Stacky curve

<h2 id="definition">Definition</h2>
A stacky curve 
 
 
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathfrak {X}}}
 
 over a field k is a <a href="/facts/Smooth_scheme/Y76o7b9T">smooth</a> <a href="/facts/Proper_morphism/EJHIgjnr">proper</a> <a href="/facts/Connected_space/5CUTxfoA">geometrically connected</a> <a href="/facts/Deligne%E2%80%93Mumford_stack/kXlptywa">Deligne–Mumford stack</a> of <a href="/facts/Dimension_of_an_algebraic_variety/sXqIHcYc">dimension</a> 1 over k that contains a dense open subscheme.<a class="footnote-ref" id="fnref:1" href="#fn:1">1</a><a class="footnote-ref" id="fnref:2" href="#fn:2">2</a><a class="footnote-ref" id="fnref:3" href="#fn:3">3</a>

<h2 id="properties">Properties</h2>
A stacky curve is uniquely determined (up to isomorphism) by its coarse space X (a smooth <a href="/facts/Quasi-projective_variety/rdBMNK0G">quasi-projective</a> curve over k), a finite set of points xi (its stacky points) and integers ni (its ramification orders) greater than 1.<a class="footnote-ref" id="fnref:4" href="#fn:4">4</a> The <a href="/facts/Canonical_divisor/nJKN8mpG">canonical divisor</a> of 
 
 
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathfrak {X}}}
 
 is <a href="/facts/Linear_system_of_divisors/L8zleDlj">linearly equivalent</a> to the sum of the canonical divisor of X and a ramification divisor R:<a class="footnote-ref" id="fnref:5" href="#fn:5">5</a>

K
          
            
              X
            
          
        
        ∼
        
          K
          
            X
          
        
        +
        R
        .
      
    
    {\displaystyle K_{\mathfrak {X}}\sim K_{X}+R.}

Letting g be the <a href="/facts/Genus_(mathematics)/o1Yhf8ty">genus</a> of the coarse space X, the degree of the <a href="/facts/Canonical_divisor/nJKN8mpG">canonical divisor</a> of 
 
 
 
 
 
 X
 
 
 
 
 {\displaystyle {\mathfrak {X}}}
 
 is therefore:<a class="footnote-ref" id="fnref:6" href="#fn:6">6</a>

d
        =
        deg
        ⁡
        
          K
          
            
              X
            
          
        
        =
        2
        −
        2
        g
        −
        
          ∑
          
            i
            =
            1
          
          
            r
          
        
        
          
            
              
                n
                
                  i
                
              
              −
              1
            
            
              n
              
                i
              
            
          
        
        .
      
    
    {\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.<a class="footnote-ref" id="fnref:7" href="#fn:7">7</a>
Although the corresponding statement of <a href="/facts/Riemann%E2%80%93Roch_theorem/TRaNyUQA">Riemann–Roch theorem</a> does not hold for stacky curves,<a class="footnote-ref" id="fnref:8" href="#fn:8">8</a> there is a generalization of <a href="/facts/Algebraic_geometry_and_analytic_geometry/k2ltJ1Ue">Riemann's existence theorem</a> that gives an <a href="/facts/Equivalence_of_categories/k6uzqAcy">equivalence of categories</a> between the <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> of stacky curves over the <a href="/facts/Complex_number/2w7mqI7e">complex numbers</a> and the category of complex orbifold curves.<a class="footnote-ref" id="fnref:9" href="#fn:9">9</a><a class="footnote-ref" id="fnref:10" href="#fn:10">10</a><a class="footnote-ref" id="fnref:11" href="#fn:11">11</a>

<h2 id="applications">Applications</h2>
The generalization of GAGA for stacky curves is used in the derivation of <a href="/facts/Ring_of_modular_forms/U2aHxQ4P">algebraic structure theory of rings of modular forms</a>.<a class="footnote-ref" id="fnref:12" href="#fn:12">12</a>
The study of stacky curves is used extensively in equivariant Gromov–Witten theory and enumerative geometry.<a class="footnote-ref" id="fnref:13" href="#fn:13">13</a><a class="footnote-ref" id="fnref:14" href="#fn:14">14</a>

