### April 2008

Ok, so with the overwhelming majority of one vote, the next thing I talk about will be algebraic surfaces and intersection theory.  However, first I need to do a bit of reading on this topic, as well as finishing up my coursework for the year, so for simplicity’s sake, I’m putting this blog on hiatus until I get back from the Cornell Topology Festival in early may, and when I get back, it’s back to writing this blog (which has turned out to be a rather good way of studying all of this material…)

I’ve generally been well-behaved about focusing on the math on this blog rather than going off into politics or whatever, but sometime while I was at my office today, someone went around campus and put up posters for Ben Stein’s “documentary” Expelled. This annoyed me, and so I’m printing up a bunch of admittedly plain (because I’m printing them myself and have no design abilities or colored ink) posters to stick up next to the advertisements promoting Expelled Exposed.

I’m mostly writing this post as something to do while they’re printing, so in the meantime, a list of links to places where the lies and hypocrisy of the producers of Expelled are more thoroughly debunked than I could ever hope to do.

Expelled Exposed

PZ Myers is Expelled

Lying for interviews

ERV gets angry

Collection of reviews

PhysioProf chimes in

Antisemitism in Expelled

More of the same

Expelled the Miniseries?

Hmm, without the witty commentary, I seem to have gone into a mini-carnival myself… But anyway, my flyers are done printing, and I’m going to go put them up next to the Expelled ones.

Ok, so last time, we discussed divisors. We’re going to keep going in that direction now, and now we’re going to talk about linear systems of divisors. Whenever we talk about linear systems, we’ll assume that our variety $X$ is nonsingular, so we can even talk about Weil divisors with no problem, though we’ll sometimes also use Cartier divisors due to how things will be handed to us.

Some people might say that the natural place for this topic is before talk of differential forms and of the canonical bundle, but I disagree. Well, really it’s fine either way, but this is my blog, so I’m going to do it my way. Today we’re going to talk about divisors and their relation to line bundles.

We’re going to need to start out the day with a bit of algebra, because we’re going to talk about differential forms. Once we have forms, we’ll make a sheaf out of them, and then we’ll use this sheaf to construct other things.

We’ve now talked about vector bundles and locally free sheaves, we’re going to specify to the nicest case: rank 1. We’re generally going to ignore the distinction between the sheaf and the line bundle.

Last time, we talked about sheaves of modules, and focused on the correspondence between sheaves of ideals and subvarieties. We were talking about the internal geometry of the variety. Today, we’ll talk a bit about more external geometry. Specifically, we’ll talk about how sheaves give us new varieties $E$ with maps to $X$ whose fibers are all vector spaces. In fact, they’ll look locally like open sets of $X$ times a vector space. Such objects are called vector bundles, and are rather closely tied to the theory of sheaves of modules on $X$.

I’ve decided to start experimenting with wordpress’s themes in order to get a different look from the other math blogs.  What do people think of this one? There’s a few others that I’ve been considering as well, and I may continue experimenting throughout the month.

So now that we have abstract varieties on hand, we’re going to do a bit more with sheaves, leading to some of the intimate connections between sheaf theory and geometry. Sadly, this often gives students a lot of trouble (I know I had a bit of trouble with it at first) because things are presented very algebraically and the geometry gets lost. So we’ll be making a point of the connections between the geometry and the algebra. (more…)

Now, I highly doubt that I have any readers that don’t read Terry Tao’s blog, but regardless, please sign his petition (here). We need to fight threats to math education everywhere, and this is a big one. I’m behind Terry 100% on this one. So really, please head over and leave a comment expressing your support.

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