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Suppose that $g: X \to Y$ is a map of varieties (perhaps flat), and we have a subvariety $V \subset X$. Important: I want $V$ to be flat over $X$, otherwise there are trivial counter examples.

Consider now the $f : Bl_V X \to Y$ as a composition $Bl_V X \to X \to Y$, where $Bl_V X$ is the blow up. Let $y$ be a point in $Y$.

Is $f^{-1}(y) \cong Bl_{V \cap g^{-1}(y)} g^{-1}(y)$. In other words, can I blow up in families?

Example: Let's say I want to understand what happens when I blow up the plane at $n$ points. Let $H$ be the hilbert scheme of length n subscheme of $P^2$. Consider $H$ with its universal family $V \subset H \times P^2$. Then blow up $H \times P^2$ along $V$, to get $Bl_V (H \times P^2) \to H$. Are the fibers of this map the blow ups of $P^2$ at the corresponding subschemes?

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    What a title... – 2017-02-10
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    I thought it is a post by isis member or some joke lol – 2017-02-10
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    Do check the case of $X=Y\mathbb{P}^1, g:X\to Y$ the projection and $V$ a point in $X$. – 2017-02-10
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    @Mohan oh yeah ...It's amazing what wishful thinking will blind one to. But what if we impose a codimension in fibers condition? – 2017-02-10
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    @Mohan Really the case I'm thinking about is like this -- we have a subvariety in $U = P^2 \times Y$ that restricts to $n$ points in the plane on each fiber over $Y$ - so $V$ is flat over $Y$, unlike your example. I want to understand blowing up the plane at all of these points by blowing up $U$ along $V$. In particular, if $Y$ is the variety parametrizing 6 generic points in the plane $((P^2)^6 \setminus \Delta)/ \Sigma_6$, I'd like to build a family of smooth cubics over it in this way...the main hypothesis I want is that $V$ be flat over $Y$, so I'll edit my question accordingly. – 2017-02-10

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