Timeline for Computing fundamental groups of the complement of plane curves
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 29, 2013 at 11:43 | comment | added | Jason Starr | Thank you for correcting the statement. The previous formulation hurt my eyes. | |
| Nov 28, 2013 at 20:27 | comment | added | pinaki | @Jason: I see I was being dense. Thanks again. | |
| Nov 28, 2013 at 20:26 | history | edited | pinaki | CC BY-SA 3.0 |
added 32 characters in body
|
| Nov 28, 2013 at 18:15 | comment | added | Jason Starr | @auniket: You did write something that is incorrect. It is not true that for $C$ a plane curve, for a general line $L$, the induced map of fundamental groups is an isomorphism. That is simply not true, as Jack already explained to you. You are incorrect, Jack is correct. | |
| Nov 28, 2013 at 17:50 | comment | added | pinaki | @Jason: I don't think I have written anything wrong. In any event, I edited the question to make it (hopefully) clearer. Thanks. | |
| Nov 28, 2013 at 17:48 | history | edited | pinaki | CC BY-SA 3.0 |
added 367 characters in body
|
| Nov 28, 2013 at 16:35 | comment | added | Ian Agol | The book by Orlik-Terao seems nice: springer.com/mathematics/geometry/book/… | |
| Nov 28, 2013 at 16:05 | answer | added | Alexandre Eremenko | timeline score: 3 | |
| Nov 28, 2013 at 13:17 | comment | added | Jason Starr | @auniket: Since Jack is correct, and since you wrote something wrong, you should either edit your question or, perhaps, delete it entirely. | |
| Nov 28, 2013 at 10:58 | comment | added | pinaki | @Jack: You are right. But I want to know how, for a fixed (not necessarily nice) $C$, to choose $D$ so that every loop in $\mathbb{P}^2-C$ can be deformed into a loop in $D-C$. Even in this case $D$ can be chosen to be a generic line. But can we take it to be a generic element in any other linear system (not necessarily containing lines)? | |
| Nov 28, 2013 at 6:59 | comment | added | Jack Huizenga | Perhaps I am missing something. $L-C$ has the homotopy type of a wedge of circles, so its fundamental group is a free group. On the other hand, for sufficiently nice curves $C$ the fundamental group of $\mathbb{P}^2 -C$ is a cyclic group. I believe is true that every loop in the later space can be deformed to a loop in $L -C$, so that $\pi_1(L-C)\to \pi_1(\mathbb{P}^2-C)$ is automatically surjective, but there are typically many more relations in the latter group, arising from the fact that we can vary the line in a pencil and obtain a nontrivial monodromy action. | |
| Nov 28, 2013 at 3:52 | history | asked | pinaki | CC BY-SA 3.0 |