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Let $\pi:\mathcal{X} \to \mathbb{P}^1$ be a flat, projective family of noetherian schemes with generic fiber a smooth, projective variety. Let $p:\mathcal{Y} \to \mathcal{X}$ be another flat, projective family such that the composed morphism $\pi \circ p:\mathcal{Y} \to \mathbb{P}^1$ satisfies the property that for a general $t \in \mathbb{P}^1$, $\mathcal{Y}_t:=(\pi \circ p)^{-1}(t)$ is isomorphic to $\mathbb{P}^2_{\mathcal{X}_t}$, where $\mathcal{X}_t:=\pi^{-1}(t)$.

Does this imply that for any point $t \in \mathbb{P}^1$,$\mathcal{Y}_t \cong \mathbb{P}^2_{\mathcal{X}_t}$? If not true in general, is there any known condition under which this holds?

Any idea/reference in this direction will be very helpful.

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    $\begingroup$ That is not true. Let $\mathcal{X}$ be $X_0\times \mathbb{P}^1$ for a projective scheme $X_0$. Let $p:\mathcal{Y}\to \mathcal{X}$ be the projective bundle associated to a locally free sheaf $\mathcal{E}$ on $X_0\times \mathbb{P}^1$ that arises from a pencil of extensions $0\to K\to E \to Q\to 0$, where $K$ and $Q$ are fixed locally free sheaves on $X_0$, where the generic such extension has $E\cong \mathcal{O}_{X_0}^{\oplus 3}$, but where $K\oplus Q$ is not isomorphic to $\mathcal{O}_{X_0}^{\oplus 3}$. $\endgroup$
    – Jason Starr
    Commented Apr 29, 2017 at 22:43
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    $\begingroup$ For instance, if $X_0$ equals $\mathbb{P}^2$, then you can use the twisted Euler sequence, $0\to \Omega_{\mathbb{P}^2}(1)\to \mathcal{O}_{\mathbb{P}^2}^{\oplus 3} \to \mathcal{O}_{\mathbb{P}^2}(1)\to 0.$ $\endgroup$
    – Jason Starr
    Commented Apr 29, 2017 at 22:46
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    $\begingroup$ Another example: take $X = P^1$ and let $Y$ be the blowup of $X \times P^2$ at one point. Of course, it is flat over $X$, but its special fiber is a union of two components. $\endgroup$
    – Sasha
    Commented Apr 30, 2017 at 8:31

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