# 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 deeply 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 xi (its stacky points) and integers ni (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]

## References

1. Voight, John; Zureick-Brown, David. The canonical ring of a stacky curve. Memoirs of the American Mathematical Society. arXiv:1501.04657.
2. ^ a b c Landesman, Aaron; Ruhm, Peter; Zhang, Robin. "Spin canonical rings of log stacky curves". Annales de l'Institut Fourier. 66 (6): 2339–2383. arXiv:1507.02643. doi:10.5802/aif.3065.
3. ^ a b c Kresch, Andrew (2009). "On the geometry of Deligne-Mumford stacks". In Abramovich, Dan; Bertram, Aaron; Katzarkov, Ludmil; Pandharipande, Rahul; Thaddeus, Michael (eds.). Algebraic Geometry: Seattle 2005 Part 1. Proc. Sympos. Pure Math. 80. Providence, RI: Amer. Math. Soc. pp. 259–271. doi:10.5167/uzh-21342. ISBN 978-0-8218-4702-2.
4. ^ Behrend, Kai; Noohi, Behrang (2006). "Uniformization of Deligne-Mumford curves". J. Reine Angew. Math. 599: 111–153. arXiv:math/0504309.
5. ^ Johnson, Paul (2014). "Equivariant GW Theory of Stacky Curves". Communications in Mathematical Physics. 327 (2): 333–386. doi:10.1007/s00220-014-2021-1. ISSN 1432-0916.