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

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]

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]

gollark: <@102436520559910912> That's just an anonymous async function, so yes, it won't run if you don't do anything with it.

gollark: To just waste CPU cycles, this has worked for me before when I wanted to do that for some evil reason or other, but it's probably not very sound:```cvolatile long long int i = 0;while (i < [a big number]) { i++; }```

gollark: Oh right, probably.

gollark: If you do just need to make it do something for a bit, maybe just```cfor (int i = 0; i < 100000000; i++) {}```or something would work, though compilers are getting crafty and might optimize it.

gollark: Is there some reason you can't just use whatever language's equivalent of `sleep(1)` or something?

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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[vzb-1] Voight, John; Zureick-Brown, David (2015). The canonical ring of a stacky curve. Memoirs of the American Mathematical Society. arXiv:1501.04657. Bibcode:2015arXiv150104657V.

[lrz-2] 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-3] 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.

[4] Behrend, Kai; Noohi, Behrang (2006). "Uniformization of Deligne-Mumford curves". J. Reine Angew. Math. 599: 111–153. arXiv:math/0504309. Bibcode:2005math......4309B.

[5] 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.