# RationalMap || Ideal -- restriction of a rational map

## Synopsis

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

## Description

 i1 : P5 = ZZ/190181[x_0..x_5] o1 = P5 o1 : PolynomialRing i2 : Phi = rationalMap {x_4^2-x_3*x_5,x_2*x_4-x_1*x_5,x_2*x_3-x_1*x_4,x_2^2-x_0*x_5,x_1*x_2-x_0*x_4,x_1^2-x_0*x_3} o2 = -- rational map -- ZZ source: Proj(------[x , x , x , x , x , x ]) 190181 0 1 2 3 4 5 ZZ target: Proj(------[x , x , x , x , x , x ]) 190181 0 1 2 3 4 5 defining forms: { 2 x - x x , 4 3 5 x x - x x , 2 4 1 5 x x - x x , 2 3 1 4 2 x - x x , 2 0 5 x x - x x , 1 2 0 4 2 x - x x 1 0 3 } o2 : RationalMap (quadratic rational map from PP^5 to PP^5) i3 : J = ideal random(1,P5); o3 : Ideal of P5 i4 : Phi' = Phi||J o4 = -- rational map -- ZZ source: subvariety of Proj(------[x , x , x , x , x , x ]) defined by 190181 0 1 2 3 4 5 { 2 2 2 x + 9702x x - 94294x - x x + 68094x x - 9702x x - 68094x x + 93593x x - 53251x + 94294x x - 93593x x + 53251x x 1 1 2 2 0 3 2 3 0 4 1 4 2 4 4 0 5 1 5 3 5 } ZZ target: subvariety of Proj(------[x , x , x , x , x , x ]) defined by 190181 0 1 2 3 4 5 { x + 16566x - 70158x - 38148x - 77864x - 71321x 0 1 2 3 4 5 } defining forms: { 2 x - x x , 4 3 5 x x - x x , 2 4 1 5 x x - x x , 2 3 1 4 2 x - x x , 2 0 5 x x - x x , 1 2 0 4 2 2 - 9702x x + 94294x - 68094x x + 9702x x + 68094x x - 93593x x + 53251x - 94294x x + 93593x x - 53251x x 1 2 2 2 3 0 4 1 4 2 4 4 0 5 1 5 3 5 } o4 : RationalMap (quadratic rational map from hypersurface in PP^5 to hypersurface in PP^5) i5 : describe Phi o5 = rational map defined by forms of degree 2 source variety: PP^5 target variety: PP^5 coefficient ring: ZZ/190181 i6 : describe Phi' o6 = rational map defined by forms of degree 2 source variety: smooth quadric hypersurface in PP^5 target variety: hyperplane in PP^5 coefficient ring: ZZ/190181