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.