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 of polynomials in some polynomial ring over some field . And we write for the set of all simultaneous roots to all these polynomials:

If we write our polynomials as coming from the ring , then the corresponding solution points will be points in the vector space . In order to emphasize that we do not care for the vector space structure of this space, we shall denote it with , or if we want to emphasize the field, with .

The first observation at this point is that if we take the polynomial , then the solution set over is empty, while the solution set over 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 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 – Gröbner bases – actually specializes to the Gauß algorithm on matrices if we specialize to linear systems of equations.

### Examples

The unit circle is an affine variety, given by the vanishing of . The hyperbola and the parabola are also affine varieties.

Another very commonly used example is the twisted cubic curve – given by the simultaneous vanishing of the parabolic sheet and the cubic curve sheet .

## Changing the defining equations

The points p at which both f and g vanish are the same points as those where f and f+g vanish. Indeed, if f(p)=0 and g(p)=0, then (f+g)(p)=0 as well, so the points in V(f,g) are all contained in V(f,f+g). On the other hand, suppose that f(p)=0 and (f+g)(p)=0. Then 0=(f+g)(p)=f(p)+g(p)=0+g(p). So g(p)=0 as well. Hence, all points in V(f,f+g) are contained in V(f,g). So the two varieties are equal.

Suppose that . Then for any h=fg we’ll see that h(p)=f(p)g(p)=0g(p)=0. Hence, if f vanishes at p, then every polynomial that f divides will also vanish at p.

Now, a set of polynomials in that is closed under addition and multiplication by elements from is an ideal of the polynomial ring. Thus, these considerations convince us that the set of all polynomials in that all vanish on all points in is in fact an ideal.

So we can define the variety from an ideal – given , we write V(I) for the set of all points such that f(p)=0 for all .

The sum of two ideals I and J are the set of all elements on the form f+g where f is from I and g is from J. If I is generated by and J is generated by , then I+J has generators . And the set of points where all the polynomials in I+J vanish are precisely the points where all polynomials in I vanish and all polynomials in J vanish. Hence .

The product IJ of two ideals I and J is generated by all products of one element from I and one from J. Hence, an element of IJ is on the form with all the coming from I and all the coming from J. Suppose p is in V(IJ). Then for all f in I and all g in J, f(p)g(p)=0. If for all f in I, f(p)=0, then p is in V(I). Otherwise we can find some such that . But then since for all g in J, we must have for all g in J. Hence p is in V(J).
Thus .
On the other hand, if p is in then specifically either p is in V(I) or it is in V(J). Either way, for all f in I and all g in J, f(p)g(p)=0 since one of the factors certainly vanishes. Hence, p is in V(IJ).

We also do have that . Which of the ideal operations works best depends a bit on what we would like to do with it.

### Examples

We can use the above discussion to actually compute things geometrically. The intersection of the circle with a line is given by the sum of the ideals – so we need to consider the ideal in . We could try to bash this out by just wrangling the corresponding equations, or we could systematize the wrangling. Systematizing it leads, basically, to the theory of Gröbner bases – and this is something I tend to avoid doing by hand if I can. Not doing it by hand looks like this:

\$ Singular
SINGULAR                             /
A Computer Algebra System for Polynomial Computations   /   version 3-0-2
0<
by: G.-M. Greuel, G. Pfister, H. Schoenemann        \   July 2006
FB Mathematik der Universitaet, D-67653 Kaiserslautern    \
> ring R=0,(x,y),dp;
> ideal I=x^2+y^2-1,2*x-y;
> std(I);
_[1]=2x-y
_[2]=5y2-4


So one good set of generators for the ideal would be . This, in turn corresponds to the conditions and . So we get two points – just as expected.

The equation wrangling, on the other hand, begins with recognizing that if both a and b are 0, then V(ax+by+c) is either empty or the entire plane. Both are rather boring, and easy to handle. So one of a and b is non-zero. Let’s say that a isn’t 0. Then we can rewrite ax+by+c=0 to . This can be inserted into the quadratic equation to form the new equation , which in turn expands to

which, in turn, we can easily solve using normal solution techniques for quadratic equations. To get rid of fractions, we’ll multiply the entire equation by yielding the new equation

Now, the discriminant

carries information about the nature of solutions. If , then the line V(ax+by+c) is a tangent to the circle, if , then the intersections are all complex and if , then the intersections are real. In both the non-tangent cases, the circle intersects the line exactly twice.

### 10 People had this to say...

• 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!

• 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.

Cool! I look forward to reading more. =)

• 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.

• 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

• 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.

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

• Ben
• February 23rd, 2008
• 2:43

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

• 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

• 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.