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 1.16691 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
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The object specialCremonaTransformation is a method function.