{{short description|Object in algebraic geometry}} 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 <math>\mathfrak{X}</math> over a field {{mvar|k}} is a smooth proper geometrically connected Deligne–Mumford stack of dimension 1 over {{mvar|k}} that contains a dense open subscheme.<ref name="vzb">{{cite book |title=The canonical ring of a stacky curve |last1=Voight |first1=John |last2=Zureick-Brown |first2=David |series=Memoirs of the American Mathematical Society |arxiv=1501.04657|bibcode=2015arXiv150104657V |year=2015 }}</ref><ref name="lrz"/><ref name="Kresch">{{cite book |first=Andrew |last=Kresch |author-link=Andrew Kresch |editor-first1=Dan |editor-last1=Abramovich |editor-link1=Dan Abramovich |editor-first2=Aaron |editor-last2=Bertram |editor-first3=Ludmil |editor-last3=Katzarkov |editor-first4=Rahul |editor-last4=Pandharipande |editor-first5=Michael |editor-last5=Thaddeus |chapter=On the geometry of Deligne-Mumford stacks |title=Algebraic Geometry: Seattle 2005 Part 1 |series=Proc. Sympos. Pure Math. |volume=80 |publisher=Amer. Math. Soc. |location=Providence, RI |year=2009 |pages=259–271 |doi=10.5167/uzh-21342|citeseerx=10.1.1.560.9644 |isbn=978-0-8218-4702-2}}</ref>
==Properties== A stacky curve is uniquely determined (up to isomorphism) by its coarse space {{mvar|X}} (a smooth quasi-projective curve over {{mvar|k}}), a finite set of points {{mvar|x<sub>i</sub>}} (its stacky points) and integers {{mvar|n<sub>i</sub>}} (its ramification orders) greater than 1.<ref name="Kresch"/> The canonical divisor of <math>\mathfrak{X}</math> is linearly equivalent to the sum of the canonical divisor of {{mvar|X}} and a ramification divisor {{mvar|R}}:<ref name="vzb"/> :<math>K_\mathfrak{X} \sim K_X + R.</math> Letting {{mvar|g}} be the genus of the coarse space {{mvar|X}}, the degree of the canonical divisor of <math>\mathfrak{X}</math> is therefore:<ref name="vzb"/> :<math>d = \deg K_\mathfrak{X} = 2g - 2 + \sum_{i=1}^r \frac{n_i - 1}{n_i}.</math> A stacky curve is called '''hyperbolic''' if {{mvar|d}} is positive, '''Euclidean''' if {{mvar|d}} is zero, and '''spherical''' if {{mvar|d}} is negative.<ref name="Kresch"/>
Although the corresponding statement of Riemann–Roch theorem does not hold for stacky curves,<ref name="vzb"/> 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.<ref name="vzb"/><ref name="lrz">{{cite journal |last1=Landesman |first1=Aaron |last2=Ruhm |first2=Peter |last3=Zhang |first3=Robin |title=Spin canonical rings of log stacky curves |journal = Annales de l'Institut Fourier |volume=66 |issue=6 |pages=2339–2383 |arxiv=1507.02643 |doi=10.5802/aif.3065|year=2016 }}</ref><ref>{{cite journal |first1=Kai |last1=Behrend |author-link1=Kai Behrend |first2=Behrang |last2=Noohi |title=Uniformization of Deligne-Mumford curves |journal=J. Reine Angew. Math. |volume=599 |year=2006 |pages=111–153 |arxiv=math/0504309|bibcode=2005math......4309B }}</ref>
==Applications== The generalization of GAGA for stacky curves is used in the derivation of algebraic structure theory of rings of modular forms.<ref name="lrz"/>
The study of stacky curves is used extensively in equivariant Gromov–Witten theory and enumerative geometry.<ref name="vzb"/><ref>{{cite journal |title=Equivariant GW Theory of Stacky Curves |last=Johnson |first=Paul |journal=Communications in Mathematical Physics |volume=327 |issue=2 |pages=333–386 |year=2014 |doi=10.1007/s00220-014-2021-1 |bibcode=2014CMaPh.327..333J |issn=1432-0916|url=http://eprints.whiterose.ac.uk/98527/1/eqorb.pdf }}</ref>
==References== {{reflist}}
Category:Moduli theory