Alg Geom: Rational curves with self-intersection -2

[bham10246]

Hi, this is a question to the members with some knowledge in algebraic geometry:

1. what are rational curves with self-intersection -2? How do they look like?

2. do you know why these correspond to the vertices of some of the Dynkin diagrams?

3. just something that's bothering me and how do canonical bundles look like (geometrically)? An example would be fine.


Thanks so much! Glad to be back!

[n_bourbaki]

I'm not sure that they 'look' like anything: it's the top exterior power of the cotangent bundle.

The rest of your question(s) appears to be about the McKay correspondence. If you just google that you'll get lots of information to help you understand the 'why' of what's going on.