In mathematics, a stacky curve is an object in algebraic geometry that is roughly an algebraic curve with potentially "fractional points" called stacky points. A stacky curve is a type of stack used in studying Gromov–Witten theory, enumerative geometry, and rings of modular forms.
Stacky curves are closely related to 1-dimensional orbifolds and therefore sometimes called orbifold curves or orbicurves.
A stacky curve 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. [3] The canonical divisor of is linearly equivalent to the sum of the canonical divisor of X and a ramification divisor R: [1]
Letting g be the genus of the coarse space X, the degree of the canonical divisor of is therefore: [1]
A stacky curve is called spherical if d is positive, Euclidean if d is zero, and hyperbolic if d is negative. [3]
Although the corresponding statement of Riemann–Roch theorem does not hold for stacky curves, [1] there is a generalization of Riemann's existence theorem that gives an equivalence of categories between the category of stacky curves over the complex numbers and the category of complex orbifold curves. [1] [2] [4]
The generalization of GAGA for stacky curves is used in the derivation of algebraic structure theory of rings of modular forms. [2]
The study of stacky curves is used extensively in equivariant Gromov–Witten theory and enumerative geometry. [1] [5]