Rational singularity

<h2 id="formulations">Formulations</h2>
<p>Alternately, one can say that 
  
    
      
        X
      
    
    {\displaystyle X}
  
 has rational singularities if and only if the natural map in the <a href="/facts/Derived_category/pBQHnFYo">derived category</a>
</p>

O
            
          
          
            X
          
        
        →
        R
        
          f
          
            ∗
          
        
        
          
            
              O
            
          
          
            Y
          
        
      
    
    {\displaystyle {\mathcal {O}}_{X}\rightarrow Rf_{*}{\mathcal {O}}_{Y}}

<p>is a <a href="/facts/Quasi-isomorphism/zJYgjTy4">quasi-isomorphism</a>. Notice that this includes the statement that 
  
    
      
        
          
            
              O
            
          
          
            X
          
        
        ≃
        
          f
          
            ∗
          
        
        
          
            
              O
            
          
          
            Y
          
        
      
    
    {\displaystyle {\mathcal {O}}_{X}\simeq f_{*}{\mathcal {O}}_{Y}}
  
 and hence the assumption that 
  
    
      
        X
      
    
    {\displaystyle X}
  
 is normal.
</p><p>There are related notions in positive and mixed <a href="/facts/Characteristic_of_a_ring/D2VzcQaG">characteristic</a> of 
</p>
<ul><li>pseudo-rational</li></ul>
<p>and
</p>
<ul><li>F-rational</li></ul>
<p>Rational singularities are in particular <a href="/facts/Cohen-Macaulay_ring/4G8wymtC">Cohen-Macaulay</a>, <a href="/facts/Normal_scheme/TQ6cB9By">normal</a> and <a href="/facts/Du_Bois_singularities/BIkQ87Ry">Du Bois</a>.  They need not be <a href="/facts/Gorenstein_scheme/RU9D5SDF">Gorenstein</a> or even <a href="/facts/Q-Gorenstein/RU9D5SDF">Q-Gorenstein</a>.
</p><p><a href="/facts/Log_terminal/cCbkz2NR">Log terminal</a> singularities are rational.<a class="footnote-ref" id="fnref:1" href="#fn:1"><sup>1</sup></a>
</p>
<h2 id="examples">Examples</h2>
<p>An example of a rational singularity is the singular point of the <a href="/facts/Quadric_cone/L3cTApd4">quadric cone</a> 
</p>

x
          
            2
          
        
        +
        
          y
          
            2
          
        
        +
        
          z
          
            2
          
        
        =
        0.
        
      
    
    {\displaystyle x^{2}+y^{2}+z^{2}=0.\,}

<p>Artin<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> showed that 
the rational <a href="/facts/Double_point/mGFKPaas">double points</a> of <a href="/facts/Algebraic_surface/2zHU0o9J">algebraic surfaces</a> are the <a href="/facts/Du_Val_singularities/pLmdkbGv">Du Val singularities</a>.
</p>
<h2 id="see-also">See also</h2>
<ul><li><a href="/facts/Elliptic_singularity/qMIvJlrd">Elliptic singularity</a></li></ul>

<ul><li><a href="/facts/Michael_Artin/Tfk0f6rK">Artin, Michael</a> (1966), "On isolated rational singularities of surfaces", <i><a href="/facts/American_Journal_of_Mathematics/lkgftcD7">American Journal of Mathematics</a></i>, 88 (1), The Johns Hopkins University Press: 129–136, <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.2307%2F2373050">10.2307/2373050</a>, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/0002-9327">0002-9327</a>, <a href="/facts/JSTOR_(identifier)/YTeVmaJ7">JSTOR</a> <a href="https://www.jstor.org/stable/2373050">2373050</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0199191">0199191</a></li>
<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/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1017%2FCBO9780511662560">10.1017/CBO9780511662560</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>Lipman, Joseph (1969), <a href="http://www.numdam.org/item?id=PMIHES_1969__36__195_0">"Rational singularities, with applications to algebraic surfaces and unique factorization"</a>, <i><a href="/facts/Publications_Math%C3%A9matiques_de_l%27IH%C3%89S/2ur5ofxL">Publications Mathématiques de l'IHÉS</a></i> (36): 195–279, <a href="/facts/ISSN_(identifier)/DPAflDvU">ISSN</a> <a href="https://search.worldcat.org/issn/1618-1913">1618-1913</a>, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0276239">0276239</a></li></ul>

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

<ol>
<li id="fn:1"><p>(Kollár & Mori 1998, Theorem 5.22.) - Kollár, János; Mori, Shigefumi (1998), Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, doi:10.1017/CBO9780511662560, ISBN 978-0-521-63277-5, MR 1658959 <a href="https://doi.org/10.1017%2FCBO9780511662560" target="_blank">https://doi.org/10.1017%2FCBO9780511662560</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>(Artin 1966) - Artin, Michael (1966), "On isolated rational singularities of surfaces", American Journal of Mathematics, 88 (1), The Johns Hopkins University Press: 129–136, doi:10.2307/2373050, ISSN 0002-9327, JSTOR 2373050, MR 0199191 <a href="https://doi.org/10.2307%2F2373050" target="_blank">https://doi.org/10.2307%2F2373050</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
</ol>

Rational singularity open-in-new

Rational singularity