<h2 id="references">References</h2>

<ol>
<li id="fn: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. <a href="/wiki/Memoirs_of_the_American_Mathematical_Society" target="_blank">/wiki/Memoirs_of_the_American_Mathematical_Society</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn: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. <a href="/wiki/Annales_de_l%27Institut_Fourier" target="_blank">/wiki/Annales_de_l%27Institut_Fourier</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
<li id="fn: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. Vol. 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. <a href="978-0-8218-4702-2" target="_blank">978-0-8218-4702-2</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></li>
<li id="fn:4">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. Vol. 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. <a href="978-0-8218-4702-2" target="_blank">978-0-8218-4702-2</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></li>
<li id="fn:5">Voight, John; Zureick-Brown, David (2015). The canonical ring of a stacky curve. Memoirs of the American Mathematical Society. arXiv:1501.04657. Bibcode:2015arXiv150104657V. <a href="/wiki/Memoirs_of_the_American_Mathematical_Society" target="_blank">/wiki/Memoirs_of_the_American_Mathematical_Society</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></li>
<li id="fn:6">Voight, John; Zureick-Brown, David (2015). The canonical ring of a stacky curve. Memoirs of the American Mathematical Society. arXiv:1501.04657. Bibcode:2015arXiv150104657V. <a href="/wiki/Memoirs_of_the_American_Mathematical_Society" target="_blank">/wiki/Memoirs_of_the_American_Mathematical_Society</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></li>
<li id="fn:7">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. Vol. 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. <a href="978-0-8218-4702-2" target="_blank">978-0-8218-4702-2</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></li>
<li id="fn:8">Voight, John; Zureick-Brown, David (2015). The canonical ring of a stacky curve. Memoirs of the American Mathematical Society. arXiv:1501.04657. Bibcode:2015arXiv150104657V. <a href="/wiki/Memoirs_of_the_American_Mathematical_Society" target="_blank">/wiki/Memoirs_of_the_American_Mathematical_Society</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></li>
<li id="fn:9">Voight, John; Zureick-Brown, David (2015). The canonical ring of a stacky curve. Memoirs of the American Mathematical Society. arXiv:1501.04657. Bibcode:2015arXiv150104657V. <a href="/wiki/Memoirs_of_the_American_Mathematical_Society" target="_blank">/wiki/Memoirs_of_the_American_Mathematical_Society</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></li>
<li id="fn:10">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. <a href="/wiki/Annales_de_l%27Institut_Fourier" target="_blank">/wiki/Annales_de_l%27Institut_Fourier</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></li>
<li id="fn:11">Behrend, Kai; Noohi, Behrang (2006). "Uniformization of Deligne-Mumford curves". J. Reine Angew. Math. 599: 111–153. arXiv:math/0504309. Bibcode:2005math......4309B. <a href="/wiki/Kai_Behrend" target="_blank">/wiki/Kai_Behrend</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></li>
<li id="fn:12">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. <a href="/wiki/Annales_de_l%27Institut_Fourier" target="_blank">/wiki/Annales_de_l%27Institut_Fourier</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></li>
<li id="fn:13">Voight, John; Zureick-Brown, David (2015). The canonical ring of a stacky curve. Memoirs of the American Mathematical Society. arXiv:1501.04657. Bibcode:2015arXiv150104657V. <a href="/wiki/Memoirs_of_the_American_Mathematical_Society" target="_blank">/wiki/Memoirs_of_the_American_Mathematical_Society</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></li>
<li id="fn:14">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. <a href="http://eprints.whiterose.ac.uk/98527/1/eqorb.pdf" target="_blank">http://eprints.whiterose.ac.uk/98527/1/eqorb.pdf</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></li>
</ol>

Stacky curve open-in-new

Stacky curve