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It is well known that if two varieties are Fourier-Mukai partners, then there are strong constraints on them. For example, if one of them is Calabi-Yau the other one must be Calabi-Yau too. Similarly for smoothness (if one is smooth the other one must be smooth).

$\textbf{Question:}$ If $\pi: X \longrightarrow B$ is a fibration (for example elliptic fibration) over $B$, and similarly $\bar{\pi}: Y\longrightarrow \bar{B}$ another fibration over $\bar{B}$, and $D^b(X)\simeq D^b(Y)$ (also assume $\bar{B}$ and $B$ are birational). Then is it correct to say, $\pi$ is a flat morphism if and only if $\bar{\pi}$ is a flat morphism?

$\textbf{Comment:}$ What I have in mind is the case where $X$ and $Y$ are both elliptically fibered threefold, or at least genus one fibration.

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  • $\begingroup$ What is the condition on $B$ and $\bar B$? Do they have the same dimension? Otherwise there are trivial counter-examples. $\endgroup$ – Chen Jiang Jan 30 at 0:30
  • $\begingroup$ As @ChenJiang said, there are trivial counterexamples. Take your favorite non-flat morphism, for instance $X = Bl_p(\mathbb{P}^2) \to \mathbb{P}^2 = B$, and take $Y = X \to Spec(k) = \tilde{B}$. $\endgroup$ – Sasha Jan 30 at 7:02
  • $\begingroup$ Assume $B$ and $\bar{B}$ are isomorphic, or at least birational... $\endgroup$ – Mohsen Karkheiran Jan 30 at 16:09
  • $\begingroup$ I do not think birational works. For example, take $B$ be blowing up a point of a smooth variety $\bar B$, and then take $X=Y=B \times E$. $E$ is an elliptic curve or any smooth variety. The fibrations are natural projections. $\endgroup$ – Chen Jiang Jan 30 at 18:21

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