Examples
- The moduli space of algebraic curves (Deligne–Mumford stack) defined as a universal family of curves of given genus g does not exist as a fine moduli space as an algebraic variety because in particular there are elliptic curves admitting nontrivial automorphisms (though there is an algebraic variety forming a coarse moduli space). For elliptic curves over the complex numbers the corresponding stack is a quotient of the upper half-plane by the action of the modular group.
Read more about this topic: Algebraic Stack
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