Timeline for Alterations factor as modification + finite map
Current License: CC BY-SA 2.5
4 events
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| Mar 1, 2011 at 13:53 | comment | added | HNuer | Thanks, I looked up Stein factorization afterward and found it in its generality after I posted that question. Unfortunately I didn't see any of these answers until after I figured out the rest on my own. | |
| Mar 1, 2011 at 13:20 | comment | added | Karl Schwede | HNuer, see EGA III, Section 4.3 for Stein factorization in greater generality (ie, proper morphism), but Hartshorne's proof will work for you, I think he only uses projective because he only proved the "projective" case of a few results earlier. Now, connected fibers doesn't always guarantee birationality, but in your particular case it is birational. This follows from the actual construction of the intermediate scheme $Z$ in this case. In particular, it is clear that both $Z$ is isomorphic to $X$ at the generic points of $Z$ (see the proof). | |
| Mar 1, 2011 at 10:56 | comment | added | HNuer | What definition are you using for modification? I'm using it to mean a proper, birational morphism, and alteration to mean a proper, surjective, generically finite. Stein factorization seems only to be for projective morphisms. Also, why does having connected fibers imply birationality? Thanks | |
| Mar 1, 2011 at 10:34 | history | answered | Sándor Kovács | CC BY-SA 2.5 |