July 2008


Ok, back to curves. We’d wandered a bit in the direction of this topic before, having discussed Bezout’s Theorem and the Riemann-Roch Theorem. Today we’ll talk about the Hurwitz formula, also called the Riemann-Hurwitz formula. It’s a rather nice result, and combined with Riemann-Roch can be used to prove a huge amount about curves and maps between curves.

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Today, we’ll link the computational thread back to the thread involving Hilbert schemes, by working out how to compute the Hilbert function (and thus polynomial) for any ideal in the ring k[x_0,\ldots,x_n]. The trick involves Groebner bases and flat families, and really ties a lot of threads together so that we can move on to something else (because my attention span is short and there are a lot of cool things to talk about.) (more…)

Let’s say that you have two polynomials, and you REALLY need to know if they have a common root. Now, if they’re quadratic, you’re in luck, because we can solve them both completely and just check. In fact, if you’re patient, you can do it whenever the Galois group is solvable, in particular, you can do it for cubics and quartics. But in general? What if I gave you a degree 100 polynomial and a degree 103 polynomial? Well, you can still do it, without having to solve anything.

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Last time we talked about Groebner bases and Buchberger’s algorithm, so today we’ll do an application of them. In fact, a few, because the Elimination Theorem and the Extension Theorem are extremely useful results, and we’ll talk a bit about their geometric consequences. But first, the algebra:

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Well, I’m taking the day off, but instead of looking for my stuff, you can all go over to the Logic Nest and read the 37th Carnival of Mathematics. Hmm…I had been intending to write something for it, but it slipped my mind. Oh well.

After all that technical work and careful but rather abstract technique developed for the construction of a bunch of moduli spaces, I’ve decided to take a break from that line of reasoning and to look to one that’s rather dear to me. It’s probably just that these methods were my first real introduction to algebraic geometry, and I use them to construct examples routinely. We’re not just dropping back to varieties here, we’re going all the way back to affine varieties over an algebraically closed field for a bit, and eventually will be stepping all the way back to characteristic zero for some stuff related to representation theory. Well, truthfully, it’s time for algebra in the coordinate ring.

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I’ve seen a lot of people playing with Wordle on their blogs lately, so here’s a wordle picture to represent the Algebraic Geometry from the Beginning series. I’ve removed some English words, removed plurals on a lot of words, and changed things like \otimes to “Tensor Product” to make it more representative. Here it is:

Now we’re done constructing \mathrm{Hilb}(\mathbb{P}^n/S), so it’s time to get the general Hilbert scheme done, and then to construct some other moduli spaces. Now, as X\subseteq\mathbb{P}^n, we can see that Hilb(X/S) is a subfunctor of Hilb(\mathbb{P}^n/S), so we want to try to find a subscheme of \mathrm{Hilb}(\mathbb{P}^n/S) to represent it.

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I promised a minipost on Nakayama when I talked about Flattening Stratifications, and I’ve got a moment now, so I’ll do it quickly. This post is all commutative algebra. So we’ll quickly state Nakyama:

Nakayama’s Lemma: Let I be an ideal contained in the Jacobson radical of a ring R and let M be a finitely generated R-module. Then if IM=M we have M=0 and if m_1,\ldots,m_n\in M have images in M/IM which generate it as an R-module, then m_1,\ldots,m_n generate M as an R-module.

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This may just be me getting hopeful, but after reading this at the LA Times blog, and catching the following quote, I think I have reason:

“We’re too busy talking about the giant Broadway adaptation, the much longer film version and the musical commentary that we’re writing now.” (As in, a sung DVD commentary.)

“But have I thought up the sequel?” he answers. “Yeah, sort of.”

This is from Joss Whedon himself.  So it looks like we can expect more out of Dr. Horrible in the near future.  Good thing I’m close enough to New York to get in to see this when it appears, if I can get tickets…

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