Alg Geom: Rational curves with self-intersection -2

In summary, the conversation discusses rational curves with self-intersection -2 in algebraic geometry and their correspondence to the vertices of some Dynkin diagrams. The topic also touches on the geometric appearance of canonical bundles, specifically in the context of the McKay correspondence.
  • #1
bham10246
62
0
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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  • #2
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.
 

1. What is the significance of self-intersection -2 for rational curves in algebraic geometry?

Self-intersection -2 is a special property that rational curves can have in algebraic geometry. It means that the curve intersects with itself in two distinct points, but with opposite orientation. This property is important because it allows for a deeper understanding of the geometry of the curve and its relationship to other curves in the same space.

2. How can rational curves with self-intersection -2 be constructed?

There are several methods for constructing rational curves with self-intersection -2, such as using rational normal curves or blowing up points on a smooth curve. These constructions involve algebraic equations and geometric transformations, and can be quite complex.

3. What are the applications of studying rational curves with self-intersection -2?

Rational curves with self-intersection -2 have important applications in algebraic geometry, such as in the study of moduli spaces and birational geometry. They also have connections to other areas of mathematics, such as differential geometry and topology.

4. Are there any open problems or conjectures related to rational curves with self-intersection -2?

Yes, there are still many open problems and conjectures in this area of algebraic geometry. Some current research focuses on the classification of rational curves with self-intersection -2 and their moduli spaces, as well as their connections to other geometric structures.

5. How does the self-intersection number of a rational curve affect its properties?

The self-intersection number of a rational curve plays a fundamental role in determining its properties and behavior. In particular, rational curves with self-intersection -2 have special properties that make them useful for studying the geometry of higher-dimensional spaces and for solving problems in algebraic geometry.

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