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Cremona :: RationalMap !

RationalMap ! -- calculates every possible thing

Synopsis

Description

This method (mainly used for tests) applies almost all the deterministic methods that are available.

i1 : QQ[x_0..x_5]; phi = rationalMap {x_4^2-x_3*x_5,x_2*x_4-x_1*x_5,x_2*x_3-x_1*x_4,x_2^2-x_0*x_5,x_1*x_2-x_0*x_4,x_1^2-x_0*x_3};

o2 : RationalMap (quadratic rational map from PP^5 to PP^5)
i3 : describe phi

o3 = rational map defined by forms of degree 2
     source variety: PP^5
     target variety: PP^5
     coefficient ring: QQ
i4 : time phi! ;
     -- used 0.0699035 seconds

o4 : RationalMap (Cremona transformation of PP^5 of type (2,2))
i5 : describe phi

o5 = rational map defined by forms of degree 2
     source variety: PP^5
     target variety: PP^5
     dominance: true
     birationality: true (the inverse map is already calculated)
     projective degrees: {1, 2, 4, 4, 2, 1}
     number of minimal representatives: 1
     dimension base locus: 2
     degree base locus: 4
     coefficient ring: QQ
i6 : QQ[x_0..x_4]; phi = rationalMap {-x_1^2+x_0*x_2,-x_1*x_2+x_0*x_3,-x_2^2+x_1*x_3,-x_1*x_3+x_0*x_4,-x_2*x_3+x_1*x_4,-x_3^2+x_2*x_4};

o7 : RationalMap (quadratic rational map from PP^4 to PP^5)
i8 : describe phi

o8 = rational map defined by forms of degree 2
     source variety: PP^4
     target variety: PP^5
     coefficient ring: QQ
i9 : time phi! ;
     -- used 0.0628639 seconds

o9 : RationalMap (quadratic rational map from PP^4 to PP^5)
i10 : describe phi

o10 = rational map defined by forms of degree 2
      source variety: PP^4
      target variety: PP^5
      image: smooth quadric hypersurface in PP^5
      dominance: false
      birationality: false
      degree of map: 1
      projective degrees: {1, 2, 4, 4, 2}
      number of minimal representatives: 1
      dimension base locus: 1
      degree base locus: 4
      coefficient ring: QQ

The command phi(*) does more or less the same thing but it uses probabilistic methods and treats them as deterministic (the user should never use this).

i11 : phi = rationalMap rationalMap map specialQuadraticTransformation(8,ZZ/33331);

o11 : RationalMap (quadratic rational map from PP^8 to PP^11)
i12 : describe phi

o12 = rational map defined by forms of degree 2
      source variety: PP^8
      target variety: PP^11
      image: complete intersection of type (2,2,2) in PP^11
      dominance: false
      birationality: false
      degree of map: 1
      projective degrees: {1, 2, 4, 8, 16, 23, 23, 16, 8}
      number of minimal representatives: 1
      dimension base locus: 3
      degree base locus: 9
      coefficient ring: ZZ/33331
i13 : time phi(*) ;
     -- used 7.249e-6 seconds

o13 : RationalMap (quadratic rational map from PP^8 to PP^11)
i14 : describe phi

o14 = rational map defined by forms of degree 2
      source variety: PP^8
      target variety: PP^11
      image: complete intersection of type (2,2,2) in PP^11
      dominance: false
      birationality: false
      degree of map: 1
      projective degrees: {1, 2, 4, 8, 16, 23, 23, 16, 8}
      number of minimal representatives: 1
      dimension base locus: 3
      degree base locus: 9
      coefficient ring: ZZ/33331

See also