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

## Synopsis

• Usage:
specialCremonaTransformation i
specialCremonaTransformation(i,K)
• Inputs:
• i, an integer, an integer between 1 and 12
• K, a ring, the ground field (optional, the default value is QQ)
• Outputs:

## 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 1.16691 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