Discussion Overview
The discussion revolves around the orbit type strata of C^3 under a 2-torus action defined by the mapping (a,b). Participants explore the implications of this action, particularly focusing on the nature of orbits and fixed points within the context of complex geometry.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
Main Points Raised
- One participant defines the action of the 2-torus on C^3 and suggests that the only fixed point is (0,0,0), proposing it as one strata.
- Another participant argues that orbits are not fixed points and asserts that every point lies in an orbit, questioning the nature of the resulting space when fixing a point (u,v,w) and applying the torus action.
- A participant clarifies that the mapping f: (a,b) → (abx, by/a, bz) is not injective under certain conditions, specifically when b=0 and a is not equal to 0, providing an example to illustrate this point.
- Further inquiry is made into the nature of the image of the map f and its relevance to the orbit structure, with a specific example of the point (1,0,0) being raised to question its orbit under the group action.
Areas of Agreement / Disagreement
Participants express differing views on the nature of orbits and the implications of the mapping. There is no consensus on the interpretation of the action or the resulting orbit structure, indicating ongoing debate and exploration of the topic.
Contextual Notes
Participants note the importance of injectivity in determining the nature of orbits, but the discussion remains unresolved regarding the specific conditions and implications of the mapping.