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Keel-Mori's theorem says an algebraic stack with a finite diagonal over a scheme S has a coarse moduli space. What is an example of an algebraic stack without coarse moduli space?

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[A^1/Gm] is one example. You can check that any Gm invariant map from A^1 to a scheme is constant. Thus the map from [A^1/Gm] to the point is universal for maps to schemes, but is not a bijection on geometric points (since [A^1/Gm] has two geometric points).

Check out Jarod Alper's thesis to learn more.

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  • $\begingroup$ This is super! I love it. $\endgroup$ – Maharana Dec 29 '09 at 13:30
  • $\begingroup$ @unknown (google): No. $\endgroup$ – David Zureick-Brown Apr 16 '10 at 19:22

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