Let $f:X \to Y$ be a flat, family of smooth projective varieties. Assume futher that $Y$ is smooth. Suppose there exists a scheme $Y'$ such that the associated reduced scheme $Y'_{\mathrm{red}} \cong Y$. Does there exist a flat family of schemes over $Y'$ such that the pullback of this family to $Y$ is isomorphic to $X$? If not always can we impose some condition on $X$ or $Y$ so as to get a positive answer?


1 Answer 1


As formulated, there are many counterexamples, although I suspect the OP has a slightly different question in mind. For instance, let $Y$ be $\mathbb{P}^2$, let $X$ be the blowing up of the diagonal in $Y\times Y$ with either of its natural morphisms to $Y$, and let $Y'$ be the nonreduced scheme from Exericse II.5.9 of Hartshorne's "Algebraic Geometry" (I hope that is the right number -- I only have a Russian edition in front of me).

However, if you assume that your scheme $Y$ is affine, then by Exercise II.8.6 (hopefully), there is a retraction, say $r:Y'\to Y$. Thus you can pull back the smooth, projective morphism $f$ by $r$.

  • 1
    $\begingroup$ Jason, what do you mean by saying "assume that $\mathbb{P}^2$ is affine"? $\endgroup$
    – Sasha
    Jan 2, 2014 at 13:50
  • $\begingroup$ @Sasha: What? I never said that. The first paragraph is an example. The second paragraph is an additional hypothesis that makes the answer positive. $\endgroup$ Jan 2, 2014 at 14:13
  • $\begingroup$ Sorry, it means that I misunderstood you. $\endgroup$
    – Sasha
    Jan 2, 2014 at 15:09
  • $\begingroup$ @Sasha: Sorry for being unclear. I edited slightly to make this clearer. $\endgroup$ Jan 2, 2014 at 19:01

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