# RationalMap | Ideal -- restriction of a rational map

## Synopsis

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

## 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 : I = ideal(random(2,P5),random(3,P5)); o3 : Ideal of P5 i4 : Phi' = Phi|I 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 2 2 2 x + 16566x x - 47468x - 70158x x + 90626x x - 24212x - 38148x x - 31779x x + 19605x x - 769x - 77864x x + 10660x x + 7410x x - 66565x x - 49192x - 71321x x + 18433x x - 46460x x + 4698x x - 76890x x - 55574x , 0 0 1 1 0 2 1 2 2 0 3 1 3 2 3 3 0 4 1 4 2 4 3 4 4 0 5 1 5 2 5 3 5 4 5 5 2 3 2 2 2 3 2 2 2 2 2 3 2 2 2 2 2 2 2 3 2 2 2 2 2 2 2 2 2 3 x x - 51610x - 80517x x x + 21123x x - 3492x x + 85424x x + 3605x + 12124x x x - 30548x x - 57743x x x - 12590x x x - 19357x x - 25505x x + 11508x x + 35026x x - 79088x + 26615x x x - 10036x x + 39457x x x + 879x x x + 78560x x + 83997x x x - 42552x x x + 69752x x x - 78713x x + 52698x x - 46597x x - 42666x x + 34140x x - 29206x - 88405x x x - 15770x x + 18059x x x - 14362x x x + 87935x x + 67371x x x - 60584x x x - 81481x x x - 61286x x + 63878x x x - 33047x x x + 53673x x x + 88603x x x - 71230x x - 45859x x + 10167x x + 8993x x - 77222x x - 4109x x - 16028x 0 1 1 0 1 2 1 2 0 2 1 2 2 0 1 3 1 3 0 2 3 1 2 3 2 3 0 3 1 3 2 3 3 0 1 4 1 4 0 2 4 1 2 4 2 4 0 3 4 1 3 4 2 3 4 3 4 0 4 1 4 2 4 3 4 4 0 1 5 1 5 0 2 5 1 2 5 2 5 0 3 5 1 3 5 2 3 5 3 5 0 4 5 1 4 5 2 4 5 3 4 5 4 5 0 5 1 5 2 5 3 5 4 5 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 } o4 : RationalMap (quadratic rational map from threefold in PP^5 to 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: complete intersection of type (2,3) in PP^5 target variety: PP^5 coefficient ring: ZZ/190181