RAT MultiprojectiveVariety -- rational map defined by a linear system of hypersurfaces through a variety

Synopsis

Operator: SPACE
Usage:

H(Z,d,m)

H(Z,d)

H(Z)
Inputs:
- the hom-set of rational maps between two varieties X and Y
- Z, a subvariety of $X$; d is a (multi-)degree and e is a multiplicity (by default, e=1 and d is the maximum degree of the generators)
Outputs:
- a multi-rational map, the map $X\dashrightarrow Y$ defined by the linear system of hypersurfaces of degree $d$ having points of multiplicity $e$ along $Z$

Description

This is another way of calling the method (rationalMap,MultiprojectiveVariety,List,ZZ).

i1 : X = random(3,0_(PP_(ZZ/17)^2));

o1 : ProjectiveVariety, curve in PP^2

i2 : Y = PP_(ZZ/17)^4;

o2 : ProjectiveVariety, PP^4

i3 : H = Hom(X,Y);

o3 : Hom(X,Y)

i4 : Z = point X + point X;

o4 : ProjectiveVariety, 0-dimensional subvariety of PP^2

i5 : H(Z,3,2) -- map defined by the cubics with double points along Z

o5 = multi-rational map consisting of one single rational map
     source variety: curve in PP^2 defined by a form of degree 3
     target variety: PP^4

o5 : MultirationalMap (rational map from X to Y)

i6 : show oo

o6 = -- multi-rational map --
                                ZZ
     source: subvariety of Proj(--[x , x , x ]) defined by
                                17  0   1   2
             {
               3     2         2     3     2                 2         2       2     3
              x  - 3x x  - 7x x  - 7x  - 4x x  - 7x x x  - 5x x  + 6x x  - 4x x  - 7x
               0     0 1     0 1     1     0 2     0 1 2     1 2     0 2     1 2     2
             }
                  ZZ
     target: Proj(--[x , x , x , x , x ])
                  17  0   1   2   3   4
     -- rational map 1/1 -- 
     map 1/1, one of its representatives:
     {
       2         2     2                   2
      x x  + 7x x  - 5x x  + 3x x x  - 3x x ,
       0 1     0 1     0 2     0 1 2     0 2
      
         2     3               2         2
      x x  + 7x  - 5x x x  + 3x x  - 3x x ,
       0 1     1     0 1 2     1 2     1 2
      
                 2         2       2     3
      x x x  + 7x x  - 5x x  + 3x x  - 3x ,
       0 1 2     1 2     0 2     1 2     2
      
       2       3               2         2
      x x  + 3x  + 3x x x  - 4x x  - 6x x ,
       0 1     1     0 1 2     1 2     1 2
      
       2       2         2       2     3
      x x  + 3x x  + 3x x  - 4x x  - 6x
       0 2     1 2     0 2     1 2     2
     }

As always, you can invoke the functions check and isWellDefined to check if the map is valid.

i7 : X = PP_(ZZ/17)^{1,1};

o7 : ProjectiveVariety, PP^1 x PP^1

i8 : H = Hom(X,image segreEmbedding X);

o8 : Hom(X,surface in PP^3)

i9 : check H(0_X,{1,1}) -- Segre embedding of X

o9 = multi-rational map consisting of one single rational map
     source variety: PP^1 x PP^1
     target variety: surface in PP^3 defined by a form of degree 2

o9 : MultirationalMap (rational map from X to surface in PP^3)

i10 : show oo

o10 = -- multi-rational map --
                   ZZ                   ZZ
      source: Proj(--[x0 , x0 ]) x Proj(--[x1 , x1 ])
                   17   0    1          17   0    1
                                 ZZ
      target: subvariety of Proj(--[t , t , t , t ]) defined by
                                 17  0   1   2   3
              {
               t t  - t t
                1 2    0 3
              }
      -- rational map 1/1 -- 
      map 1/1, one of its representatives:
      {
       x0 x1 ,
         0  0
       
       x0 x1 ,
         0  1
       
       x0 x1 ,
         1  0
       
       x0 x1
         1  1
      }

Ways to use this method: