Category Archives: Big Theorems

Monodromy and Moduli

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

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

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

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

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

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

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 | 4 Comments