Alg Geom: Rational curves with self-intersection -2

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SUMMARY

Rational curves with self-intersection -2 are significant in algebraic geometry, particularly in relation to Dynkin diagrams. These curves do not have a specific geometric appearance but are defined through the top exterior power of the cotangent bundle. The discussion also touches on the McKay correspondence, which provides insight into the relationship between these curves and their geometric properties. Understanding canonical bundles geometrically is essential for grasping their role in algebraic geometry.

PREREQUISITES
  • Algebraic geometry fundamentals
  • Understanding of Dynkin diagrams
  • Knowledge of cotangent bundles
  • Familiarity with the McKay correspondence
NEXT STEPS
  • Research the properties of rational curves with self-intersection -2
  • Study the relationship between rational curves and Dynkin diagrams
  • Explore the geometric interpretation of canonical bundles
  • Investigate the McKay correspondence in detail
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Algebraic geometers, mathematicians studying rational curves, and anyone interested in the connections between algebraic geometry and Dynkin diagrams.

bham10246
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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!
 
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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.
 

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