Timeline for Embed the normalization of a curve in a larger space
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
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| Feb 26, 2014 at 16:20 | comment | added | FedeB | (The up vote is mine). Thank you @JasonStarr, I was just being stupid. | |
| Feb 26, 2014 at 13:16 | comment | added | Jason Starr | "My question is about ..." Yes, of course. Denote by $\kappa:C^N\to \mathbb{P}^N_X$ any closed immersion of $X$-schemes. Then the morphism $(\iota\circ \phi,\kappa):C^N\to \mathbb{P}^n_X\times_X \mathbb{P}^N_X$ is also a closed immersion. | |
| Feb 26, 2014 at 11:03 | comment | added | FedeB | Of course $\phi$ is finite and surjective (all schemes are excellent here) and one can embed $C^N$ in some projective space over $X$, in general. My question is about the possibility of realizing this embedding so that the resulting morphism $\mathbb{P}^M_X\to\mathbb{P}^n_X$ is given by the actual projection on the first factors. | |
| Feb 26, 2014 at 10:59 | comment | added | Jason Starr | I recommend you look up "finiteness of integral closure" in a commutative algebra textbook -- Eisenbud deduces this as a corollary of the Noether normalization theorem. Then you might look at the exercises in Hartshorne, Chapter III around the Chevalley theorems regarding affiness and ampleness being stable for finite surjective morphisms (such as $\phi$). | |
| Feb 26, 2014 at 10:55 | comment | added | FedeB | @JasonStarr Yes, everything is of finite type over the base field $k$. | |
| Feb 26, 2014 at 10:53 | comment | added | Jason Starr | Is your $X$ finite type? If $X$ is a finite type $k$-scheme, there should be such a morphism. If $X$ is not finite type (i.e., not quasi-compact), I am pretty sure there are counterexamples, roughly the "strongly projective" versus "weakly projective" problem. | |
| Feb 26, 2014 at 10:52 | history | edited | FedeB | CC BY-SA 3.0 |
Fixed a typo.
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| Feb 26, 2014 at 10:25 | history | asked | FedeB | CC BY-SA 3.0 |