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Michi’s blog » Introduction to Algebraic Geometry (1 in a series)

 Introduction to Algebraic Geometry (1 in a series)

  • February 21st, 2008
  • 12:33 pm

I’m growing embarrassed by my lack of understanding for the sheaf-theoretic approaches to algebraic (and differential) geometry. I’ve tried to deal with it several times before, and I’m currently reading up on Algebraic Geometry again to fill the void that the finished thesis, soon arriving travels and non-existent job application responses produce.

So, why not learn by teaching? It’s an approach that has been pretty darn good in the past. So I thought I’d write a sequence of posts on algebraic geometry, introducing what it’s supposed to be about and how the main viewpoints develop more or less naturally from the approaches taken.

Varieties

The basic objective of algebraic geometry is to study solution sets to systems of polynomial equations. That is, we take some set f_1,\dots,f_r of polynomials in some polynomial ring k[x_1,\dots,x_n] over some field k. And we write V(f_1,\dots,f_r) for the set of all simultaneous roots to all these polynomials:
V(f_1,\dots,f_r)=\{p\in k^n:f_1(p)=0, \dots, f_r(p)=0\}

If we write our polynomials as coming from the ring k[x_1,\dots,x_n], then the corresponding solution points will be points in the vector space k^n. In order to emphasize that we do not care for the vector space structure of this space, we shall denote it with \mathbb A^n, or if we want to emphasize the field, with \mathbb A^n_k.

The first observation at this point is that if we take the polynomial x^2+1, then the solution set over \mathbb R is empty, while the solution set over \mathbb C is not. So, in order to set all solution sets on an equal footing – and also to make the later occurring correspondences work out – we shall require k to be an algebraically closed field. In other words, we can always find a root to any polynomial.

We call the solution sets varieties (or – in order to distinguish from everything else we might encounter, we shall call them affine algebraic varieties).

So, the study of solutions to systems of polynomial equations is the study of varieties. And hence geometry. This neatly expands on the classical linear algebra viewpoint – where we study systems of linear equations as intersections of planes. It turns out that the main computational approach – Gr

10 People had this to say...

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  • Charles
  • February 21st, 2008
  • 14:03

Good post. You’re going a bit more into the guts of the theory, but you might be interested in my series of posts http://rigtriv.wordpress.com/category/algebraic-geometry/algebraic-geometry-from-the-beginning/ which is more about the intuition behind everything…and I should get back to that, as I’ve just introduced sheaves…

Oh, and if you’re going to talk about Groebner Bases, that’d be great, and I’ll just link back to you whenever I want to do a computation.

Good luck!

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  • Michi
  • February 21st, 2008
  • 14:07

Thanks Charles,

I actually wasn’t planning on talking particularly much about Gröbner bases – mainly because that’s an aspect I actually DO understand. I was planning on building up a setting where heading into sheaves looks natural pretty swiftly, and then go on from there to discuss sheaves and schemes.

However, I’m always happy to consider requests for material my audience wants me to cover. :)

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Cool! I look forward to reading more. =)

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  • Charles
  • February 21st, 2008
  • 19:19

Ahh, well, no particular request. And if you’re going to talk about schemes (especially in general) then our viewpoints will diverge, because I’m focusing on varieties (albeit abstract varieties with potentially embedded components). I wish you good luck, and I’m going to keep reading.

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  • John
  • February 21st, 2008
  • 23:22

I was always under the impression that a set of common solutions to polynomials is called an algebraic set.

A variety is only an irreducible algebraic set.

Wikipedia seems to agree with me…
http://en.wikipedia.org/wiki/Algebraic_variety

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  • Michi
  • February 21st, 2008
  • 23:33

John: Sure, that -is- a completely reasonable definition. And one made by, for instance, Shafarevich’s rather seminal work.

However, regardless of what the Wikipedia says, the literature does show more definitions than this. For instance, Smith-Kahanpää-Kekäläinen-Traves does define an affine algebraic variety in my way; and I do recall having seen it that way elsewhere as well.

It all really boils down to what you want the language you define to work out for. With my definition of varieties, closed sets in the Zariski topologies are precisely subvarieties of the affine space, and open sets are precisely complements thereof. In your language, distinction would have to be made between reducible and irreducible closed sets – which, I felt, made the exposition slightly more awkward.

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[...] Difakis (won the ACM Turing award). 21 – Michi finished his thesis, and found the time to post an introduction to algebraic geometry (in two [...]

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  • Ben
  • February 23rd, 2008
  • 2:43

Have you tried Mumford’s Red Book? It’s the best I’ve found…..

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  • mwanahamisi
  • February 23rd, 2008
  • 16:20

I am maths student in tanzania, and you summary on algebraic geometry is attracting my interest in the subject. Allow me to contact you whenever I need elaboration on this complex area

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  • Michi
  • February 23rd, 2008
  • 17:08

mwanahamisi: I’m glad you’re interested in the subject. I cannot guarantee neither that I know the answers to any questions you might have nor that I’ll have time to elaborate on request. But you’re free to ask.

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