### February 2009

Ok, to continue our quest toward the statement of the theorem, we need to explain a few more ideas.  Today, we’ll talk about the structure of the Grothendieck group K(X) of coherent sheaves on an algebraic variety X.  (Remember the ongoing convention that varieties are all (quasi) projective, defined over an algebraically closed field).  In fact, this group has a ring structure and, given a map $f: X \rightarrow Y$, there are maps between K(X) and K(Y) (in both directions).  It turns out that the most general statement of the Riemann Roch Theorem is an equality inside this group, so we had better have an idea of how it works!  There are some homological methods today, a topic which seems to me particularly ill-suited to this forum given my difficulty creating complicated commutative diagrams.  At a few instances I will sketch how certain facts are to be proven without going into all the details.

Ok, today we start our march towards Schubert Calculus.  Before we start, we’ll review the Grassmannian variety itself, because it’s central to the story.  A lot of this will consist of setting up notation, and there will be two different notations for the “same” Grassmannians, so the notation will describe how we’re going to think of the object.  For this series, we’ll assume that everything is done over $\mathbb{C}$, because we’re going to use a bit of algebraic topology along the way.

Clicking around wordpress, I’ve found two new math blogs that seem to have started in the last couple of months.  So, go and check out Motivic Stuff and Embûches tissues and welcome them to the blathosphere.

Today we continue our review/introduction of background material en route to stating and proving the Riemann Roch theorem. This theorem involves the relationship between proper maps and sheaves on the domain and target, so we need to understand how they are related. Really, we need to understand how to move from sheaves on the domain to sheaves on the target – so we’ll explain the (higher) direct image sheaves. Recall that all varieties are (quasi) projective and defined over algebraically closed fields.

Today I’m going to talk a bit about an important paper from 1969.  This one.  It’s a bit hard to read at some points, but it was revolutionary.  In it, Pierre Deligne and David Mumford prove that the moduli space of curves is always irreducible.   They proved this fact in two ways: first by using the Stable Reduction Theorem, which is a hard theorem on Abelian varieties.  The second way was by using stacks.

Today I want to talk about morphisms between curves and the special properties they have. This is sort of a broad-based topic, given that the study of morphisms and their properties is essentially the whole of algebraic geometry. I decided that perhaps the best way to narrow my focus is to push towards proving that if $\phi: E_1 \to E_2$ is a non-constant isogeny of elliptic curves over a field $k$ with a kernel of order $n$ then there exists a dual isogeny (more…)

I’m back!  So let’s get down to business.

What I really want to talk about is the Riemann Roch Theorem.  You may wonder why, since it has already been discussed here , but there have been vast generalizations of this theorem throughout the last century.  I want to focus on Grothendieck’s version, but I plan to start slow.  The material I’m following will mostly be taken from the classic Serre-Borel paper (recently pointed out to me).  Today, we’ll content ourselves with talking about proper maps.   I know that Charles already talked about a complete variety here and even proved Chow’s Lemma which is great.  It tells us, that all complete varieties are “close” to being projective, so today we’ll focus on the projective ones.

Today we begin by talking about elliptic curves. We define an elliptic curve to be an abstract nonsingular genus one curve over a field $k$ with a given rational point. I’m told the theory of genus one curves without rational points is perhaps even more interesting than that of elliptic curves, but that may or may not be a topic for the future(when I know more about them)
Conjecture: For every positive integer $d$, a general quintic hypersurface in $\mathbb{P}^4$ will contain only finitely many rational curves of degree $d$.