TITLE: Blow-up along an ideal sheaf
QUESTION [5 upvotes]: Let $k^2=\operatorname{Spec} \; k[x,y]$ where $k$ is an algebraically closed field. Let $\mathcal{I}$ be the ideal sheaf defined by $(x,y)$. Then 
$$
Bl_{\mathcal{I}}k^2 
$$
is covered by two open charts  $\operatorname{Spec} \; k[x, y/x] \cup \operatorname{Spec}\; k[y,x/y]$. 
Q1: Why can each chart be described by 
$$
\operatorname{Spec} \; k[x,y][t]/(tx-y) \mbox{ and } \operatorname{Spec} \; k[x,y][t]/(ty-x)?  
$$
Q2: Isn't $Bl_{\mathcal{I}}k^2=\operatorname{Proj}(\oplus_{i\geq 0} (Rx\oplus Ry)^i t^i)$? 
Q3.a: Now let $k^3 =\operatorname{Spec} \; k[x,y,z]$ with $\mathcal{I}$ being defined by $(x,y,z).$ Then 
isn't 
$$
Bl_{\mathcal{I}}k^3 = \operatorname{Spec} \; k[x,y/x,z/x] \cup \operatorname{Spec}\; k[y,x/y,z/y] \cup \operatorname{Spec}\; k[z,x/z,y/z]? 
$$
Q3.b: How can one see that the charts 
$$
         \operatorname{Spec}\; k[x,y,z][t_1, t_2]/(t_1 x - y, t_2 x-z) $$
$$\operatorname{Spec}\; k[x,y,z][t_1, t_2]/(t_1 y - x, t_2 y-z) 
$$ 
$$\operatorname{Spec}\; k[x,y,z][t_1, t_2]/(t_1 z - x, t_2 z-y) 
$$ 
also cover $Bl_{\mathcal{I}}k^3$? 
$$
$$

REPLY [2 votes]: Let $A=k[x,y]$, then $$k[x,y/x]=k[x,y,y/x]=A[y/x]=A[t]/(y-tx).$$ For the last equality, see the comments below. Similar equalities apply to the other $k[y,x/y]$.
By definition the blowing-up of $k^2$ along the origin should be
$$\operatorname{Proj}(\oplus_{i\geq0}I^i)\longrightarrow k^2,$$
where $I=(x,y)$.
The blowing-up of the affine space $k^n$ along the origin is (see Liu,8.1.13)
$$\operatorname{Proj}(k[t_1,...,t_n][T_1,...,T_n]/(t_iT_j-t_jT_i))\longrightarrow k^n.$$
For your $n=3$ case, we have
$$D_+(T_1)=\operatorname{Spec}(k[t_1,t_2,t_3][\frac{T_2}{T_1},\frac{T_3}{T_1}]/(t_2-t_1\frac{T_2}{T_1},t_3-t_1\frac{T_3}{T_1})),$$
and similar for $D_+(T_2),D_+(T_3)$. By re-writting the symbols carefully these are the three open charts you mentioned exactly.