Orbit Closures
summary
Gives a concrete algorithm that decides whether one linear G-orbit lies in the closure of another by producing explicit regular functions from a dominant parametrization, computing the degree d of the image variety (via Kazarnovsky / spherical-variety intersection theory), forming a polynomial with algebraically independent coefficient variables, substituting the parametrized matrix entries to get linear forms, and checking incompatibility of the resulting linear system (rank test). It requires knowing d computed from intersection theory, and produces very large systems (e.g. tens of millions of variables for small binary-form cases).
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chapters (10)
- 0:00Orbit closures in algebraic geometry
- 1:47Orbit containment problem
- 4:16Linear group actions reduction
- 8:47Classifying algebras via orbit closures
- 15:11Applications in algebraic geometry
- 21:48Algorithm input requirements
- 30:37Binary forms example
- 42:08Deciding orbit containment
- 49:15Degree of representation
- 55:04Critique and future directions
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