Timeline for "General position" on $\mathbb{P}^1\times\mathbb{P}^1$
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Feb 25 at 0:45 | comment | added | fp1 | @Sasha Thanks for the answer and comments. Do you happen to know any literature or paper that either explains or applies this? I just want to study this in more details | |
| Feb 21 at 8:55 | comment | added | Sasha | For instance, you can project to the plane from one of these points (say $P_0$) and write down the genericity assumptions on the images of the other four points (say, $P_i$, $1 \le i \le 4$) and the images (say $Q_1$ and $Q_2$) of the two rulings through $P_0$. Say, if the images of $P_1$, $P_2$, and $P_3$ are colinear, then $P_0$, $P_1$, $P_2$, and $P_3$ are coplanar, hence lie on a conic. If $Q_1$ and the images of $P_1$ and $P_2$ are colinear then $P_0$, $P_1$, and $P_2$ lie on a conic that contains a ruling, hence $P_1$ and $P_2$ lie on the other ruling. | |
| Feb 21 at 8:21 | comment | added | user347489 | It'd be great if you could give a brief explanation on why this is the case, or at least a reference. | |
| Feb 21 at 7:41 | comment | added | Sasha | You don't need to consider all $(a,b)$; only small ones. | |
| Feb 21 at 7:19 | comment | added | Zach Teitler | I guess the idea is that the linear system of curves of bidegree $(a,b)$ has (projective) dimension $(a+1)(b+1)-1$. So a set of $(a+1)(b+1)$ points imposes independent conditions on curves of bidegree $(a,b)$ if and only if the points don't lie on a curve of bidegree $(a,b)$. And so the "general position" here is really something like "every subset imposes independent conditions on linear systems". | |
| Feb 21 at 6:00 | history | answered | Sasha | CC BY-SA 4.0 |