# Derived category and flatness

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.

• What is the condition on $B$ and $\bar B$? Do they have the same dimension? Otherwise there are trivial counter-examples. – Chen Jiang Jan 30 at 0:30
• 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 $Y = X \to Spec(k) = \tilde{B}$. – Sasha Jan 30 at 7:02
• Assume $B$ and $\bar{B}$ are isomorphic, or at least birational... – Mohsen Karkheiran Jan 30 at 16:09
• 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. – Chen Jiang Jan 30 at 18:21