I am reading the book "Algebraic Geometry and Statistical Learning Theory" by Sumio Watanabe and have a question regarding Remark 3.16 (1) on page 95.
He defines the blow-up of $ \mathbb{R}^2$ with center $V=\lbrace 0\rbrace$ as $B_V(\mathbb{R}^2)=\overline{\lbrace (x,y,(x:y))\in\mathbb{R}^2\times\mathbb{P}^1\vert (x,y)\in\mathbb{R}^2\setminus V\rbrace }$ where $\mathbb{P}^1$ denotes the real projective line. Then after using the typical identification of $(x:y)$ with $(1:z)$ where \begin{align} z=\begin{cases}\frac{y}{x}\,,\quad x\neq 0\\ \infty\,,\quad x=0\end{cases}\end{align} some easy computation shows that $B_V(\mathbb{R}^2) = \lbrace (x,y,z)\vert y=zx\rbrace\subset\mathbb{R}^2\times\mathbb{P}^1$. This approach is understandable for me. However in the remark the author writes that the above blow-up is equivalent to the substitution \begin{align} x=u=st\,,\quad y=uv=s \end{align} and then gluing the two coordinates $(u,v)$ and $(s,t)$. Unfortunately there are no more comments regarding this procedure. As far as I understand this, he defines two sets
\begin{align} U_1=\lbrace (u,v)\vert u,v\in\mathbb{R}\rbrace\,\quad U_2=\lbrace(s,t)\vert s,t\in\mathbb{R}\rbrace \end{align}
and then sets $U=U_1\sqcup U_2/\sim$ where $\sim$ is defined by the above two equations. But I have no idea how $U$ is the same as $B_V(\mathbb{R}^2)$. Could someone please explain me this approach?
Best regards
You can define the blow-up by $$ Bl_0(\mathbb{R}^2) = \{(x,y;u:t) \in \mathbb{R}^2\times \mathbb{P}^1 \mid xt=yu \} $$ Note that this implies that $(x:y) = (u:t)$.
You'll get two charts: $\{ u=1 \}$ and $\{t=1\}$. For the first you get $$Bl_0(\mathbb{R}^2) \cap \{ u=1 \}= \{(x,y;1:t) \in \mathbb{R}^2\times \mathbb{P}^1 \mid xt=y \} = \{(x,y,t) \in \mathbb{R}^3 \mid xt=y \}$$ with coordinates $(x,t)$ and the blow-up map is given by the projection to $\mathbb{R}^2$: $$ \pi(x,y;u:t) = (x,y). $$ In coordinates this means the composition $$ (x,t) \mapsto (x,xt;1:t) \mapsto (x,xt) $$ The other chart is similar. The gluing data comes from the gluing data of $\mathbb{P}^1$, $u=1/t$.
Let me know if you need to discuss it further.