Timeline for relations between 4 plane conics
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 2, 2017 at 22:13 | vote | accept | Dima Pasechnik | ||
| Jul 25, 2017 at 15:47 | comment | added | Robert Bryant | @DimaPasechnik:Take the 2-dimensional space (i.e., pencil) of conics $P^\perp\subset S^2(V)$ annihilated by $P\subset S^2(V^*)$. Generically, this pencil will be the set of conics that pass through $4$ points in general position in $\mathbb{P}V^*\simeq \mathbb{P}^2$; take these points to be $[x_1],[x_2],[x_3], [x_1+x_2+x_3]$. Then $P^\perp$ is spanned by $x^1x^2-x^2x^3$ and $x^2x^3-x^3x^1$, so $P\subset S^2(V^*)$ has the basis claimed. When the pencil is degenerate, there are special cases to consider, but not very many, and your hypotheses rule out the pencil having nonzero perfect squares. | |
| Jul 25, 2017 at 15:11 | comment | added | Dima Pasechnik | This is a totally unexpected and mysterious for me. In what sense $P$ can be chosen as the span of $x_i^2$ etc. as above? Is there a group acting on $S^2(V^*)$, and you pick an orbit representative? | |
| Jul 25, 2017 at 13:11 | history | answered | Robert Bryant | CC BY-SA 3.0 |