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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 2.13006 seconds

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

o1 : Sequence

See also

Ways to use specialCremonaTransformation :

For the programmer

The object specialCremonaTransformation is a method function.