A Bit of Variety

Time to introduce some real mathematics! Today we’ll be talking about algebraic varieties. Gosh, that already sounds pretty heavy going. Part of the problem with starting algebraic geometry is the none of the nomenclature makes any intuitive sense. So it’s probably worth going on a bit of a historical digression to find out where this term originated.

Back in the 19th century, a good deal of algebraic geometry was done by French mathematicians. So it’s not surprising that much of the terminology of basic algebraic geometry has been borrowed from French. The word variety is one example. In 19th century French, a variété was an umbrella term for a geometrical object in space. A typical example of a 19th century variété would be a manifold, that is a space that looks locally like \mathbb{R}^n everywhere.

As time passed, the word variété caught on in English, despite the fact it seemed linguistically arcane. Mathematicians rarely worry about such things, it seems. As maths became increasingly formalised and rigorous, new terms like manifold and surface were introduced to describe particular types of varieties. By a combination of French stubbornness and historical accident, the word variety eventually came to refer to an abstract class of geometrical ‘things’.

(If you want more information, read this wonderful account of the history of Algebraic Geometry).

Hopefully the concept of algebraic presents fewer difficulties. As I mentioned in an earlier post, algebra is essentially the study of solutions to (mostly polynomial) equations. So what’s an algebraic variety? You got it – it’s a geometrical object which can be represented as a solution of (one or many) polynomial equations.

I’d properly better formalise all that as a definition. But first we need to know what kind of space we are working in. In other words, where do we allow our algebraic varieties to exist? The naive answer is in n-dimensional Euclidean space. This is indeed a good suggestion, and yields many informative examples, but there is too much loss of generality. Instead we’ll work in n-dimensional affine space which I’ll define shortly. Keep the idea of n-dimensional Euclidean space in mind as an intuition, though!

Definition 1.1 Let k be a field. We say affine space of dimension n over k is the set \mathbb{A}^n:=k^n=\{(a_1,\dots,a_n):a_i\in k\}.

You might think this is a bit of an odd notation. After all it takes more time to write \mathbb{A}^n than k^n and they are the same as sets by definition. However there is a subtlety. Mathematicians often think of k^n as being endowed with a natural vector space structure, with an origin and addition operation. Affine space \mathbb{A}^n is to be regarded merely as a geometrical blank canvas, with no associated operations or distinguished points. In fact we’ll see later that the right way to think about \mathbb{A}^n is as a topological space.

Since this post has an historical bent, I’ll digress a little to explain why we use the word affineThe word has its roots in Latin – affinis, meaning ‘related’. Mathematical usage seems to have been introduced by Euler to describe a type of geometry that studies how geometric objects are ‘related’ by slanting and scaling. Absolute notions of length and angle cease to make sense in this setting. Rather affine geometry is concerned more with the concepts of parallelism and ratios of lengths.

This might all seem a bit abstract, so let me put it another way. Affine geometry is the study of shapes which remain unchanged when they are transformed in such a way as to preserve straight lines. These so called affine transformations crop up all the time – translation, expansion, rotation are all examples we meet in everyday life. Affine geometry tries to make sense of all these in one geometrical space, affine space.

Definition 1.2 Let T \subset k[x_1,\dots,x_n] be a subset of the polynomial ring. We define the zero locus of T to be the set V(T):=\{P\in \mathbb{A}^n : f(P) = 0 \forall f \in T\}.

Sorry if that definition was a bit out of the blue. You may have to get used to me moving fast as this blog evolves. Remember that the polynomial ring is just the set of all polynomials in the variables x_1,\dots,x_n endowed with the obvious addition operation allowing you to add two polynomials. In plain English this definition is saying, ‘the zero locus of a set of polynomials, is all the points that make all the polynomials zero’. Sensible, huh?

Let’s do some examples. If we fix k=\mathbb{R} and work in \mathbb{A}^2 we have V(x^2+y^2) is a circle. Can you see what V(y-x^2) is? (Hint: the answer is in an earlier post)! Now if we work in \mathbb{A}^3 we can get some familiar surfaces. V(x^2+y^2-z^2) is a cone. V(x^2-y, x^3 - z) is a weird shape called a twisted cubic, pictured below.

Have some fun trying to think up some more wacky and wonderful shapes that can be represented as the zero locus of a subset of k[\mathbb{A}^n]. (Note: we’ll sometimes use the terminology k[\mathbb{A}^n ]= k[x_1,\dots ,x_n] as I have done here). Do leave me a comment if you come up with something fun!

Finally we’re ready to say what an algebraic variety is. Here we go.

Definition 1.3 A subset Y \subset \mathbb{A}^n is called an affine algebraic variety if there exists some subset T\subset k[x_1,\dots,x_n] of the polynomial ring such that Y = V(T).

Read that a couple of times and make sure you understand it. This really is the bedrock on which the subject stands. In plain English this merely says that ‘an affine algebraic variety is a geometrical shape which can be represented as the zero locus of some polynomials’. That’s exactly what we said it should mean above. (Note: I’ll often call affine algebraic varieties just affine varieties for short).

Right, that’s quite enough for one night. Next time I’ll talk about what Hilbert had to say about affine varieties. We’ll also start to see a surprisingly deep connection between algebra and geometry. Oh and if someone reminds me I’ll throw in an amusing video, like this marvellous CassetteBoy offering.

One more thing to do before you go – think about what kinds of shapes aren’t affine varieties.  Answers in the comments please!  Looking for such examples is something mathematicians like to do. It’ll hopefully give you a better understanding of what the concepts really mean! I’ll touch on this properly next time.

Apologies that this post is a little late – I have been struggling with some technical issues! I hope now they are sorted, thanks to the kindness of the IT Department at New College, Oxford. 

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