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Required fields*

3
  • $\begingroup$ For (i) does "generic smoothness" not suffice? This is in Hartshorne, the chapter on smooth morphisms. $\endgroup$ Commented Feb 2, 2011 at 22:43
  • $\begingroup$ If the initial space is smooth, then yes, certainly. But it didn't seem that mmm was assuming this. $\endgroup$ Commented Feb 3, 2011 at 0:21
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    $\begingroup$ The result in Hartshorne (Lemma 10.5) only assumes that you have a dominant morphism of integral schemes of finite type over an algebraically closed field of charactertistic zero. Indeed, in this case there is a open dense subset where your schemes are non-singular varieties. If the morphism is also proper then we can use Ehresmann's theorem to deduce that it is a fibration, but I guess you need more in the non-proper case. $\endgroup$ Commented Feb 3, 2011 at 11:17