Q:

Non-separated quotient of separated scheme

I am reading Mumford's GIT book. I found the following claim there.
Let $X$ be an algebraic variety. Let $G$ be an algebraic group acting on $X$. Then the categorical quotient of $X$ by $G$ may be non-separated.
Question:  Could you construct an example?

A:

This answer is given by Takumi Murayama. I am just writing down the details.
Consider the quotient of $X = \mathbb{C}^2 \backslash \{ 0 \}$ by $\mathbb{C}^*$ action $\lambda (x, y) = ( \lambda x, \lambda^{-1} y )$.
We can consider two charts 
$U_1 = \{ (x,y) \in \mathbb{C}^2 \backslash \{ 0 \}$   such that  $x \neq 0 \}$
$U_2 = \{ (x,y) \in \mathbb{C}^2 \backslash \{ 0 \}$   such that  $y \neq 0 \}$
Let me consider $z= xy$ and $t=x$ as coordinates on $U_1$. 
 $U_1 = \mathbb{C} \times \Big( \mathbb{C}  \backslash \{ 0 \} \Big) $ where $z \in \mathbb{C}$ and $t \in \mathbb{C} \backslash \{ 0 \}$. The action is given by $\lambda(z, t ) = (z, \lambda t)$. So the quotient $U_1 / G$ is $\mathbb{C}$ with coordinate $z = xy$.
If one swap $x$ and $y$, the same words could be repeated for $U_2$. So the quotient $U_2 / G$ is also $\mathbb{C}$ with coordinate $xy$. Quotient of $( U_1 \cap U_2 ) / G$ is $\mathbb{C} \backslash \{ 0 \}$. 
Quotient of whole thing $(U_1 \cup U_2)/G$ cab be obtained by gluing $U_1 / G$ and $U_2 / G$ along $(U_1 \cap U_2)/G$. So we get a line two origins indeed.

