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MultiprojectiveVarieties :: parametrize(MultiprojectiveVariety)

parametrize(MultiprojectiveVariety) -- try to get a parametrization of a multi-projective variety

Synopsis

Description

Currently, this function works in particular for linear varieties, quadrics, varieties of minimal degree, Grassmannians, Severi varieties, del Pezzo fivefolds, and some types of Fano fourfolds.

i1 : K = ZZ/65521;
i2 : X = PP_K^{2,4,1,3};

o2 : ProjectiveVariety, PP^2 x PP^4 x PP^1 x PP^3
i3 : f = parametrize X;

o3 : MultirationalMap (rational map from PP^10 to X)
i4 : Y = random({{1,0,0,0},{0,1,0,0},{0,1,0,0},{0,0,0,1}},0_X);

o4 : ProjectiveVariety, 6-dimensional subvariety of PP^2 x PP^4 x PP^1 x PP^3
i5 : g = parametrize Y;

o5 : MultirationalMap (rational map from PP^6 to Y)
i6 : Z = random({{1,1,0,0},{0,1,0,0},{0,0,1,0},{0,0,0,1},{0,0,0,1}},0_X);

o6 : ProjectiveVariety, 5-dimensional subvariety of PP^2 x PP^4 x PP^1 x PP^3
i7 : h = parametrize Z;

o7 : MultirationalMap (rational map from PP^5 to Z)
i8 : describe h

o8 = multi-rational map consisting of 4 rational maps
     source variety: PP^5
     target variety: 5-dimensional subvariety of PP^2 x PP^4 x PP^1 x PP^3 cut out by 5 hypersurfaces of multi-degrees (0,0,0,1)^2 (0,0,1,0)^1 (0,1,0,0)^1 (1,1,0,0)^1 
     base locus: threefold in PP^5 cut out by 6 hypersurfaces of degrees 4^5 2^1 
     dominance: true
     multidegree: {1, 6, 15, 31, 50, 50}
     degree: 1
     degree sequence (map 1/4): [2]
     degree sequence (map 2/4): [2]
     degree sequence (map 3/4): [0]
     degree sequence (map 4/4): [2]
     coefficient ring: K
i9 : describe inverse h

o9 = multi-rational map consisting of one single rational map
     source variety: 5-dimensional subvariety of PP^2 x PP^4 x PP^1 x PP^3 cut out by 5 hypersurfaces of multi-degrees (0,0,0,1)^2 (0,0,1,0)^1 (0,1,0,0)^1 (1,1,0,0)^1 
     target variety: PP^5
     base locus: threefold in PP^2 x PP^4 x PP^1 x PP^3 cut out by 23 hypersurfaces of multi-degrees (0,0,1,0)^1 (1,1,0,0)^1 (1,2,0,0)^3 (1,1,0,1)^6 (1,0,0,2)^1 (0,1,0,0)^1 (0,0,0,1)^2 (2,1,0,0)^2 (2,0,0,1)^2 (0,2,0,1)^3 (0,1,0,2)^1 
     dominance: true
     multidegree: {50, 50, 31, 15, 6, 1}
     degree: 1
     degree sequence (map 1/1): [(1,1,0,1)]
     coefficient ring: K
i10 : W = projectiveVariety pfaffians(4,matrix pack(5,for i to 24 list random(1,ring PP_K^8)));

o10 : ProjectiveVariety, 5-dimensional subvariety of PP^8
i11 : parametrize W

o11 = multi-rational map consisting of one single rational map
      source variety: PP^5
      target variety: 5-dimensional subvariety of PP^8 cut out by 5 hypersurfaces of degree 2
      dominance: true
      degree: 1

o11 : MultirationalMap (birational map from PP^5 to W)
i12 : parametrize (W ** (point W))

o12 = multi-rational map consisting of 2 rational maps
      source variety: PP^5
      target variety: 5-dimensional subvariety of PP^8 x PP^8 cut out by 13 hypersurfaces of multi-degrees (0,1)^8 (2,0)^5 

o12 : MultirationalMap (rational map from PP^5 to 5-dimensional subvariety of PP^8 x PP^8)

See also