the blowup of $\Bbb C^n$ at the origin is the subvariety of $ \Bbb P^{n-1} \times \Bbb C^n$ given by
$B = V(x_{i-1} y_j - x_{j-1} y_i \mid 1 \le i < j \le n)$.$\ \ \ $ (1)
I am interested in computing $B\cap (U_{i-1} \times \Bbb C^n)$ for $i\in \{1,...,n\}$, where $U_{i-1}=\Bbb P^{n-1} \setminus V(x_{i-1}) = \operatorname{Spec} \Bbb C [\frac{x_0}{x_{i-1}},...,\frac{x_{i-2}}{x_{i-1}},\frac{x_{i}}{x_{i-1}},...,\frac{x_{n-1}}{x_{i-1}}]$.
I see in the literature the following and I don't see where it comes from:
$B\cap (U_{i-1}\times \Bbb C^n) = \operatorname{Spec} \Bbb C[\frac{x_0}{x_{i-1}},...,\frac{x_{i-2}}{x_{i-1}},\frac{x_{i}}{x_{i-1}},...,\frac{x_{n-1}}{x_{i-1}}, y_i]$ (2)
I guess we have only $n$ coordinates because in the following expansion for expression (1), we have $B=\operatorname{Spec} \Bbb C[x_0,...,x_{i},\frac{y_{i+1}}{y_{i}},...,\frac{y_{n}}{y_{i}}]$ but I don't see how we get (2)
Thank you for your help
Here is the case of $n=2$, and one can extend this calculation in an obvious way to the general case.
In this case, $B$ in $P^1 \times C^2$ is defined by the single function $x_0y_1- x_1y_0$. In the chart $U_0 \times C^2$, where $U_0 = P^1 - V(x_0) = \operatorname{Spec} C[x_1/x_0]$, the equation $x_0y_1- x_1y_0 = 0$ can be written as $y_1 = (x_1/x_0) y_0$. Thus, the coordinate ring of $U_0 \times C^2$ is $$ C[x_1/x_0,y_0,y_1] = C[x_1/x_0,y_0]. $$
In the general case, on $U_0 \subset P^n$, use the equations $x_0y_j - x_jy_0 = 0$ to see $y_j = (x_j/x_0) y_0$ for $j>0$.