Let $k$ be a field of characteristic zero.

Let $X$ be a finite type algebraic stack over $k$ with a coarse (or good) moduli space $M$.

Suppose that $M$ is isomorphic to a point, i.e., $M = Spec k$.

Examples of such stacks are classifying stacks $BG$, with $G$ a finite type group scheme over $k$.

Are there any other examples of such stacks? What if we impose extra conditions on $X$ such as smoothness, affine diagonal, etc?

  • $\begingroup$ What about $\mathbb C$ mod $\mathbb C^*$? $\endgroup$ – YZhou Jun 27 '15 at 8:57

I think you can make weird, degenerate examples. Starting with a scheme $X$, take the coequalizer of all the points of $X$, viewed as a maps from a point to $X$. I believe when you do this the colimit is nontrivial - given the original map from $X$ to $X$, there is no fpqc cover that forces it to be trivial, since the neigborhoods of any non-open point must include the other points.

A nicer example are the ones that locally look like $BG$. For instance $H^2(k,G)$ classifies gerbes on $k$ that are locally isomorphic to $BG$. For some $k$ and $G$ this is nontrivial and then this stack will have coarse moduli space a point.

One example that I'm not sure whether it can happen is a stack that is isomorphic over some larger field $k'$ to $BG$ for a group $G$ defined only over $k'$.


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