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Let $X$ be a Noetherian integral scheme. When does $X$ have a cover by affine open subschemes that are isomorphic as abstract schemes?

Does there exist an example when we have a cover by $n$ affine opens and a cover by some number of isomorphic affine opens but no cover by $n$ isomorphic affine opens?

Arbitrary Noetherian integral schemes are pretty non-geometric (to my taste), so we might restrict to integral separated schemes of finite type over a field.

Some thoughts

  • such a cover exists for affine schemes;
  • such a cover exists for projective spaces;
  • such a cover probably exists for smooth curves of genus 1 (I think the automorphism group should act transitively on the set of closed points);
  • I think that a proper curve over a field with no automorphisms should not have this property (the isomorphism between the affines should extend to a morphism between the curves themselves, which has to be bijective because the image is closed, right?).
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