Monthly Archives: July 2008

Hurwitz’s Theorem

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, … Continue reading

Posted in AG From the Beginning, Algebraic Geometry, Big Theorems, Curves | 2 Comments

Computing Hilbert Functions

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 . The trick involves Groebner bases and flat families, … Continue reading

Posted in AG From the Beginning, Algebraic Geometry, Computational Methods | 1 Comment

Resultants

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 … Continue reading

Posted in AG From the Beginning, Algebraic Geometry, Computational Methods | 8 Comments

Elimination and Extension Theorems

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 … Continue reading

Posted in AG From the Beginning, Algebraic Geometry, Big Theorems, Computational Methods | 11 Comments

Carnival of Math 37

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 … Continue reading

Posted in Uncategorized | Leave a comment

Groebner Bases and Buchberger’s Algorithm

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 … Continue reading

Posted in AG From the Beginning, Algebraic Geometry, Big Theorems, Computational Methods | 7 Comments

Wordle

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 … Continue reading

Posted in Uncategorized | Leave a comment

Examples of Moduli Spaces

Now we’re done constructing , so it’s time to get the general Hilbert scheme done, and then to construct some other moduli spaces. Now, as , we can see that is a subfunctor of , so we want to try … Continue reading

Posted in AG From the Beginning, Algebraic Geometry, Deformation Theory, Examples, Hilbert Scheme | 3 Comments

Nakayama’s Lemma

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 be an ideal … Continue reading

Posted in AG From the Beginning, Algebraic Geometry, Big Theorems | 13 Comments

Dr. Horrible goes to Broadway?

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 … Continue reading

Posted in Uncategorized | Leave a comment