Brief summary of the question:
Let $C\subset \mathbb A^n$ be a singular curve and $\pi:X=Bl_C\mathbb A^n\to \mathbb A^n$ be the blow-up along $C$.
1) Is there a reference showing that $\pi^{-1}(C)=\mathbb P(\mathcal N_{C/\mathbb A^n})$?
2) Is $\pi^{-1}(C)\to C$ a locally trivial algebraic $\mathbb P^{n-2}$-bundle, i.e. locally looking like $C\times \mathbb P^{n-2}$? Or is it rather looking like $Bl_pC\times \mathbb P^{n-2}$ where $p$ is the singular point of $C$?
I was reading about blow-ups along sub varieties recently in Shafarevich's book and have a question concerning it.
Let us take a curve $C$ in $\mathbb A^n$ and consider $X=Bl_C\mathbb A^n$. Since $C$ is one-dimensional, we need (locally) $m=n-1$ equations to define it and thus, according to my textbook, $$X\subset \mathbb A^n\times \mathbb P^{m-1}=\mathbb A^n\times \mathbb P^{n-2}.$$
Back to the curve case, locally the situation can be described explicitly and over each point of $C$ we find a $\mathbb P^{n-2}$ in $X$. In particular, $\pi^{-1}(C)\to C$ is an algebraic fibre bundle with fibres $\mathbb P^{n-2}$. By $\pi$ I of course mean the blow-up map. Shafarevich further gives the nice global description that $$\pi^{-1}(C)=\mathbb P(N_{C/\mathbb A^n}),$$ where $N_{C/ \mathbb A^n}$ denotes the normal bundle.
So far so good, I think I understood that, at least the parts from the book. Now, if $C$ is a curve with a singular point, say $p\in C$, I think we have a similar description and that $$\pi^{-1}(C)=\mathbb P(\mathcal N_{C/ \mathbb A^n}),$$ where $\mathcal N_{C/ \mathbb A^n}$ denotes the normal sheaf. Is this correct? The only reference I have for blow-ups along general subschemes I could find was the Algebraic geometry book by Hartshorne and I can't translate his description of blowing up into the one by Shafarevich. So if you can give me some reference you would really help me.
Up to this point I think myself to be more or less on the safe side. However, I was thinking about how $\mathbb P (\mathcal N_{C/ \mathbb A^n})$ will look like in general and how I can interpret such a blow-up along $C$. My naive thinking is that we kind of blow-up all point on $C$ at once. But then we would in particular blow-up the singular point of $C$. So does $\mathcal N_{C/ \mathbb A^n}$ locally look like $C\times \mathbb P^{n-2}$ or rather like $Bl_pC\times \mathbb P^{n-2}$? And if the latter is the case, what happens to the divisor from the blow-up of $C$ at $p$?
I hope this question is not too vague. If there is a way to improve it, please let me know. I'm aware that a question is supposed to show "research effort" but since these questions just came to me whilst reading a textbook, I don't really have a clue of how to start and was unable to say more than what I just wrote.
EDIT: Here are some examples:
1) $C=\{x^2+y^2+z=z=0\}\subset \mathbb A^3$. The blow-up is given in the respective affine charts by $a_0(x^2+y^2)=0$ and $a_1(x^2+y^2)=0$. So here it looks like $C\times \mathbb P^1$.
2) $C=\{xy+z^2=x^3+y^3=0\}\subset \mathbb A^3$. In the respective charts we obtain $$a_0(xy+z^2)=x^3+y^3=0\quad\text{and}\quad xy+z^2=a_1(x^3+y^3)=0.$$ And here I am unable to proceed. Probably more light can be shed on the situation by computing the normal sheaf explicitly. I asked a related question Compute normal sheaf from equations some time ago but I can't manage to adapt the answer to the new situation here.
Best Answer
1) In general when you blow up $X$ along $Y$, the pre-image of $Y$ is the projectivisation of the normal cone of $Y$ in $X$ (essentially by definition of the normal cone). When $Y$ is regularly embedded in $X$, then the normal cone is the same as its normal bundle, and you recover the usual notion that the exceptional divisor is the projectivisation of the normal bundle.
You can find this in the appendix of Fulton's intersection theory.
2) As above, if $C$ is regularly embedded then you'll find that the exceptional divisor is a projective bundle over $C$ (you should see this in your examples!). If $C$ is not regularly embedded, then what you'll have is a projective cone over $C$.