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Let $T$ be a smooth curve over $\mathbb{C}$ and $p:\mathbb{P}^n \times T \to T$ the natural projection. Let $V \subset \mathbb{P}^n_T$ be a $T$-flat subscheme of codimension at least $2$ and $\pi: \mathrm{Bl}_V \mathbb{P}^n_T \to X$ be the blow-up of $\mathbb{P}^n \times T$ along $V$. Under what conditions on $V$ can we say that for all $t \in T$, the fiber over $t$ of the morphism $\pi$, is the blow-up of $\mathbb{P}^n$ along $V_t$ (for example, if $V$ is a regular embedding in $\mathbb{P}^n_T$)? Any idea or referece will be most welcome. The example that I have in mind, for every $t$, the fiber $V_t$ is zero dimensional.

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    $\begingroup$ I recommend that you search the key phrase “normal flatness”. $\endgroup$ – Jason Starr Jun 26 at 5:29

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