Let $X$ be a variety with a $G$-action by an algebraic group on it.

My question refers to a motivating example from

https://web.maths.unsw.edu.au/~danielch/thesis/mbrassil.pdf

Here the relevant excerpt:

Here the author discusses an example of $X/G$ in order to explaine that it is neccessary to form $X/G$ as categorical quotient and not the topological one.

We consider following motivating example introduced at page 27:

Here we take $X:= \mathbb{C}^2$ with action by $G:=\mathbb{C}^x$ via multiplication $\lambda \cdot (x,y) \mapsto (\lambda x, \lambda y)$.

Obviously the "naive" topological quotient consists set theoretically of the lines $\{(\lambda x, \lambda y) \vert \lambda \in \mathbb{C}^x \}$ and the origin $\{(0,0)\}$.

Topologically the origin lies in the closure of every line.

So the QUESTION is why does this argument already imply that $Y:=X/G$ cannot have a strucure of a variety? I don't understand the argument given by the author.

If we denote by $p:\mathbb{C}^2 \to Y$ the canonical projection map and by (continuity?) this map can't separate orbits, why does this already imply that $Y$ doesn't have structure of a variety as stated in the excerpt?

Especially which role does here play the fact that we can't separate the lines from the origin (in pure topologically way)? Does it cause an obstacle in order to form a variety structure on $X/G$?

Remark: I know that there are different ways to deduce that if we define $X/G$ pure topologically then it cannot have a structure of a variety. The most common argument is to introduce the invariant ring $R^G$ and to calculate it explicitely here. But the main issue of this question is it made me curious that the given argumentation seems to be a bit more "elementary" in sense that he doesn't explicitely work in this example with the concept of the invariant ring $R^G$.

Let $X$ be a variety with a $G$-action by an algebraic group.

My question refers to a motivating example from:

https://web.maths.unsw.edu.au/~danielch/thesis/mbrassil.pdf

Here is the relevant excerpt:

Here the author discusses an example of $X/G$ in order to explain that it is necessary to form $X/G$ as categorical quotient and not the topological one.

We consider this motivating example introduced at page 27:

Take $X:= \mathbb{C}^2$ with action by $G:=\mathbb{C}^\times$ via multiplication $\lambda \cdot (x,y) \mapsto (\lambda x, \lambda y)$.

Obviously the "naive" topological quotient consists set theoretically of the lines $\{(\lambda x, \lambda y) \vert \lambda \in \mathbb{C}^x \}$ and the origin $\{(0,0)\}$.

Topologically the origin lies in the closure of every line.

So the QUESTION is why does this argument already imply that $Y:=X/G$ cannot have the structure of a variety? I don't understand the argument given by the author.

If we denote by $p:\mathbb{C}^2 \to Y$ the canonical projection map and by (continuity?) this map can't separate orbits, why does this already imply that $Y$ doesn't have structure of a variety as stated in the excerpt?

Especially which role does the fact that we can't separate the lines from the origin play (in a purely topological way)? Does it cause an obstruction to form a variety structure on $X/G$?

Remark: I know that there are different ways to deduce that if we define $X/G$ purely topologically then it cannot have a structure of a variety. The most common argument is to introduce the invariant ring $R^G$ and to calculate it explicitly. It seems to me that the given argument is a bit too "elementary" in the sense that the author doesn't explicitly work in this example with the concept of the invariant ring $R^G$.

Let $X$ be a variety with a $G$-action by an algebraic group on it.

My question refers to a motivating example from

https://web.maths.unsw.edu.au/~danielch/thesis/mbrassil.pdf

Here the relevant excerpt:

Here the author discusses an example of $X/G$ in order to explaine that it is neccessary to form $X/G$ as categorical quotient and not the topological one.

We consider following motivating example introduced at page 27:

Here we take $X:= \mathbb{C}^2$ with action by $G:=\mathbb{C}^x$ via multiplication $\lambda \cdot (x,y) \mapsto (\lambda x, \lambda y)$.

Obviously the "naive" topological quotient consists set theoretically of the lines $\{(\lambda x, \lambda y) \vert \lambda \in \mathbb{C}^x \}$ and the origin $\{(0,0)\}$.

Topologically the origin lies in the closure of every line.

So the QUESTION is why does this argument already imply that $Y:=X/G$ cannot have a strucure of a variety? I don't understand the argument given by the author.

Which role does here play the fact that we can't separate the lines from the origin (in pure topologically way)? Does it cause an obstacle in order to form a variety structure on $X/G$?

If we denote by $p:\mathbb{C}^2 \to Y$ the canonical projection map and by (continuity?) this map can't separate orbits, why does this already imply that $Y$ doesn't have structure of a variety as stated in the excerpt?

Remark: I know that there are different ways to deduce that if we define $X/G$ pure topologically then it cannot have a structure of a variety. The most common argument is to introduce the invariant ring $R^G$ and to calculate it explicitely here. But the main issue of this question is it made me curious that the given argumentation seems to be a bit more "elementary" in sense that he doesn't explicitely work in this example with the concept of the invariant ring $R^G$.

Let $X$ be a variety with a $G$-action by an algebraic group on it.

My question refers to a motivating example from

https://web.maths.unsw.edu.au/~danielch/thesis/mbrassil.pdf

Here the relevant excerpt:

Here the author discusses an example of $X/G$ in order to explaine that it is neccessary to form $X/G$ as categorical quotient and not the topological one.

We consider following motivating example introduced at page 27:

Here we take $X:= \mathbb{C}^2$ with action by $G:=\mathbb{C}^x$ via multiplication $\lambda \cdot (x,y) \mapsto (\lambda x, \lambda y)$.

Obviously the "naive" topological quotient consists set theoretically of the lines $\{(\lambda x, \lambda y) \vert \lambda \in \mathbb{C}^x \}$ and the origin $\{(0,0)\}$.

Topologically the origin lies in the closure of every line.

So the QUESTION is why does this argument already imply that $Y:=X/G$ cannot have a strucure of a variety? I don't understand the argument given by the author.

If we denote by $p:\mathbb{C}^2 \to Y$ the canonical projection map and by (continuity?) this map can't separate orbits, why does this already imply that $Y$ doesn't have structure of a variety as stated in the excerpt?

Especially which role does here play the fact that we can't separate the lines from the origin (in pure topologically way)? Does it cause an obstacle in order to form a variety structure on $X/G$?

Remark: I know that there are different ways to deduce that if we define $X/G$ pure topologically then it cannot have a structure of a variety. The most common argument is to introduce the invariant ring $R^G$ and to calculate it explicitely here. But the main issue of this question is it made me curious that the given argumentation seems to be a bit more "elementary" in sense that he doesn't explicitely work in this example with the concept of the invariant ring $R^G$.