Posts Tagged ‘math.AG’

Filtrations in Algebraic Geometry

August 28, 2007

    When learning scheme theory, the first important functor one learns about is probably \mathbf{Spec}, and the second is likely \mathbf{Proj}\mathbf{Spec} is important but bland, since scheme theory can’t say anything new; for instance, the cohomology of any coherent module must vanish.  \mathbf{Proj} is a whole different story, where graded algebras are turned into spaces with interesting topology and invariants that can be useful in solving some very fundamental problems (see, for example, the Jacobian of an elliptic curve).  I have always found it a little bit mysterious that something as simple as a grading, which I usually regard as a computational tool, can have such important geometric implications.

    A natural question then is, “How far can I weaken things and still construct interesting schemes?”  The particular weakening I have in mind is a filtered algebra.  Instead of the algebra being the direct sum of vector spaces with compatible multiplication, we will look at algebras which are the union of a chain of nested vector spaces with compatible multiplication.  Then, we will look at a couple of associated schemes, and discover that familiar things like projective compactifications arise naturally in this setting.



Murphy’s Law for Moduli Spaces

August 20, 2007

    I’m finally back in Ithaca after about a month of travelling about.  Hopefully this will alleviate the lull the blog has been in the last few weeks.  Though, Jim Pivarski filled some of the otherwise dead-space with some excellent physics posts.  Kudos to him!

    The most mathy thing I did while I was gone was the aforementioned mini-course at MSRI on Deformation Theory.  It was very well put together, and the lectures were a cut above what I went in expecting.  However, I couldn’t help but feel a little dissappointed.  The conference was mostly on the nitty-gritty of making stacks work, and very few concepts I hadn’t seen before were introduced.  It was good for me to work out lots of examples and discover computational tricks, but I was hoping to have my imagination sparked by some nifty new ideas.

    There were some great moments, though.  Ravi Vakil, one of the organizers, gave a fun talk on Murphy’s Law for moduli spaces.  It was on a recent paper of his which gave a concrete meaning and proof to an old folklore meta-theorem called Murphy’s Law for the Hilbert scheme.  I first read it in Morrison and Harris’s wonderful book “Moduli of Curves“, as follows:

There is no geometric possibility so horrible that it cannot be found generically on some component of the Hilbert scheme.

Apparently, this idea goes back to Mumford and his paper “Further Pathologies in Algebraic Geometry”, where he showed that there is a component of the Hilbert scheme that is everywhere non-reduced.  The meta-theorem was meant to stop mathematicians from wasting their time trying to find a line the Hilbert scheme wouldn’t cross.


Fixing Bezout’s Theorem

July 16, 2007

  Bezout’s theorem in algebraic geometry is one of those simple facts that manages to capture the heart and style of its field.  It states that any two irreducible curves C_1 and C_2 in \mathbb{C}^2 usually intersect in deg(C_1)\cdot deg(C_2) points (where the degree of a curve is the degree of the polynomial that defines it).

  Now, very few mathematicians will stand for a ‘usually’ in their theorems, and the most basic form of Bezout’s theorem is typically stated differently – so as to be a real theorem.  However, my favorite aspect of the theorem is that figuring out how to fix the ‘usually’ has repeatedly foreshadowed the development of algebraic geometry as a whole.