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.
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$\begingroup$ is this a question not about research mathematics within the scope defined by the community? $\endgroup$– user141498Jun 7, 2019 at 10:45
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4$\begingroup$ 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. $\endgroup$– Piotr AchingerJun 7, 2019 at 10:46
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$\begingroup$ @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. $\endgroup$– user141498Jun 7, 2019 at 10:47
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3$\begingroup$ Possible duplicate of The Jouanolou trick $\endgroup$– Denis NardinJun 7, 2019 at 11:30
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2$\begingroup$ 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. $\endgroup$– nafJun 7, 2019 at 12:34
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