I'm teaching myself a little algebraic geometry and I was hoping you could help me with an exercise. I have my head around affine spaces alright but I am having a little more trouble with projective space. The exercise I am confused about is to show that the curve given by set of zeros $W \subset \mathbb{P}^2$ of $X_0^3 X_2^2 + X_2^3 X_1^2 + X_1^3 X_0^2$ is irreducible (over field of zero characteristic).
More generally, I would like to know if there is a sensible method to approach a problem like this: my first thought was to pick some point which must be in at least 1 irreducible component, and then show any component containing that point must be the whole of W: I got nowhere with this, however. For this and in general zeros of other reasonably simple polynomials, how should I try to show irreducibility? Please keep answers simple, I haven't been doing this for very long! -K