# Blowup of a line and a point

I need to construct a morphism $f:X\to \mathbb{A}^3$ which is surjective, $X$ needs to be irreducible as does each fiber, and the dimension of $f^{-1}(0,0,0)$ must be $2$ while the dimension of the fiber of line minus a point $(x=y=0,\,z\neq 0)\subset \mathbb{A}^3$ must be $1$.

The suggestion is to use the blowup of the point $(0,0)$ in $\mathbb{A}^2$ and then consider the morphism $BL_0(\mathbb{A}^2)\times Id:\mathbb{A^2}\times \mathbb{P}^1\times \mathbb{A}^1\to \mathbb{A}^3$, but I can't see how to go from there to blowing up the origin of $\mathbb{A}^3$ to get something which is both irreducible and satisfies the requirements of the dimensions. Any suggestions?

Edit: Forgot to mention the fibers outside of the line and the origin described above must be dimension zero.

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So, I think the general idea is to generalize cleverly the blowup of a point of affine space, which is essentially some sort of incidence relation (specifically, the relation $\{(P,L)\mid P\in \mathbb{A}^n,\,L\in \mathbb{P}^{n-1},\,P\in L\}$. More generally, we can think about a tautological bundle over some projective variety, so we basically put one bundle (blowup of origin of $\mathbb{A}^3$) on top of another bundle (blowup of the origin of $\mathbb{A}^2$). – rondo9 Dec 3 '12 at 22:23