Category Archives: Big Theorems

Monodromy and Moduli

Posted on March 2, 2010 by Charles Siegel

Today, we’re going to prove a BIG theorem, but only in the characteristic zero case (we’ll be working over as usual). The theorem is rather tough, and to do it in positive characteristics it’s best done through stacks. Specifically, we’ll … Continue reading →

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

B-N-R Part 5: Spectral Curves

Posted on September 10, 2009 by Charles Siegel

Today we’re back to some material from the first post in this series, and going to prove an actual theorem about vector bundles.  Next time, we’ll be getting into the heart of the paper, and that may be my last … Continue reading →

Posted in Abelian Varieties, Big Theorems, Curves, Vector Bundles | 4 Comments

B-N-R Part 4: Prym Varieties

Posted on September 9, 2009 by Charles Siegel

The last post was on the generalities of Abelian varieties, and constructing a map.  This time, we’re going to do it for a specific one, and the maps involved will all be useful later.  We start out with a finite … Continue reading →

Posted in Abelian Varieties, Big Theorems, Curves, Vector Bundles | Leave a comment

B-N-R Part 3: PPAV’s and some details

Posted on September 9, 2009 by Charles Siegel

Now, we continue our tour through Beauville, Narasimhan, Ramanan.  We’ve talked about Twisted Endomorphisms and we’ve talked about the Generalized Theta Divisor on the Moduli Space of Vector Bundles.  So today we’ll talk a bit about Abelian Varieties, Principal Polarizations, … Continue reading →

Posted in Abelian Varieties, Algebraic Geometry, Big Theorems, Curves, Vector Bundles | 2 Comments

B-N-R Part 2: Moduli of Vector Bundles

Posted on August 25, 2009 by Charles Siegel

Last time, we talked about twisted endomorphisms.  Now, we’re moving on to the second paragraph of the paper: generalized theta divisors.  In the meantime, we’re going to have to talk a bit about vector bundles and their moduli.

Posted in Abelian Varieties, Big Theorems, Curves, Vector Bundles | 2 Comments

B-N-R Part 1: Twisted Endomorphisms

Posted on August 24, 2009 by Charles Siegel

Alright, I’m back, and newly married (thus the long hiatus from posting).  And now it’s time to get back to math.  I’m currently attempting to read a paper by Beauville, Narasimhan and Ramanan titled “Spectral Curves and the Generalized Theta … Continue reading →

Posted in Abelian Varieties, Big Theorems, Curves, Vector Bundles | 5 Comments

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

Deligne and Mumford on the Moduli of Curves

Posted on February 18, 2009 by Charles Siegel

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

Posted on December 23, 2008 by Charles Siegel

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