Monthly Archives: February 2009

The Grothendieck Group of Coherent Sheaves on a Variety

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 … Continue reading

Posted in AG From the Beginning, Algebraic Geometry | 12 Comments

Grassmannians, Redux

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 … Continue reading

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

Some new math blogs

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.

Posted in Uncategorized | Leave a comment

Direct Image Sheaves Under Proper Maps

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 … Continue reading

Posted in AG From the Beginning, Algebraic Geometry, Cohomology | 7 Comments

Deligne and Mumford on the Moduli of Curves

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 … Continue reading

Posted in Algebraic Geometry, Big Theorems, Curves, Deformation Theory | 3 Comments

The Dual Isogeny

Dear Readers, 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. … Continue reading

Posted in Uncategorized | 5 Comments

Proper Maps and (Quasi) Projective Varieties

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 … Continue reading

Posted in AG From the Beginning, Algebraic Geometry | 12 Comments

Elliptic Curves and Jacobians

Dear Readers, Today we begin by talking about elliptic curves. We define an elliptic curve to be an abstract nonsingular genus one curve over a field with a given rational point. I’m told the theory of genus one curves without … Continue reading

Posted in Abelian Varieties, Curves | 3 Comments

Wiki Troubles

So, I’ve been tinkering with having a local wiki on my laptop for notes and stuff, but I’ve run into the problem that, although pictures seem to be stored in the right location, and pages are created for them in … Continue reading

Posted in Uncategorized | 6 Comments

The Clemens Conjecture

This is a draft of a talk I’m giving on Thursday, mostly going over the work of Katz in this paper. Let’s begin with an informal statement of the conjecture: Conjecture: For every positive integer , a general quintic hypersurface … Continue reading

Posted in Algebraic Geometry, Curves, Enumerative Geometry, Hilbert Scheme, Intersection Theory, Talks | Leave a comment