This function applies a general algorithm to calculate the inverse map passing through the computation of the graph.
i1 : -- map defined by the quadrics through a rational normal quartic curve ZZ/65521[a..e], f = rationalMap minors(2,matrix {{a,b,c,d},{b,c,d,e}}); |
i2 : Phi = multirationalMap {f}; o2 : MultirationalMap (rational map from PP^4 to PP^5) |
i3 : -- we see Phi as a dominant map Phi = multirationalMap(Phi,image Phi); o3 : MultirationalMap (dominant rational map from PP^4 to hypersurface in PP^5) |
i4 : time inverse Phi; -- used 0.402053 seconds o4 : MultirationalMap (birational map from hypersurface in PP^5 to PP^4) |
i5 : Psi = last graph Phi; o5 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to hypersurface in PP^5) |
i6 : time inverse Psi; -- used 1.02319 seconds o6 : MultirationalMap (birational map from hypersurface in PP^5 to 4-dimensional subvariety of PP^4 x PP^5) |
i7 : Eta = first graph Psi; o7 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5) |
i8 : time inverse Eta; -- used 7.64635 seconds o8 : MultirationalMap (birational map from 4-dimensional subvariety of PP^4 x PP^5 to 4-dimensional subvariety of PP^4 x PP^5 x PP^5) |
i9 : describe oo! o9 = multi-rational map consisting of 3 rational maps source variety: 4-dimensional subvariety of PP^4 x PP^5 cut out by 9 hypersurfaces of degrees ({0, 2},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1},{1, 1}) target variety: 4-dimensional subvariety of PP^4 x PP^5 x PP^5 cut out by 34 hypersurfaces of degrees ({0, 1, 1},{0, 0, 2},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{1, 0, 1},{1, 0, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{1, 0, 1},{1, 0, 1},{1, 0, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{0, 1, 1},{1, 0, 1},{1, 0, 1},{1, 0, 1},{0, 2, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0},{1, 1, 0}) base locus: empty subscheme of PP^4 x PP^5 dominance: true birationality: true -- rational map (1/3) defined by multiforms of degree {1, 0} target variety: PP^4 dominance: true birationality: true projective degrees: {51, 23, 9, 3, 1} number of minimal representatives: 2, with degrees ({1, 0},{0, 2}) dimension base locus: -1 -- rational map (2/3) defined by multiforms of degree {0, 1} target variety: PP^5 image: smooth quadric hypersurface in PP^5 dominance: false birationality: false degree of map: 1 projective degrees: {51, 28, 14, 6, 2} number of minimal representatives: 2, with degrees ({0, 1},{2, 0}) dimension base locus: -1 -- rational map (3/3) defined by multiforms of degree {0, 1} target variety: PP^5 image: smooth quadric hypersurface in PP^5 dominance: false birationality: false degree of map: 1 projective degrees: {51, 28, 14, 6, 2} number of minimal representatives: 2, with degrees ({0, 1},{2, 0}) dimension base locus: -1 -- coefficient ring: ZZ/65521 |
i10 : assert(Phi * Phi^-1 == 1 and Phi^-1 * Phi == 1) |
i11 : assert(Psi * Psi^-1 == 1 and Psi^-1 * Psi == 1) |
i12 : assert(Eta * Eta^-1 == 1 and Eta^-1 * Eta == 1) |