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.

Definition

A stacky curve ${\displaystyle {\mathfrak {X}}}$ 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]

Properties

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 ${\displaystyle {\mathfrak {X}}}$ is linearly equivalent to the sum of the canonical divisor of X and a ramification divisor R:^[1]

${\displaystyle K_{\mathfrak {X}}\sim K_{X}+R.}$

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

${\displaystyle d=\deg K_{\mathfrak {X}}=2-2g-\sum _{i=1}^{r}{\frac {n_{i}-1}{n_{i}}}.}$

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]

Applications

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]