rope up — October 19, 2012

rope up

I have a new challenge for myself… since we are getting towards the end of Hartshorne’s book in class, and based on what I learned over fall break (not finished / proofread) and based on what I need for commutative algebra class:
1. Finish my Hartshorne notes chapters II.6, II.7, III, IV, V
2. Write up of Langlands-Tunnell
3. Read / Take Notes on “Hyman Bass and Ubiquity: Gorenstein Rings”, by Huneke
To be posted: November 15

So Hartshorne starts out defining intersections for divisors \left(C.D\right) on surfaces. The idea is that for divisors which are curves intersecting transversally, then we define \left(C.D\right) to be the number of points in the intersection. Writing divisors as differences of curves gives us a general way to define the operation \left(C.D\right) . But I guess you can define these more generally. Over the complex numbers, you take the cup product of the first chern class of the sheaf associated to each of C and D. On a projective scheme, you can look at a certain coefficient of the Euler characteristic to define the intersection of Cartier divisors. An interesting result I found not in Hartshorne uses spectral sequence to show that the degree of a morphism times the intersection number is the intersection of the pullbacks of the divisors.

After intersection theory, Hartshorne talks about ruled surfaces, which are basically schemes with a morphism to a curve whose fibers are \mathbb{P}^{1} , and which have a section. It turns out that these can be written as the projective bundle of some sheaf. The section from the surface corresponds to a surjection with said sheaf, and classification results about X, can be derived from an exact sequence resulting from the surjection.

Monoidal transformations are basically blowing up a point and it seems like a lot of the result of the book was about blowing up certain points and contracting curves on surfaces in certain ways to classify different surfaces. A common technique is to contract a curve, and then note that some invariant, such as genus will drop, and keep contracting curves on the surface until you get some nonsingular surface. Next up for algebraic geometry: understand the Weil conjectures better and finish reading Debarre’s lecture notes on Mori theory. Possibly I want to find another topic to learn in the algebraic geometry area as well.

As far as the earlier parts of Hartshorne which I wanted to review, I found some notes typed by someone from a class given by Hartshorne at Berkeley (sp?), which were more lively, but less comprehensive yet still served as a good contrast to read through.

As far as the bits of modularity I wanted to read through, it seems that the book by Cornell has the general ideas. I’m just letting them sink in on my Anki deck and hoping to read the chapter on Gorenstein/ Complete intersections next? I don’t think any of the notes I typed are original enough for posting. A lot of the theorems which had really nice distilled proofs (in Hartshorne for example) I just rewrote word-for-word if I understood them but if you want a pdf of bits of modularity + hartshorne notes send a message.