Stacky curve explained

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

ak{X} over a field is a smooth proper geometrically connected Deligne–Mumford stack of dimension 1 over that contains a dense open subscheme.^{[1] [2]}

Properties

A stacky curve is uniquely determined (up to isomorphism) by its coarse space (a smooth quasi-projective curve over), a finite set of points (its stacky points) and integers (its ramification orders) greater than 1.^[2] The canonical divisor of

ak{X} is linearly equivalent to the sum of the canonical divisor of and a ramification divisor :^[1] K _{ak{X}\sim} K _{X} + R. Letting be the genus of the coarse space, the degree of the canonical divisor of ak{X} is therefore:^[1] d = \deg K _{ak{X}=} 2 - 2 g - \sum ^{r} _{i =1} (n _{i} -1)/(n _{i}) . A stacky curve is called spherical if is positive, Euclidean if is zero, and hyperbolic if is negative.^[2]

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] [3] [4]}

Applications

The generalization of GAGA for stacky curves is used in the derivation of algebraic structure theory of rings of modular forms.^[3]

The study of stacky curves is used extensively in equivariant Gromov–Witten theory and enumerative geometry.^{[1] [5]}

Notes and References

1. Book: The canonical ring of a stacky curve . Voight . John . Zureick-Brown . David . . 1501.04657. 2015arXiv150104657V . 2015 .
2. Book: Kresch, Andrew . Dan . Abramovich . Dan Abramovich . Aaron . Bertram . Ludmil . Katzarkov . Rahul . Pandharipande . Michael . Thaddeus . On the geometry of Deligne-Mumford stacks . Algebraic Geometry: Seattle 2005 Part 1 . Proc. Sympos. Pure Math. . 80 . Amer. Math. Soc. . Providence, RI . 2009 . 259–271 . 10.5167/uzh-21342. 10.1.1.560.9644 . 978-0-8218-4702-2.
3. Landesman . Aaron . Ruhm . Peter . Zhang . Robin . Spin canonical rings of log stacky curves . . 66 . 6 . 2339–2383 . 1507.02643 . 10.5802/aif.3065. 2016 .
4. Kai . Behrend . Kai Behrend . Behrang . Noohi . Uniformization of Deligne-Mumford curves . . 599 . 2006 . 111–153 . math/0504309. 2005math......4309B .
5. Equivariant GW Theory of Stacky Curves . Johnson . Paul . Communications in Mathematical Physics . 327 . 2 . 333–386 . 2014 . 10.1007/s00220-014-2021-1 . 2014CMaPh.327..333J . 1432-0916.