Timeline for Family of elliptic curves in $\mathbb P^3$
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Apr 11, 2020 at 18:57 | comment | added | Sasha | A general curve in this linear system is the strict transform of a plane cubic passing through 4 out of 6 blowup points, a general such curve is smooth, hence so is its strict transform. Its degree is 5 because $(3\ell - e_1 - e_2 - e_3 - e_4)(3\ell - e_1 - e_2 - e_3 - e_4 - e_5 - e_6) = 9 - 4 = 5$. | |
| Apr 11, 2020 at 16:13 | comment | added | Libli | @Sasha : I don't see how you find a smooth elliptic quintic in the linear system $3 \ell -e_1-e_2-e_3-e_4$ | |
| Apr 11, 2020 at 11:38 | comment | added | Sasha | I am using the standard notation for the blowup of a $\mathbb{P}^2$ at 6 points ($\ell$ is the line class, $e_i$ are the exceptional divisors). I don't claim that the deformation can be realized by cubic equations; instead, I say that one can deform a curve within a linear system on the fixed cubic surface. | |
| Apr 11, 2020 at 10:16 | comment | added | Libli | @Sasha : you are right for the quadric, it is true only if the projection is from a point $P \in Sec(C) $ (so that the curve in $\mathbb{P}^3$ is not smooth but nodal/cuspidal), I will correct that. As for the second part of your comment, I am not sure I understand the notations and the argument. In any case, I don't think that the smoothing in $\mathbb{P}^3$ of a pentagonal configuration to an elliptic curve can be realized only with cubic equations. Though, if it were true, it would certainly be intresting, it doesn't appear in Ellingsrud-Laksov. | |
| Apr 11, 2020 at 10:13 | history | edited | Libli | CC BY-SA 4.0 |
wrong statement about qudric surface corrected
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| Apr 11, 2020 at 4:29 | comment | added | Sasha | Why do you say that a smooth quintic elliptic curve in $\mathbb{P}^3$ lies on a smooth quadric surface? I guess it does not (what its bidegree would be?}. Instead, it lies on a smooth cubic surface. As such, it can be thought of as a curve of type $3\ell - e_1 - e_2 - e_3 - e_4$. In the same linear system you can find the penthagon of lines $e_1, \ell - e_1 - e_2, e_2, \ell - e_2 - e_3, \ell - e_1 - e_4$. Of course, one is the deformation of the other. | |
| Apr 11, 2020 at 0:11 | history | edited | Libli | CC BY-SA 4.0 |
wrong power
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| Apr 11, 2020 at 0:05 | history | edited | Libli | CC BY-SA 4.0 |
missing power
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| Apr 10, 2020 at 23:55 | history | edited | Libli | CC BY-SA 4.0 |
added complete answer
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| Apr 10, 2020 at 17:03 | history | answered | Libli | CC BY-SA 4.0 |