Timeline for $\mathbb{P}^n$ is simply connected
Current License: CC BY-SA 3.0
7 events
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| Apr 22, 2011 at 22:57 | history | edited | Sándor Kovács | CC BY-SA 3.0 |
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| Apr 22, 2011 at 22:56 | comment | added | Sándor Kovács | @Martin: The pull back of an ample divisor via a finite map is ample. This is an exercise in Hartshorne and can be proved using pretty much any characterization of ampleness. As for the degree: $f$ is unramified, so the degree is equal to the number of preimages of any (closed) point. This remains the same for any subvariety. | |
| Apr 22, 2011 at 20:11 | comment | added | Martin Brandenburg | How can we show that $f^* H$ is ample? And how the degree of $f$ is related with the degree of the restriction to $H$? | |
| Apr 19, 2011 at 21:23 | history | edited | Sándor Kovács | CC BY-SA 3.0 |
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| Apr 19, 2011 at 21:21 | comment | added | Sándor Kovács | @Martin: sorry, I meant to say that you can assume at the start that $X$ is connected. | |
| Apr 19, 2011 at 20:00 | comment | added | Martin Brandenburg | @Sandor: You cannot conclude that $f$ is an isomorphism, just a finite disjoint union of isomorphisms.. | |
| Apr 19, 2011 at 18:41 | history | answered | Sándor Kovács | CC BY-SA 3.0 |