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 \( {\mathfrak {X}} \) 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 \( {\mathfrak {X}} \) 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 \( {\mathfrak {X}} \) 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
Voight, John; Zureick-Brown, David (2015). The canonical ring of a stacky curve. Memoirs of the American Mathematical Society. arXiv:1501.04657. Bibcode:2015arXiv150104657V.
Landesman, Aaron; Ruhm, Peter; Zhang, Robin (2016). "Spin canonical rings of log stacky curves". Annales de l'Institut Fourier. 66 (6): 2339–2383. arXiv:1507.02643. doi:10.5802/aif.3065.
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. CiteSeerX 10.1.1.560.9644. doi:10.5167/uzh-21342. ISBN 978-0-8218-4702-2.
Behrend, Kai; Noohi, Behrang (2006). "Uniformization of Deligne-Mumford curves". J. Reine Angew. Math. 599: 111–153. arXiv:math/0504309. Bibcode:2005math......4309B.
Johnson, Paul (2014). "Equivariant GW Theory of Stacky Curves" (PDF). Communications in Mathematical Physics. 327 (2): 333–386. Bibcode:2014CMaPh.327..333J. doi:10.1007/s00220-014-2021-1. ISSN 1432-0916.
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