Timeline for Elementary question: Cubic 4-fold and rational quartic scroll
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 15, 2020 at 7:27 | history | edited | CommunityBot |
Commonmark migration
|
|
| Dec 8, 2015 at 11:32 | comment | added | Jason Starr | Above when I wrote "Segre cubic threefold", I should have written "image of $\mathbb{P}^1\times \mathbb{P}^2$ under a Segre embedding". The atlas just says "Segre threefold". Sorry. | |
| Dec 7, 2015 at 18:38 | vote | accept | HLC | ||
| Dec 7, 2015 at 18:38 | vote | accept | HLC | ||
| Dec 7, 2015 at 18:38 | |||||
| Dec 7, 2015 at 18:36 | comment | added | HNuer | Yes a cubic 4-fold containing 2 skewed planes lies in $\mathcal C_{14}$. You can see that the virtual class of a quartic scroll is $2h^2-P_1-P_2$ where $P_1$ and $P_2$ are skew planes. So such a cubic fourfold contains the usual rank 2 sublattice of discriminant 14 associated with containing a quartic scroll. | |
| Dec 7, 2015 at 17:09 | answer | added | abx | timeline score: 3 | |
| Dec 7, 2015 at 16:52 | answer | added | Jason Starr | timeline score: 1 | |
| Dec 7, 2015 at 16:30 | comment | added | HLC | Thank you very much. One question: Does a cubic 4-fold containing two skewed planes lie in the class $\mathcal{C}_{14}$ just like a cubic 4-fold containing a quartic scroll ($\mathcal{C}_{14}$ are the cubic 4-folds having a discriminant-$14$ lattice in the sense of Hassett)? | |
| Dec 7, 2015 at 16:20 | comment | added | Jason Starr | I may have misattributed the result about Pfaffian cubic fourfolds being precisely those that contain a quintic del Pezzo surface. According to the MO answer that I linked, it appears the following is the correct citation: Arnaud Beauville; Determinantal Hypersurfaces. Michigan Math. J. 48, 2000. | |
| Dec 7, 2015 at 16:18 | comment | added | Jason Starr | By Beauville and Donagi, a cubic fourfold that contains a quintic del Pezzo surface is Pfaffian: Beauville, Arnaud; Donagi, Ron; La variété des droites d'une hypersurface cubique de dimension 4, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), no. 14, 703–706. Please see the following MO question for an explanation why every $X$ containing a quartic scroll contains a quintic del Pezzo: mathoverflow.net/questions/78228/… | |
| Dec 7, 2015 at 16:08 | comment | added | Jason Starr | According to the atlas that Joe Harris produced, it appears that if a cubic fourfold $X$ contains a quartic scroll $\Sigma$, then the residual to $\Sigma$ in the intersection of $X$ with a Segre cubic threefold $Y$ is either a quintic del Pezzo surface or a quintic scroll. Presumably the difference is whether the quintic rational normal curve whose secant variety equals $Y$ sits in $\Sigma$ as a curve with self-intersection number $4$ or $6$. | |
| Dec 7, 2015 at 15:56 | comment | added | abx | 2/ is false. For a cubic fourfold $X$, let $\rho $ denote the rank of $H^4(X,\mathbb{Z})_{\mathrm{alg}}$. A general cubic fourfold containing $S$ will have $\rho =2$, while cubics containing two skew planes have $\rho =3$. | |
| Dec 7, 2015 at 15:50 | review | First posts | |||
| Dec 7, 2015 at 15:57 | |||||
| Dec 7, 2015 at 15:44 | history | asked | HLC | CC BY-SA 3.0 |