Author Archives: Matt DeLand

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

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

The Chow Ring and Chern Classes

Posted on March 22, 2009 by Matt DeLand

Hopefully this will be the last background information post before we state and begin the proof of the Riemann Roch Theorem.  This post will be a brief overview of Cycles, Chow Rings, and Chern Classes and their properties.  The briefness … Continue reading →

Posted in Uncategorized | 3 Comments

The Grothendieck Group of Coherent Sheaves on a Variety

Posted on February 27, 2009 by Matt DeLand

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

Direct Image Sheaves Under Proper Maps

Posted on February 18, 2009 by Matt DeLand

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

Proper Maps and (Quasi) Projective Varieties

Posted on February 10, 2009 by Matt DeLand

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

When is a variety not rational?

Posted on November 6, 2008 by Matt DeLand

I also have an excuse for my absence… I’ve also been working on grant applications, job applications, my thesis, etc!  I’ll take a little time now to write more about rational varieties.  In general, it can be quite difficult to … Continue reading →

Posted in Algebraic Geometry | 2 Comments

Rational Varieties — An Introduction Through Quadrics

Posted on September 27, 2008 by Matt DeLand

Projective spaces are the most basic algebraic varieties we know (at least in some sense) and rational varieties are those that are as close as possible to being projective spaces.  Informally, a rational variety is one admitting a parametrization by projective space.  The project of determining which … Continue reading →

Posted in AG From the Beginning, Algebraic Geometry | 4 Comments

Group Schemes and Moduli (IV)

Posted on September 9, 2008 by Matt DeLand

Unless there is some specific interest, I think this will be the final post in the series about taking quotients of varieties by actions of group schemes.  Recall that everything is being done over an algebraically closed field, .  I’ll … Continue reading →

Posted in AG From the Beginning, Algebraic Geometry, Examples | 3 Comments

Group Schemes and Moduli (III)

Posted on September 1, 2008 by Matt DeLand

Finally we’re ready to discuss what sorts of quotients exist when group schemes act on (some) other schemes.  Recall that for simplicity, every scheme/variety/morphism in sight is assumed to be over the spectrum of a fixed algebraically closed field, . … Continue reading →

Posted in AG From the Beginning, Algebraic Geometry, Uncategorized | Leave a comment