Stacky curve

A stacky curve is a type of stack used in studying Gromov–Witten theory, enumerative geometry, and rings of modular forms.

over a field k is a smooth proper geometrically connected Deligne–Mumford stack of dimension 1 over k that contains a dense open subscheme.

[1][2][3] A stacky curve is uniquely determined (up to isomorphism) by its coarse space X (a smooth quasi-projective curve over k), a finite set of points xi (its stacky points) and integers ni (its ramification orders) greater than 1.

is therefore:[1] A stacky curve is called spherical if d is positive, Euclidean if d is zero, and hyperbolic if d is negative.

[1][2][4] The generalization of GAGA for stacky curves is used in the derivation of algebraic structure theory of rings of modular forms.