# MultirationalMap | MultiprojectiveVariety -- restriction of a multi-rational map

## Synopsis

• Operator: |
• Usage:
Phi | Z
• Inputs:
• Phi, , $\Phi:X \dashrightarrow Y$
• Z, , a subvariety of $X$
• Outputs:
• , the restriction of $\Phi$ to $Z$, $\phi|_{Z}: Z \dashrightarrow Y$

## Description

 i1 : ZZ/33331[x_0..x_3], f = rationalMap {x_2^2-x_1*x_3,x_1*x_2-x_0*x_3,x_1^2-x_0*x_2}, g = rationalMap {x_2^2-x_1*x_3,x_1*x_2-x_0*x_3}; i2 : Phi = last graph rationalMap {f,g}; o2 : MultirationalMap (rational map from threefold in PP^3 x PP^2 x PP^1 to PP^2 x PP^1) i3 : Z = (source Phi) * projectiveVariety ideal random({1,1,2},ring ambient source Phi); o3 : ProjectiveVariety, surface in PP^3 x PP^2 x PP^1 i4 : Phi' = Phi|Z; o4 : MultirationalMap (rational map from Z to PP^2 x PP^1) i5 : source Phi' o5 = Z o5 : ProjectiveVariety, surface in PP^3 x PP^2 x PP^1 i6 : assert(image Phi' == Phi Z)

The following is a shortcut to take restrictions on random hypersurfaces as above.

 i7 : Phi|{1,1,2}; o7 : MultirationalMap (rational map from surface in PP^3 x PP^2 x PP^1 to PP^2 x PP^1)