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Given a connected scheme $X$ proper over $\mathbb{C}$, does there exist a scheme $X'$ affine and of finite type over $\mathbb{C}$, and a Zariski locally trivial $\mathbb{C}$-morphism $X'\rightarrow X$ with fibers isomorphic to affine spaces? I believe that this is true if $X$ is either smooth or projective.

  • is this a question not about research mathematics within the scope defined by the community? –  Jun 07 '19 at 10:45
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    First, the conditions imply that $X$ is smooth. Second, your question seems premature since if you google Jouanolou's trick you quickly learn that it works for any qcqs scheme by work of Thomason. – Piotr Achinger Jun 07 '19 at 10:46
  • @PiotrAchinger Wikipedia seems to say "Let X be a quasicompact and quasiseparated scheme with an ample family of line bundles. Then an affine vector bundle torsor over X exists." Not every qcqs scheme. –  Jun 07 '19 at 10:47
  • @PiotrAchinger I had a slightly wrong formulation, now corrected. –  Jun 07 '19 at 10:49
  • The new version of the question does not follow from Thomason and therefore seems suitable for MO. I wonder if the existence of $X'$ implies existence of an ample family of line bundles. – Piotr Achinger Jun 07 '19 at 10:50
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    Possible duplicate of The Jouanolou trick – Denis Nardin Jun 07 '19 at 11:30
  • I do not think that any of the answers there precisely address this question. –  Jun 07 '19 at 11:31
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    A simple way to construct proper varieties $X$ without an ample family of bundles is to take two disjoint isomorphic curves of different degrees in some projective space $Y$ and then identify points on one curve with points on the other (using any isomorphism). It seems quite likely to me that such varieties give a counterexample, but I don't see any simple way of proving this. – naf Jun 07 '19 at 12:34

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