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MultiprojectiveVarieties :: isIsomorphism(MultirationalMap)

isIsomorphism(MultirationalMap) -- whether a birational map is an isomorphism

Synopsis

Description

i1 : -- map defined by the quadrics through a twisted cubic curve
     ZZ/33331[a..d]; f = rationalMap {c^2-b*d,b*c-a*d,b^2-a*c};

o2 : RationalMap (quadratic rational map from PP^3 to PP^2)
i3 : Phi = multirationalMap {f,f};

o3 : MultirationalMap (rational map from PP^3 to PP^2 x PP^2)
i4 : time isIsomorphism Phi
     -- used 0.000012163 seconds

o4 = false
i5 : Psi = first graph Phi;

o5 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 to PP^3)
i6 : time isIsomorphism Psi
     -- used 0.939026 seconds

o6 = false
i7 : Eta = first graph Psi;

o7 : MultirationalMap (birational map from threefold in PP^3 x PP^2 x PP^2 x PP^3 to threefold in PP^3 x PP^2 x PP^2)
i8 : time isIsomorphism Eta
     -- used 3.29297 seconds

o8 = true
i9 : describe Eta!

o9 = multi-rational map consisting of 3 rational maps
     source variety: threefold in PP^3 x PP^2 x PP^2 x PP^3 cut out by 17 hypersurfaces of degrees ({1, 0, 0, 1},{0, 0, 1, 1},{0, 1, 0, 1},{1, 0, 0, 1},{1, 0, 0, 1},{0, 0, 1, 1},{0, 1, 0, 1},{1, 0, 0, 1},{1, 0, 0, 1},{1, 0, 0, 1},{0, 1, 1, 0},{0, 1, 1, 0},{0, 1, 1, 0},{1, 0, 1, 0},{1, 0, 1, 0},{1, 1, 0, 0},{1, 1, 0, 0})
     target variety: threefold in PP^3 x PP^2 x PP^2 cut out by 7 hypersurfaces of degrees ({0, 1, 1},{0, 1, 1},{0, 1, 1},{1, 0, 1},{1, 0, 1},{1, 1, 0},{1, 1, 0})
     base locus: empty subscheme of PP^3 x PP^2 x PP^2 x PP^3
     dominance: true
     birationality: true
     --
     rational map (1/3) defined by multiforms of degree {0, 0, 0, 1}
     target variety: PP^3
     dominance: true
     birationality: true
     projective degrees: {80, 24, 6, 1}
     number of minimal representatives: 2, with degrees ({0, 0, 0, 1},{1, 0, 0, 0})
     dimension base locus: -1
     --
     rational map (2/3) defined by multiforms of degree {0, 0, 1, 0}
     target variety: PP^2
     dominance: true
     birationality: false
     degree of map: 0
     projective degrees: {80, 16, 2, 0}
     number of minimal representatives: 5, with degrees ({0, 0, 1, 0},{0, 1, 0, 0},{0, 0, 0, 2},{1, 0, 0, 1},{2, 0, 0, 0})
     dimension base locus: -1
     --
     rational map (3/3) defined by multiforms of degree {0, 0, 1, 0}
     target variety: PP^2
     dominance: true
     birationality: false
     degree of map: 0
     projective degrees: {80, 16, 2, 0}
     number of minimal representatives: 5, with degrees ({0, 0, 1, 0},{0, 1, 0, 0},{0, 0, 0, 2},{1, 0, 0, 1},{2, 0, 0, 0})
     dimension base locus: -1
     --
     coefficient ring: ZZ/33331
i10 : describe (inverse Eta)!

o10 = multi-rational map consisting of 4 rational maps
      source variety: threefold in PP^3 x PP^2 x PP^2 cut out by 7 hypersurfaces of degrees ({0, 1, 1},{0, 1, 1},{0, 1, 1},{1, 0, 1},{1, 0, 1},{1, 1, 0},{1, 1, 0})
      target variety: threefold in PP^3 x PP^2 x PP^2 x PP^3 cut out by 17 hypersurfaces of degrees ({1, 0, 0, 1},{0, 0, 1, 1},{0, 1, 0, 1},{1, 0, 0, 1},{1, 0, 0, 1},{0, 0, 1, 1},{0, 1, 0, 1},{1, 0, 0, 1},{1, 0, 0, 1},{1, 0, 0, 1},{0, 1, 1, 0},{0, 1, 1, 0},{0, 1, 1, 0},{1, 0, 1, 0},{1, 0, 1, 0},{1, 1, 0, 0},{1, 1, 0, 0})
      base locus: empty subscheme of PP^3 x PP^2 x PP^2
      dominance: true
      birationality: true
      --
      rational map (1/4) defined by multiforms of degree {1, 0, 0}
      target variety: PP^3
      dominance: true
      birationality: true
      projective degrees: {25, 13, 5, 1}
      number of minimal representatives: 1
      dimension base locus: -1
      --
      rational map (2/4) defined by multiforms of degree {0, 0, 1}
      target variety: PP^2
      dominance: true
      birationality: false
      degree of map: 0
      projective degrees: {25, 6, 1, 0}
      number of minimal representatives: 3, with degrees ({0, 0, 1},{0, 1, 0},{2, 0, 0})
      dimension base locus: -1
      --
      rational map (3/4) defined by multiforms of degree {0, 0, 1}
      target variety: PP^2
      dominance: true
      birationality: false
      degree of map: 0
      projective degrees: {25, 6, 1, 0}
      number of minimal representatives: 3, with degrees ({0, 0, 1},{0, 1, 0},{2, 0, 0})
      dimension base locus: -1
      --
      rational map (4/4) defined by multiforms of degree {1, 0, 0}
      target variety: PP^3
      dominance: true
      birationality: true
      projective degrees: {25, 13, 5, 1}
      number of minimal representatives: 1
      dimension base locus: -1
      --
      coefficient ring: ZZ/33331
i11 : assert(o8 and (not o6) and (not o4))

See also