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Let $X = [P/G]$ be a smooth finite type separated DM-stack over $\mathbb C$ given as the quotient of a smooth projective scheme $P$ by the action of a smooth (finite type separated) reductive group scheme $G$.

At atlas of $X$ is an etale morphism $U\to X$ with $U$ an algebraic space.

I'm interested in the converse of the following statement:

Thm. If the coarse moduli space of $X$ is a scheme (i.e., not just an algebraic space) then every atlas $U$ of $X$ is a scheme.

Proof. The map from an atlas $U$ to the coarse moduli space $X_c$ of $X$ is quasi-finite separated. Therefore $U$ is a scheme by Knutson Cor. II.6.16., p. 138 QED

So the converse would read as follows:

Q. Suppose that every atlas $U$ of $X$ is a scheme. Is the coarse moduli space of $X$ a scheme?

I expect the answer to be negative, but can't find a good example. Note that a counterexample can not be an algebraic space, as an algebraic space with the property that every atlas is a scheme is itself a scheme (the identity morphism being an atlas).

Edit: I have an application in mind of the above question, and in the context of my application the stack $X$ is even generically a scheme. Not sure if that helps...

Let $X = [P/G]$ be a smooth finite type separated DM-stack over $\mathbb C$ given as the quotient of a smooth quasi-projective scheme $P$ by the action of a smooth (finite type separated) reductive group scheme $G$. Since $X$ has finite inertia, the coarse space $X^c$ of $X$ exists as an algebraic space.

At atlas of $X$ is an étale morphism $U\to X$ with $U$ an algebraic space.

I sm interested in the converse of the following statement:

Thm. If the coarse moduli space of $X$ is a scheme,then every atlas $U$ of $X$ is a scheme.

Proof. The map from an atlas $U$ to the coarse moduli space $X^c$ of $X$ is quasi-finite separated. Therefore $U$ is a scheme by Knutson Cor. II.6.16., p. 138 QED

So the converse would read as follows:

Q. Suppose that every atlas $U$ of $X$ is a scheme. Is the coarse moduli space of $X$ a scheme?

I expect the answer to be negative, but can't find a good example. Note that a counterexample can not be an algebraic space, as an algebraic space with the property that every atlas is a scheme is itself a scheme (the identity morphism being an atlas).

Edit: I have an application in mind of the above question, and in the context of my application the stack $X$ is even generically a scheme. Not sure if that helps...

Let $X = [P/G]$ be a smooth finite type separated DM-stack over $\mathbb C$ given as the quotient of a smooth projective scheme $P$ by the action of a smooth (finite type separated) reductive group scheme $G$.

At atlas of $X$ is an etale morphism $U\to X$ with $U$ an algebraic space.

I'm interested in the converse of the following statement:

Thm. If the coarse moduli space of $X$ is a scheme (i.e., not just an algebraic space) then every atlas $U$ of $X$ is a scheme.

Proof. The map from an atlas $U$ to the coarse moduli space $X_c$ of $X$ is quasi-finite separated. Therefore $U$ is a scheme by Knutson Cor. II.6.16., p. 138 QED

So the converse would read as follows:

Q. Suppose that every atlas $U$ of $X$ is a scheme. Is the coarse moduli space of $X$ a scheme?

I expect the answer to be negative, but can't find a good example. Note that a counterexample can not be an algebraic space, as an algebraic space with the property that every atlas is a scheme is itself a scheme (the identity morphism being an atlas).

Let $X = [P/G]$ be a smooth finite type separated DM-stack over $\mathbb C$ given as the quotient of a smooth projective scheme $P$ by the action of a smooth (finite type separated) reductive group scheme $G$.

At atlas of $X$ is an etale morphism $U\to X$ with $U$ an algebraic space.

I'm interested in the converse of the following statement:

Thm. If the coarse moduli space of $X$ is a scheme (i.e., not just an algebraic space) then every atlas $U$ of $X$ is a scheme.

Proof. The map from an atlas $U$ to the coarse moduli space $X_c$ of $X$ is quasi-finite separated. Therefore $U$ is a scheme by Knutson Cor. II.6.16., p. 138 QED

So the converse would read as follows:

Q. Suppose that every atlas $U$ of $X$ is a scheme. Is the coarse moduli space of $X$ a scheme?

I expect the answer to be negative, but can't find a good example. Note that a counterexample can not be an algebraic space, as an algebraic space with the property that every atlas is a scheme is itself a scheme (the identity morphism being an atlas).

Edit: I have an application in mind of the above question, and in the context of my application the stack $X$ is even generically a scheme. Not sure if that helps...

Let $X = [P/G]$ be a smooth finite type separated DM-stack over $\mathbb C$ given as the quotient of a smooth projective scheme $P$ by the action of a smooth (finite type separated) reductive group scheme $G$.

At atlas of $X$ is an etale morphism $U\to X$ with $U$ an algebraic space.

I'm interested in the converse of the following statement:

Thm. If the coarse moduli space of $X$ is a scheme (i.e., not just an algebraic space) then every atlas $U$ of $X$ is a scheme.

Proof. The map from an atlas $U$ to the coarse moduli space $X_c$ of $X$ is quasi-finite separated. Therefore $U$ is a scheme by Knutson Cor. II.6.16., p. 138 QED

So the converse would read as follows:

Q. Suppose that every atlas $U$ of $X$ is a scheme. Is the coarse moduli space of $X$ a scheme?

I expect the answer to be negative, but can't find a good example. Note that a counterexample can not be an algebraic space, as an algebraic space with the property that every atlas is a scheme is itself a scheme (the identity morphism being an atlas).

Let $X = [P/G]$ be a smooth finite type separated DM-stack over $\mathbb C$ given as the quotient of a smooth projective scheme $P$ by the action of a smooth (finite type separated) reductive group scheme $G$.

At atlas of $X$ is an etale morphism $U\to X$ with $U$ an algebraic space.

I'm interested in the converse of the following statement:

Thm. If the coarse moduli space of $X$ is a scheme (i.e., not just an algebraic space) then every atlas $U$ of $X$ is a scheme.

Proof. The map from an atlas $U$ to the coarse moduli space $X_c$ of $X$ is quasi-finite separated. Therefore $U$ is a scheme by Knutson Cor. II.6.16., p. 138 QED

So the converse would read as follows:

Q. Suppose that every atlas $U$ of $X$ is a scheme. Is the coarse moduli space of $X$ a scheme?

I expect the answer to be negative, but can't find a good example. Note that a counterexample can not be an algebraic space, as an algebraic space with the property that every atlas is a scheme is itself a scheme (the identity morphism being an atlas).