Stacky curve

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 X {\displaystyle {\mathfrak {X}}} 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 x_i (its stacky points) and integers n_i (its ramification orders) greater than 1.^[3] The canonical divisor of X {\displaystyle {\mathfrak {X}}} is linearly equivalent to the sum of the canonical divisor of X and a ramification divisor R:^[1]

K X ∼ K X + R . {\displaystyle K_{\mathfrak {X}}\sim K_{X}+R.}

Letting g be the genus of the coarse space X, the degree of the canonical divisor of X {\displaystyle {\mathfrak {X}}} is therefore:^[1]

d = deg ⁡ K X = 2 g − 2 + ∑ i = 1 r n i − 1 n i . {\displaystyle d=\deg K_{\mathfrak {X}}=2g-2+\sum _{i=1}^{r}{\frac {n_{i}-1}{n_{i}}}.}

A stacky curve is called hyperbolic if d is positive, Euclidean if d is zero, and spherical 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]