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Let $X$ be a projective variety and $Y$ an Artin stack. Suppose that $f:X\to Y$ is a morphism of Artin stacks. Is $f(X)$ necessarily a closed substack of $Y$?

This seems like it should be true and probably one can find it somewhere in the stacks project, but I cannot locate a good source.

I am especially interested in the case where $Y$ is the quotient of an affine variety by the action of a reductive group (in particular, a torus). I have asked a separate question here asking whether or not this condition may force $Y$ to be a variety.

Let $X$ be a projective variety and $Y$ an Artin stack. Suppose that $f:X\to Y$ is a morphism of Artin stacks. Is $f(X)$ necessarily a closed substack of $Y$?

This seems like it should be true and probably one can find it somewhere in the stacks project, but I cannot locate a good source.

I am especially interested in the case where $Y$ is the quotient of an affine variety by the action of a reductive group (in particular, a torus). I have asked a separate question here asking whether or not this condition may force $Y$ to be a variety.

Let $X$ be a projective variety and $Y$ an Artin stack. Suppose that $f:X\to Y$ is a morphism of Artin stacks. Is $f(X)$ necessarily a closed substack of $Y$?

This seems like it should be true and probably one can find it somewhere in the stacks project, but I cannot locate a good source.

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LAGC
LAGC
  • 143
  • 4

Let $X$ be a projective variety and $Y$ an Artin stack. Suppose that $f:X\to Y$ is a morphism of Artin stacks. Is $f(X)$ necessarily a closed substack of $Y$?

This seems like it should be true and probably one can find it somewhere in the stacks project, but I cannot locate a good source.

I also have a finer question which would answer this question trivially. I am happy to assume thatespecially interested in the case where $Y$ is the quotient of an affine variety by the action of a reductive group (in particular, a torus). Does this automatically imply that $Y$ isI have asked a variety (in which case the answer to my firstseparate question is obviously yes)? Doeshere asking whether or not this imply thatcondition may force $Y$ is a scheme (in which case I don't know the answer to my first question)?be a variety.

Let $X$ be a projective variety and $Y$ an Artin stack. Suppose that $f:X\to Y$ is a morphism of Artin stacks. Is $f(X)$ necessarily a closed substack of $Y$?

This seems like it should be true and probably one can find it somewhere in the stacks project, but I cannot locate a good source.

I also have a finer question which would answer this question trivially. I am happy to assume that $Y$ is the quotient of an affine variety by the action of a reductive group (in particular, a torus). Does this automatically imply that $Y$ is a variety (in which case the answer to my first question is obviously yes)? Does this imply that $Y$ is a scheme (in which case I don't know the answer to my first question)?

Let $X$ be a projective variety and $Y$ an Artin stack. Suppose that $f:X\to Y$ is a morphism of Artin stacks. Is $f(X)$ necessarily a closed substack of $Y$?

This seems like it should be true and probably one can find it somewhere in the stacks project, but I cannot locate a good source.

I am especially interested in the case where $Y$ is the quotient of an affine variety by the action of a reductive group (in particular, a torus). I have asked a separate question here asking whether or not this condition may force $Y$ to be a variety.

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LAGC
LAGC
  • 143
  • 4

Let $X$ be a projective variety and $Y$ an Artin stack. Suppose that $f:X\to Y$ is a morphism of Artin stacks. Is $f(X)$ necessarily a closed substack of $Y$?

This seems like it should be true and probably one can find it somewhere in the stacks project, but I cannot locate a good source.

I also have a finer question which would answer this question trivially. I am happy to assume that $Y$ is the quotient of an affine variety by the action of a reductive group scheme(in particular, a torus). Other assumptionsDoes this automatically imply that would make$Y$ is a variety (in which case the answer to my first question affirmative are also interesting.is obviously yes)? Does this imply that $Y$ is a scheme (in which case I don't know the answer to my first question)?

Let $X$ be a projective variety and $Y$ an Artin stack. Suppose that $f:X\to Y$ is a morphism of Artin stacks. Is $f(X)$ necessarily a closed substack of $Y$?

This seems like it should be true and probably one can find it somewhere in the stacks project, but I cannot locate a good source. I am happy to assume that $Y$ is the quotient of an affine variety by the action of a group scheme. Other assumptions that would make the answer to my question affirmative are also interesting.

Let $X$ be a projective variety and $Y$ an Artin stack. Suppose that $f:X\to Y$ is a morphism of Artin stacks. Is $f(X)$ necessarily a closed substack of $Y$?

This seems like it should be true and probably one can find it somewhere in the stacks project, but I cannot locate a good source.

I also have a finer question which would answer this question trivially. I am happy to assume that $Y$ is the quotient of an affine variety by the action of a reductive group (in particular, a torus). Does this automatically imply that $Y$ is a variety (in which case the answer to my first question is obviously yes)? Does this imply that $Y$ is a scheme (in which case I don't know the answer to my first question)?

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LAGC
LAGC
  • 143
  • 4