Wednesday, 30 November 2011

Blowing up

Algebraic geometry is a subject I have had quite a lot of use for in recent years, in my work on string compactifications, and I have come to love it, although I am still very much a novice. It has a reputation for being difficult and very abstract, and I think modern algebraic geometry thoroughly deserves this. But the roots of the subject are simple enough: it is a systematic study of the solutions of polynomial equations (this only really becomes interesting in more than one variable, where the solution sets can form interesting geometric spaces). On the other hand, thanks to the 'GAGA' principle, fairly elementary techniques of algebraic geometry are capable of solving complicated analytic problems arising in complex geometry.

To show you that it's not all so scary, I want to present an example of something quite familiar to physicists, and explain how it corresponds to a common operation in algebraic geometry.

Suppose I have two variables $x$ and $y$, and I am interested in what happens (to various quantities) as I take them both to $0$. Sometimes, the answer is unambiguous; for example if $p$ is any polynomial in $x$ and $y$, then $$\lim_{x,y \to 0}~ p(x, y) = p(0,0) ~,$$ which is just a well-defined constant. But things can be more complicated than this. Suppose I want to know the limit of $f(x,y) = \frac{y^2}{y^2 + x^2}$ as $x \to 0, y \to 0$. This obviously depends on how I take the limit: if I first take $x$ to $0$, then $f$ becomes equal to $1$, independent of y. If on the other hand I first take $y$ to $0$, then $f$ goes to $0$, independent of $x$. So $f(x,y)$ is simply not defined at all at the point $(x,y) = (0,0)$ — it doesn't even take the value $\infty$.

In this sort of situation, we often do the following: take $x, y \to 0$, while keeping $\frac{x}{y}$ constant. In other words, let $x = ky$ for some constant $k$, and then let $y \to 0$. The limiting value of $f(x,y)$ then depends on $k$. It's almost as if the 'function' $f(x,y)$ sees a whole line, parametrised by $k$, in place of the origin. (In fact, it sees a real line with infinity adjoined, corresponding to taking $y$ to zero much faster than $x$). So actually, $f(x,y)$ is not a function defined on the plane parametrised by $x, y$; it is defined on this different space, where the origin is replaced by a line. This new space is called the "blow up" of the plane at the origin — it is obtained by replacing the origin with the set of all tangent directions at that point. It might seem like we obtain a weird, singular space by this process, but that's actually not true — the new space is perfectly smooth.

Blowing up is the simplest version of something which algebraic geometers call a birational transformation, and it is an important concept in the subject. To those without a background in pure maths, its definition in textbooks might seem somewhat abstract, but as I hope you can now see, it's a very simple idea.