Category Archives: Big Theorems

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

Posted in Algebraic Geometry, Big Theorems, Cohomology, Intersection Theory | 5 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

Kontsevich’s Formula

This is my last post of 2008, so happy holidays to everyone!. This one shouldn’t take too long, it’s just applying the lessons of the last few posts to compute some numbers. We start by talking like physicists. We’ll write … Continue reading

Posted in Algebraic Geometry, Big Theorems, Cohomology, Computational Methods, Curves, Enumerative Geometry | 6 Comments

Request: Projective Elimination Theory

We talked before about elimination theory, doing it entirely in the affine case. The question was asked about how to do it projectively. There are a couple of subtleties to it, but the idea is simple: we eliminate in each … Continue reading

Posted in AG From the Beginning, Algebraic Geometry, Big Theorems, Computational Methods | 2 Comments

Geometric Form of Riemann-Roch

Now, the way that the Riemann-Roch theorem was phrased before, the geometry wasn’t obvious. We had to extract it in terms of rational functions with poles given by a divisor. Now that we’ve talked about canonical curves, we can use … Continue reading

Posted in AG From the Beginning, Algebraic Geometry, Big Theorems, Curves | Leave a comment

Hurwitz’s Theorem on Automorphisms

This is this blog’s 100th post. Now, not quite my hundredth, nor nearly the hundredth with actual math content, but still, it’s a number which, when expressed in base ten, happens to have some zeros. More importantly, however, tomorrow marks … Continue reading

Posted in AG From the Beginning, Algebraic Geometry, Big Theorems, Curves | 1 Comment

Hurwitz’s Theorem

Ok, back to curves. We’d wandered a bit in the direction of this topic before, having discussed Bezout’s Theorem and the Riemann-Roch Theorem. Today we’ll talk about the Hurwitz formula, also called the Riemann-Hurwitz formula. It’s a rather nice result, … Continue reading

Posted in AG From the Beginning, Algebraic Geometry, Big Theorems, Curves | 1 Comment