Quot scheme | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-03-28

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SUMMARY
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In algebraic geometry, the Quot scheme is a scheme parametrizing locally free sheaves on a projective scheme. More specifically, if X is a projective scheme over a Noetherian scheme S and if F is a coherent sheaf on X, then there is a scheme 
  
    
      
        
          Quot
          
            F
          
        
        ⁡
        (
        X
        )
      
    
    {\displaystyle \operatorname {Quot} _{F}(X)}
   whose set of T-points 
  
    
      
        
          Quot
          
            F
          
        
        ⁡
        (
        X
        )
        (
        T
        )
        =
        
          Mor
          
            S
          
        
        ⁡
        (
        T
        ,
        
          Quot
          
            F
          
        
        ⁡
        (
        X
        )
        )
      
    
    {\displaystyle \operatorname {Quot} _{F}(X)(T)=\operatorname {Mor} _{S}(T,\operatorname {Quot} _{F}(X))}
   is the set of isomorphism classes of the quotients of 
  
    
      
        F
        
          ×
          
            S
          
        
        T
      
    
    {\displaystyle F\times _{S}T}
   that are flat over T. The notion was introduced by Alexander Grothendieck.It is typically used to construct another scheme parametrizing geometric objects that are of interest such as a Hilbert scheme. (In fact, taking F to be the structure sheaf 
  
    
      
        
          
            
              O
            
          
          
            X
          
        
      
    
    {\displaystyle {\mathcal {O}}_{X}}
   gives a Hilbert scheme.)