## 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 … Continue reading

Posted in Algebraic Geometry, Big Theorems, Cohomology | 5 Comments

## 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 … Continue reading

## The Chow Ring and Chern Classes

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

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## 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

## 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

## 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

## When is a variety not rational?

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

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)

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