Timeline for Degree $6$ curve in $\mathbb{P}^3$
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 24, 2020 at 16:00 | comment | added | user130022 | Yes. I realized that. Sorry for silly question. However if $Q_1 \cap Q_2$ is contained in a cubic which is singular along the line $l$, then there is a quadric containing $Q_1 \cap Q_2$ and $l$. | |
| May 24, 2020 at 15:27 | comment | added | abx | Certainly not in general! But the space of quadrics containing $Q_1\cap Q_2$ is indeed the pencil spanned by $Q_1$ and $Q_2$ — this follows from an easy Koszul complex argument. | |
| May 24, 2020 at 15:23 | comment | added | pinaki | @abx: "The quadric must belong to the pencil spanned by Q1 and Q2" - can you please explain why? Is it true that the space of quadrics containing a given curve is at most two dimensional? | |
| May 24, 2020 at 15:05 | review | Close votes | |||
| May 25, 2020 at 17:03 | |||||
| May 24, 2020 at 14:49 | comment | added | abx | No. The quadric must belong to the pencil spanned by $Q_1$ and $Q_2$; if a line is contained in such a quadric, it intersects $E=Q_1\cap Q_2$ in 2 points. Just take for $l$ a general line through a point of $E$. | |
| May 24, 2020 at 14:35 | history | asked | user130022 | CC BY-SA 4.0 |