Contraction morphism

<h2 id="birational-perspective">Birational perspective</h2>
<p>The following perspective is crucial in <a href="/facts/Birational_geometry/hsWqRhJ1">birational geometry</a> (in particular in <a href="/facts/Mori%27s_minimal_model_program/f15RyS0J">Mori's minimal model program</a>).
</p><p>Let 
  
    
      
        X
      
    
    {\displaystyle X}
  
 be a projective variety and 
  
    
      
        
          
            
              N
              S
            
            ¯
          
        
        (
        X
        )
      
    
    {\displaystyle {\overline {NS}}(X)}
  
 the closure of the span of irreducible curves on 
  
    
      
        X
      
    
    {\displaystyle X}
  
 in 
  
    
      
        
          N
          
            1
          
        
        (
        X
        )
      
    
    {\displaystyle N_{1}(X)}
  
 = the real vector space of numerical equivalence classes of real 1-cycles on 
  
    
      
        X
      
    
    {\displaystyle X}
  
. Given a face 
  
    
      
        F
      
    
    {\displaystyle F}
  
 of 
  
    
      
        
          
            
              N
              S
            
            ¯
          
        
        (
        X
        )
      
    
    {\displaystyle {\overline {NS}}(X)}
  
, the contraction morphism associated to <i>F</i>, if it exists, is a contraction morphism 
  
    
      
        f
        :
        X
        →
        Y
      
    
    {\displaystyle f:X\to Y}
  
 to some projective variety 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 such that for each irreducible curve 
  
    
      
        C
        ⊂
        X
      
    
    {\displaystyle C\subset X}
  
, 
  
    
      
        f
        (
        C
        )
      
    
    {\displaystyle f(C)}
  
 is a point if and only if 
  
    
      
        [
        C
        ]
        ∈
        F
      
    
    {\displaystyle [C]\in F}
  
.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a> The basic question is which face 
  
    
      
        F
      
    
    {\displaystyle F}
  
 gives rise to such a contraction morphism (cf. <a href="/facts/Cone_theorem/GD2L8a1k">cone theorem</a>).
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Castelnuovo%27s_contraction_theorem/SFraYv3z">Castelnuovo's contraction theorem</a></li>
<li><a href="/facts/Flip_(mathematics)/a9z0V5Y0">Flip (mathematics)</a></li></ul>

<ul><li>Kollár, János; Mori, Shigefumi (1998), <i>Birational geometry of algebraic varieties</i>, Cambridge Tracts in Mathematics, vol. 134, <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>, <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-63277-5, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1658959">1658959</a></li>
<li><a href="/facts/Robert_Lazarsfeld/etSRlf6k">Robert Lazarsfeld</a>, <i>Positivity in Algebraic Geometry I: Classical Setting</i> (2004)</li></ul>

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

<ol>
<li id="fn:1"><p>Kollár & Mori 1998, Definition 1.25. - Kollár, János; Mori, Shigefumi (1998), Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, ISBN 978-0-521-63277-5, MR 1658959 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1658959" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1658959</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Contraction morphism open-in-new

Contraction morphism