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

## Synopsis

• Operator: ||
• Usage:
Phi || Z
• Inputs:
• Phi, , $\Phi:X \dashrightarrow Y$
• Z, , a subvariety of $Y$
• Outputs:
• , the restriction of $\Phi$ to ${\Phi}^{(-1)} Z$, ${{\Phi}|}_{{\Phi}^{(-1)} Z}: {\Phi}^{(-1)} Z \dashrightarrow Z$

## 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 = projectiveVariety ideal random({1,2},ring target Phi); o3 : ProjectiveVariety, surface in PP^2 x PP^1 i4 : Phi' = Phi||Z; o4 : MultirationalMap (rational map from surface in PP^3 x PP^2 x PP^1 to Z) i5 : target Phi' o5 = Z o5 : ProjectiveVariety, surface in PP^2 x PP^1 i6 : assert(source Phi' == Phi^* Z)

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

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