Timeline for Finite map from quasi-projective variety
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
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| Mar 1, 2012 at 9:27 | comment | added | Olivier Benoist | You are right : this is much simpler. Moreover, it constructs an explicit ample line bundle on $Y$ (I don't think it is possible to get one with the arguments I gave). On the other hand, Kollar's result says something even if $Y$ is not normal. For example, if $Y$ were only known to be an algebraic space, then it is automatically a scheme. | |
| Mar 1, 2012 at 5:18 | comment | added | Torsten Wedhorn | I agree. Although there is more elementary way to see this by a nice and easy construction in EGA2, 6.5: To simplify things I assume that all schemes are of finite type over some noetherian ring $R$. In EGA2 it is explained that for a finite morphism of schemes $f: X \to Y$ you can construct a group homomorphism $N_{X/Y}: {\rm Pic}(X) \to {\rm Pic}(Y)$ if $f$ is flat of if $Y$ is normal. Moreover in EGA2, 5.6, it is then shown that if $L$ is an ample line bundle on $X$, then $N_{X/Y}(L)$ is an ample line bundle on $Y$. This shows that $Y$ is quasi-projective if $f$ is flat or $Y$ is normal. | |
| Feb 29, 2012 at 19:54 | comment | added | Karl Schwede | I just want to say wow, that's really interesting. | |
| Feb 29, 2012 at 17:42 | history | answered | Olivier Benoist | CC BY-SA 3.0 |