Timeline for Are del-Pezzo surfaces complete intersections?
Current License: CC BY-SA 3.0
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| May 21, 2015 at 22:14 | comment | added | Ricardo Andrade |
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| May 21, 2015 at 21:59 | review | Close votes | |||
| May 22, 2015 at 8:26 | |||||
| May 21, 2015 at 7:07 | comment | added | user5117 | @Ritwik: yes, that's correct. Only those three are complete intersections. | |
| May 20, 2015 at 14:14 | comment | added | Ritwik | @Rene: If I understand your (and Smith's) remark correctly, the answer to my question is no in general; in particular if $k=1,2,3,4$, $7$ or $8$? The answer is only yes if $k=0,5$ or $6$. | |
| May 20, 2015 at 11:23 | comment | added | user5117 | @René: good point (about quadrics)! | |
| May 20, 2015 at 11:20 | comment | added | R.P. | To add to Artie Prendergast-Smith's comment: this means that $X_k$ is a complete intersection iff $k = 0,5,$ or $6$ (the case $n=3,d=2$ he mentioned does correspond to a del Pezzo, but not to a blow-up of $\mathbb{P}^2$). | |
| May 20, 2015 at 11:08 | comment | added | user5117 | "I think": yes, any projective variety can be embedded in some $\mathbf P^n$. "Can $X_k...$ ?" Assume wlog that all the polynomials have degree $\geq 2$. Use adjunction to calculate the anticanonical bundle of a complete intersection of $n-2$ things in $\mathbf P^n$. Observe that the answer is positive only for $n=3,d=2,3$ or $n=4,(d_1,d_2)=(2,2).$ | |
| May 20, 2015 at 11:08 | comment | added | Lev Borisov | Of course, $X_6$ is a cubic in $\mathbb P^3$. If you allow weighted projective spaces, then $X_7$ is a degree $4$ hypersurface in $\mathbb P(1,1,1,2)$. I can't think of other constructions off the top of my head. | |
| May 20, 2015 at 10:39 | history | asked | Ritwik | CC BY-SA 3.0 |