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# MultirationalMap << MultiprojectiveVariety -- force the change of the target in a multi-rational map

- Operator: <<
- Usage:
    `Phi << Y`
- Inputs:
  - Phi, a multi-rational map, whose image is a subvariety $X\subseteq\mathbb{P}^{k_1}\times\cdots\times\mathbb{P}^{k_n}$
  - Y, a multi-projective variety, a subvariety of $\mathbb{P}^{l_1}\times\cdots\times\mathbb{P}^{l_n}$ with $k_i\leq l_i$ for $i=1,\ldots,n$
- Outputs:
  - a multi-rational map, the composition of Phi with an inclusion of $X$ into $Y$ (if this is possible and easy, otherwise an error is generated)

## Description

    i1 : Phi = parametrize PP_(ZZ/65521)^({1,3},{2,1});

    o1 : MultirationalMap (rational map from PP^4 to hypersurface in PP^2 x PP^3)

    i2 : X = image Phi;

    o2 : ProjectiveVariety, hypersurface in PP^2 x PP^3

    i3 : describe X

    o3 = ambient:.............. PP^2 x PP^3
         dim:.................. 4
         codim:................ 1
         degree:............... 8
         multidegree:.......... 2*T_0
         generators:........... (2,0)^1
         purity:............... true
         dim sing. l.:......... -1
         Segre embedding:...... map to PP^11

    i4 : Y = PP^{3,5};

    o4 : ProjectiveVariety, PP^3 x PP^5

    i5 : Psi = Phi << Y;

    o5 : MultirationalMap (rational map from PP^4 to Y)

    i6 : describe image Psi

    o6 = ambient:.............. PP^3 x PP^5
         dim:.................. 4
         codim:................ 4
         degree:............... 8
         multidegree:.......... 2*T_0^2*T_1^2
         generators:........... (0,1)^2 (1,0)^1 (2,0)^1
         purity:............... true
         dim sing. l.:......... -1
         Segre embedding:...... map to PP^11 ⊂ PP^23

The inclusion $j:X\to Y$ such that Phi * j == Psi can be obtained as follows:

    i7 : j = X << Y;

    o7 : MultirationalMap (rational map from X to Y)

    i8 : assert(Phi * j == Psi and j == (1_X << Y))

## See also

- multirationalMap(MultirationalMap,MultiprojectiveVariety) -- change the target of a multi-rational map

## Ways to use this method:

- `MultiprojectiveVariety << MultiprojectiveVariety`
- MultirationalMap << MultiprojectiveVariety -- force the change of the target in a multi-rational map

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/MultiprojectiveVarieties.m2:3906:0.