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

Synopsis

Operator: ||
Usage:

Phi || Z
Inputs:
- Phi, a multi-rational map, $\Phi:X \dashrightarrow Y$
- Z, a multi-projective variety, a subvariety of $Y$
Outputs:
- a multi-rational map, the restriction of $\Phi$ to ${\Phi}^{(-1)} Z$, ${{\Phi}|}_{{\Phi}^{(-1)} Z}: {\Phi}^{(-1)} Z \dashrightarrow Z$

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)