Dedekind Domains, Krull-Akizuki and the Picard Group

As has been hinted in many previous posts, many facts about algebraic number theory tell us about geometric objects like elliptic curves. For instance, if you are working on a problem which primarily uses the affine geometry of a curve like the semistable reduction theorem for elliptic curves, the scheme you’re working on is opposite to what’s called a Dedekind Domain. We begin a series of posts on Dedekind Domains, beginning today with the very abstract and progressing to the concrete(which would of course be terrible for teaching this material but I mean these posts as more of a reference work).

 

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 “easy” in the sense that the most is made of a relatively simple computation on projecive space.  Last time we saw that it is enough to prove the formula separately for an injection and a projection.  We’ll see here how to carry these two steps through and how the first may be reduced to the inclusion of a divisor.  Though last time I said that I wanted to go into each step in more detail, I realized that 1) probably very few people are (still?) following along, 2) for those who are, they will get more by seeing an outline and reading the paper or looking at Fulton’s Intersection Theory themselves, and 3) this way we can illustrate the power of the theorem with some applications.

 

Summer School: Day 1

As I mentioned, I’m participating in a summer school on the Geometry of Quantum Fields at Penn.  I’m in Katrin Wendland’s mentoring session this week, which means conformal fields and vertex algebras.

 

The next two weeks

I know I’ve been fairly bad about posting recently. Started teaching my first course. But that SHOULD end on Monday. Not the course, the silence. That’s when the Geometry of Quantum Fields summer school starts here at Penn.  For the first week, I’m going to be attempting to learn what a Conformal Field is with Katrin Wendland, and I’ll be attempting to blog about it.  The next week, I’m with Eric Sharpe, talking about Heterotic Compactifications and Quantum Cohomology.  The posts these next two weeks will be rather technical, but afterwards, I’ll probably attempt to distill them and provide some background and motivation beyond whatever else is covered.  Might also blog on some of the talks, but those are the two mentoring sessions I’m in, so they’ll be the most in-depth.

 

Endomorphisms of Elliptic Curves and the Tate module

Dear Readers,

We’ve now talked about quaternion algebras, and today I’ll talk about the surprisingly close connection between quaternion algebras and elliptic curves.

We first recall a fundamental fact about isogenies:

 

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 Grassmannians, so what we’re going to do is talk about what the word “generic” means in some very precise contexts, and then we’re going to count a bunch of things.

 

Quaternion Algebras and Modular Forms

Dear readers,

I know I promised a post  on modular curves, but I had to devote more time to my end of semester project. Since it’s strongly related to the topic of modular curves and I present on it tomorrow, I decided to make this post in the seminar talks category.

 

Pieri and Giambelli Formulas

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 cases of the Pieri rule and the Giambelli formula.

 

The Grothendieck-Riemann-Roch Theorem, Stated

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 you could do.  The first thing is that you could push forward this element to the Grothendieck Group on , and then take its chern character, to arrive at an element in the Chow Ring of .  The second option you have is to first take the chern character of in the Chow Ring of , and then push it forward to the Chow Ring of .  Now we have two elements in the Chow Ring of and we could wonder if we have the same element.  The Grothendieck Riemann Roch (GRR) Theorem tells us we don’t quite have the same element, but it tells us exactly by how much we are off!

 

Schubert Classes and Cellular Cohomology

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 work out the cup product in this cohomology ring, and then finally we’ll use it to solve some problems.  So today, we’ll discuss Cellular Homology, Poincaré Duality, and the cohomology of the Grassmannian.