, 2010 by Charles Siegel

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

Posted on February 4, 2010 by Charles Siegel

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

Posted on June 9, 2009 by Matt DeLand

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 | 5 Comments

Applications of the Schubert Calculus

Posted on May 12, 2009 by Charles Siegel

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

Pieri and Giambelli Formulas

Posted on April 20, 2009 by Charles Siegel

It’s been a few weeks, but now I’m back and today we’ll talk about the multiplication in the cohomology ring of Grassmannians.  Though we won’t talk about the Littlewood-Richardson rule in its full glory, we will howver discuss the special … Continue reading →

Posted in Algebraic Geometry, Algebraic Topology, Cohomology, Combinatorics, Computational Methods, Enumerative Geometry, Intersection Theory, Uncategorized | 1 Comment

The Grothendieck-Riemann-Roch Theorem, Stated

Posted on April 5, 2009 by Matt DeLand

Suppose you have a proper map between smooth (quasi) projective varieties.  Then suppose you have a coherent sheaf on .  After viewing that sheaf as an element of the Grothendieck Group of coherent sheaves on , there are two things … Continue reading →

Posted in Algebraic Geometry, Big Theorems, Cohomology, Intersection Theory | 6 Comments

Schubert Classes and Cellular Cohomology

Posted on March 27, 2009 by Charles Siegel

So, as of the last post in the series, we defined Schubert cells.  We’re going to use them to discuss the Cohomology of the Grassmannian, and to write down an explicit basis.  With an eye looking forward, next time, we’ll … Continue reading →

Posted in Algebraic Geometry, Algebraic Topology, Cohomology, Combinatorics, Computational Methods, Enumerative Geometry, Intersection Theory, Uncategorized | 6 Comments