Timeline for The closure $\overline{Gx}$ for an affine variety on which an reductive algebraic group acts
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Jun 27, 2017 at 14:53 | vote | accept | Hang | ||
| Jun 27, 2017 at 14:50 | history | edited | Hang | CC BY-SA 3.0 |
added 214 characters in body
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| Jun 22, 2017 at 11:27 | answer | added | Friedrich Knop | timeline score: 6 | |
| Jun 22, 2017 at 4:55 | comment | added | Victor Protsak | You can add stability conditions "to make it correct". The GIT quotient only behaves well on semistable points. In the opposite case of nullcone (for a linear action), very complicated behavior is possible. This is illustrated in the standard counterexample to Q1 presented in my answer, where the nullcone is the whole space $X$. | |
| Jun 22, 2017 at 4:44 | answer | added | Victor Protsak | timeline score: 4 | |
| Jun 22, 2017 at 2:36 | comment | added | Hang | Thank you. But, can we add some further conditions to make it correct? | |
| Jun 22, 2017 at 2:28 | comment | added | Jason Starr | Question 1 has a negative answer. The counterexample that I love the best is when $G$ equals $\textbf{GL}_5$ and $X$ is the affine cone over a (projective) parameter space of rational normal curves of degree $4$ in $\mathbb{P}^4$, e.g., the Hilbert scheme, the Chow variety, or the space of Kontsevich stable maps of genus $0$. There is one dense orbit parameterizing the rational normal curves. However, these can specialize to a union of 4 concurrent lines. The usual cross-ratio for 4 points on $\mathbb{P}^1$ give continuous moduli of orbits in the orbit closure. | |
| Jun 22, 2017 at 2:23 | history | undeleted | Hang | ||
| Jun 22, 2017 at 2:11 | history | deleted | Hang | via Vote | |
| Jun 22, 2017 at 0:34 | history | asked | Hang | CC BY-SA 3.0 |