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# MultirationalMap MultiprojectiveVariety -- direct image via a multi-rational map

- Operator: SPACE
- Usage:
    `Phi X`
- Inputs:
  - Phi, a multi-rational map
  - X, a multi-projective variety, a subvariety of the source of Phi
- Outputs:
  - a multi-projective variety, the (closure of the) direct image of X via Phi

## Description

    i1 : ZZ/65521[x_0..x_4];

    i2 : f = last graph rationalMap {x_2^2-x_1*x_3, x_1*x_2-x_0*x_3, x_1^2-x_0*x_2, x_0*x_4, x_1*x_4, x_2*x_4, x_3*x_4, x_4^2};

    o2 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^7 to PP^7)

    i3 : Phi = rationalMap {f,f};

    o3 : MultirationalMap (rational map from 4-dimensional subvariety of PP^4 x PP^7 to PP^7 x PP^7)

    i4 : Z = source Phi;

    o4 : ProjectiveVariety, 4-dimensional subvariety of PP^4 x PP^7

    i5 : time Phi Z;
     -- used 0.0967813s (cpu); 0.0957622s (thread); 0s (gc)

    o5 : ProjectiveVariety, 4-dimensional subvariety of PP^7 x PP^7

    i6 : dim oo, degree oo, degrees oo

    o6 = (4, 80, {({0, 2}, 5), ({1, 1}, 33), ({2, 0}, 5)})

    o6 : Sequence

    i7 : time Phi (point Z + point Z + point Z)
     -- used 1.71356s (cpu); 1.27437s (thread); 0s (gc)

    o7 = 0-dimensional subvariety of PP^7 x PP^7 cut out by 22 hypersurfaces of multi-degrees (0,1)^5 (0,2)^3 (1,0)^5 (1,1)^6 (2,0)^3

    o7 : ProjectiveVariety, 0-dimensional subvariety of PP^7 x PP^7

    i8 : dim oo, degree oo, degrees oo

    o8 = (0, 3, {({0, 1}, 5), ({0, 2}, 3), ({1, 0}, 5), ({1, 1}, 6), ({2, 0},
         ------------------------------------------------------------------------
         3)})

    o8 : Sequence

## See also

- image(MultirationalMap) -- image of a multi-rational map
- MultirationalMap ^* -- inverse image via a multi-rational map
- RationalMap Ideal -- direct image via a rational map

## Ways to use this method:

- MultirationalMap MultiprojectiveVariety -- direct image via a multi-rational map
- `RationalMap MultiprojectiveVariety`

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MultiprojectiveVarieties.m2:3066:0.