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LSpice
LSpice
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I am trying to understand algebraic stacks and I have a newbie question. Let $X$ be an affine variety over an algebraically closed field to keep things simple and let $G$ be a reductive group acting on $X$. Then the categorical quotient $X//G$ does not necessarily "classify" all $G$-orbits in $X$. (The simplest example being $G=GL(2,\mathbb K)$$G=\operatorname{GL}(2,\mathbb K)$ action on $X=\mathbb K^2$ where $X//G$ is a point, while there are two orbits of $G$ in $X$.) Do algebraic stacks resolve that "shortcomming""shortcoming" of categorical quotients?

I am trying to understand algebraic stacks and I have a newbie question. Let $X$ be an affine variety over an algebraically closed field to keep things simple and let $G$ be a reductive group acting on $X$. Then the categorical quotient $X//G$ does not necessarily "classify" all $G$-orbits in $X$. (The simplest example being $G=GL(2,\mathbb K)$ action on $X=\mathbb K^2$ where $X//G$ is a point, while there are two orbits of $G$ in $X$.) Do algebraic stacks resolve that "shortcomming" of categorical quotients?

I am trying to understand algebraic stacks and I have a newbie question. Let $X$ be an affine variety over an algebraically closed field to keep things simple and let $G$ be a reductive group acting on $X$. Then the categorical quotient $X//G$ does not necessarily "classify" all $G$-orbits in $X$. (The simplest example being $G=\operatorname{GL}(2,\mathbb K)$ action on $X=\mathbb K^2$ where $X//G$ is a point, while there are two orbits of $G$ in $X$.) Do algebraic stacks resolve that "shortcoming" of categorical quotients?

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Adam
Adam
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I am trying to understand algebraic stacks and I have a newbie question. Let $X$ be an affine variety over an algebraically closed field to keep things simple and let $G$ be a reductive group acting on $X$. Then the categorical quotient $X//G$ does not necessarily "classify" all $G$-orbits in $X$. (The simplest example being $G=GL(2,\mathbb K)$ action on $X=\mathbb K^2$ where $X//G$ is a point, while there are two orbits of $G$ in $X$.) Do algebraic stacks resolve that "shortcomming" of categorical quotients?

I am trying to understand algebraic stacks and I have a newbie question. Let $X$ be an affine variety over algebraically closed field to keep things simple and let $G$ be a reductive group acting on $X$. Then the categorical quotient $X//G$ does not necessarily "classify" all $G$-orbits in $X$. (The simplest example being $G=GL(2,\mathbb K)$ action on $X=\mathbb K^2$ where $X//G$ is a point, while there are two orbits of $G$ in $X$.) Do algebraic stacks resolve that "shortcomming" of categorical quotients?

I am trying to understand algebraic stacks and I have a newbie question. Let $X$ be an affine variety over an algebraically closed field to keep things simple and let $G$ be a reductive group acting on $X$. Then the categorical quotient $X//G$ does not necessarily "classify" all $G$-orbits in $X$. (The simplest example being $G=GL(2,\mathbb K)$ action on $X=\mathbb K^2$ where $X//G$ is a point, while there are two orbits of $G$ in $X$.) Do algebraic stacks resolve that "shortcomming" of categorical quotients?

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Adam
Adam
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Do quotient stacks help classify the orbits of group actions on varieties?

I am trying to understand algebraic stacks and I have a newbie question. Let $X$ be an affine variety over algebraically closed field to keep things simple and let $G$ be a reductive group acting on $X$. Then the categorical quotient $X//G$ does not necessarily "classify" all $G$-orbits in $X$. (The simplest example being $G=GL(2,\mathbb K)$ action on $X=\mathbb K^2$ where $X//G$ is a point, while there are two orbits of $G$ in $X$.) Do algebraic stacks resolve that "shortcomming" of categorical quotients?