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

## Synopsis

• Operator: <<
• Usage:
Phi << Y
• Inputs:
• Phi, , whose image is a subvariety $X\subseteq\mathbb{P}^{k_1}\times\cdots\times\mathbb{P}^{k_n}$
• Y, , 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:
• , 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))