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

- Operator: SPACE
- Usage:
    `H(Z,d,m)`
    `H(Z,d)`
    `H(Z)`
- Inputs:
  - a hom-set, the hom-set of rational maps between two varieties X and Y
  - Z, a multi-projective variety, 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
          }

## See also

- Hom(MultiprojectiveVariety,MultiprojectiveVariety) -- get the hom-set of rational maps between two multi-projective varieties
- shortcuts -- Some convenient shortcuts for multi-rational maps consisting of a single rational map
- rationalMap(Ideal,ZZ,ZZ) -- makes a rational map from an ideal

## Ways to use this method:

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

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MultiprojectiveVarieties.m2:4136:0.