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Cremona :: specialCremonaTransformation

specialCremonaTransformation -- special Cremona transformations whose base locus has dimension at most three

Synopsis

Description

A Cremona transformation is said to be special if the base locus scheme is smooth and irreducible. To ensure this condition, the field K must be large enough but no check is made.

i1 : time apply(1..12,i -> describe specialCremonaTransformation(i,ZZ/3331))
     -- used 1.22712 seconds

o1 = (rational map defined by forms of degree 3,
      source variety: PP^3                      
      target variety: PP^3                      
      dominance: true                           
      birationality: true                       
      projective degrees: {1, 3, 3, 1}          
      number of minimal representatives: 1      
      dimension base locus: 1                   
      degree base locus: 6                      
      coefficient ring: ZZ/3331                 
     ------------------------------------------------------------------------
     rational map defined by forms of degree 2,
     source variety: PP^4                      
     target variety: PP^4                      
     dominance: true                           
     birationality: true                       
     projective degrees: {1, 2, 4, 3, 1}       
     number of minimal representatives: 1      
     dimension base locus: 1                   
     degree base locus: 5                      
     coefficient ring: ZZ/3331                 
     ------------------------------------------------------------------------
     rational map defined by forms of degree 3,
     source variety: PP^4                      
     target variety: PP^4                      
     dominance: true                           
     birationality: true                       
     projective degrees: {1, 3, 4, 2, 1}       
     number of minimal representatives: 1      
     dimension base locus: 2                   
     degree base locus: 5                      
     coefficient ring: ZZ/3331                 
     ------------------------------------------------------------------------
     rational map defined by forms of degree 4,
     source variety: PP^4                      
     target variety: PP^4                      
     dominance: true                           
     birationality: true                       
     projective degrees: {1, 4, 6, 4, 1}       
     number of minimal representatives: 1      
     dimension base locus: 2                   
     degree base locus: 10                     
     coefficient ring: ZZ/3331                 
     ------------------------------------------------------------------------
     rational map defined by forms of degree 2,
     source variety: PP^5                      
     target variety: PP^5                      
     dominance: true                           
     birationality: true                       
     projective degrees: {1, 2, 4, 4, 2, 1}    
     number of minimal representatives: 1      
     dimension base locus: 2                   
     degree base locus: 4                      
     coefficient ring: ZZ/3331                 
     ------------------------------------------------------------------------
     rational map defined by forms of degree 2,
     source variety: PP^6                      
     target variety: PP^6                      
     dominance: true                           
     birationality: true                       
     projective degrees: {1, 2, 4, 8, 9, 4, 1} 
     number of minimal representatives: 1      
     dimension base locus: 2                   
     degree base locus: 7                      
     coefficient ring: ZZ/3331                 
     ------------------------------------------------------------------------
     rational map defined by forms of degree 2,
     source variety: PP^6                      
     target variety: PP^6                      
     dominance: true                           
     birationality: true                       
     projective degrees: {1, 2, 4, 8, 8, 4, 1} 
     number of minimal representatives: 1      
     dimension base locus: 2                   
     degree base locus: 8                      
     coefficient ring: ZZ/3331                 
     ------------------------------------------------------------------------
     rational map defined by forms of degree 5,
     source variety: PP^5                      
     target variety: PP^5                      
     dominance: true                           
     birationality: true                       
     projective degrees: {1, 5, 10, 10, 5, 1}  
     number of minimal representatives: 1      
     dimension base locus: 3                   
     degree base locus: 15                     
     coefficient ring: ZZ/3331                 
     ------------------------------------------------------------------------
     rational map defined by forms of degree 2         ,
     source variety: PP^8                               
     target variety: PP^8                               
     dominance: true                                    
     birationality: true                                
     projective degrees: {1, 2, 4, 8, 16, 20, 14, 5, 1} 
     number of minimal representatives: 1               
     dimension base locus: 3                            
     degree base locus: 12                              
     coefficient ring: ZZ/3331                          
     ------------------------------------------------------------------------
     rational map defined by forms of degree 2         ,
     source variety: PP^8                               
     target variety: PP^8                               
     dominance: true                                    
     birationality: true                                
     projective degrees: {1, 2, 4, 8, 16, 19, 13, 5, 1} 
     number of minimal representatives: 1               
     dimension base locus: 3                            
     degree base locus: 13                              
     coefficient ring: ZZ/3331                          
     ------------------------------------------------------------------------
     rational map defined by forms of degree 3  ,
     source variety: PP^6                        
     target variety: PP^6                        
     dominance: true                             
     birationality: true                         
     projective degrees: {1, 3, 9, 13, 11, 5, 1} 
     number of minimal representatives: 1        
     dimension base locus: 3                     
     degree base locus: 14                       
     coefficient ring: ZZ/3331                   
     ------------------------------------------------------------------------
     rational map defined by forms of degree 3  )
     source variety: PP^6
     target variety: PP^6
     dominance: true
     birationality: true
     projective degrees: {1, 3, 9, 14, 12, 5, 1}
     number of minimal representatives: 1
     dimension base locus: 3
     degree base locus: 13
     coefficient ring: ZZ/3331

o1 : Sequence

See also

Ways to use specialCremonaTransformation :

For the programmer

The object specialCremonaTransformation is a method function.