Timeline for Cubic hypersurfaces of complex projective space
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Feb 17, 2012 at 17:45 | history | edited | gio | CC BY-SA 3.0 |
added 145 characters in body
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| Jun 17, 2011 at 13:50 | answer | added | Remke Kloosterman | timeline score: 6 | |
| Jun 17, 2011 at 13:26 | comment | added | Francesco Polizzi | Sorry yiou are right, I misread your comment. Now I understand what you say. Well, usually "factorial" means that every codimension 1 subvariety is cut out by an hypersurface in the ambient space, and my answer shows that it is not always the case. But, as you remark, if C is not smooth this is not equivalent to $\textrm{Pic}(C)=Z$ . | |
| Jun 17, 2011 at 13:04 | comment | added | Parsa | Why do you need smoothness? In the case $D$ is reduced, you have the Lefschetz hyperplane theorem, and if $D$ is reduced means its exponential sequence is exact, so you use the Kodaira vanishings on $X$ to get the vanishings you need on $D$, and the 5-lemma on the long exact sequence associated to the exponential sequences of $X$ and $D$ gives you the isomorphism $H^1(X,\mathcal {O_X}^*) \cong H^1(D,\mathcal {O_D}^*)$. | |
| Jun 17, 2011 at 12:55 | answer | added | Francesco Polizzi | timeline score: 4 | |
| Jun 17, 2011 at 12:39 | comment | added | Francesco Polizzi | Parsa, this is true when $C$ is smooth. | |
| Jun 17, 2011 at 12:36 | comment | added | Parsa | By the Grothendieck-Lefschetz theorem on Picard groups, the restriction map $Pic(X) \rightarrow Pic(D)$ is an isomorphism when $D$ is an ample effective divisor on $X$ and the dimension of $X$ is at least $4$. So the second part of your question is always the case. See Ample Subvarieties of Alg Var's by Hartshorne (Corollary IV.3.3) and Positivity in Alg. Geom. I Remark (3.1.26.) | |
| Jun 17, 2011 at 9:47 | history | asked | gio | CC BY-SA 3.0 |