How to construct a blowup $f$: X $\rightarrow$ Y between non-singular quasi-projective varieties? This is an exercise from Shafarevich "Basic Algebraic Geometry 1".
I want to construct for any $n \geq$ 2 a regular birational map $f$: X $\rightarrow$ Y between $n$-dimensional quasi-projective varieties having an exceptional divisor $Z \subset X$ such that $f(Z) \subset Y$ has codimension 2.
Now: how to use properly the local systems of parameters in $\xi \in Y$ point of the blowup?
I have also problems to define X, with a right use of local systems of parameters, as a subset of $Y x \mathbb{P^{n-1}}$.