# Fibers of blow up in families

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.

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