# ideal(RationalMap) -- base locus of a rational map

## Synopsis

• Function: ideal
• Usage:
ideal phi
• Inputs:
• phi,
• Outputs:
• an ideal, the ideal of the base locus of phi

## Description

This is generally difficult, but in some cases it is equivalent to ideal matrix phi, which does not perform any computation.

 i1 : y = gens(QQ[x_0..x_5]/(x_2^2-x_2*x_3+x_1*x_4-x_0*x_5)); i2 : phi = rationalMap {y_4^2-y_3*y_5,-y_2*y_4+y_3*y_4-y_1*y_5, -y_2*y_3+y_3^2-y_1*y_4, -y_1*y_2+y_1*y_3-y_0*y_4, y_1^2-y_0*y_3} o2 = -- rational map -- source: subvariety of Proj(QQ[x , x , x , x , x , x ]) defined by 0 1 2 3 4 5 { 2 x - x x + x x - x x 2 2 3 1 4 0 5 } target: Proj(QQ[y , y , y , y , y ]) 0 1 2 3 4 defining forms: { 2 x - x x , 4 3 5 - x x + x x - x x , 2 4 3 4 1 5 2 - x x + x - x x , 2 3 3 1 4 - x x + x x - x x , 1 2 1 3 0 4 2 x - x x 1 0 3 } o2 : RationalMap (quadratic rational map from hypersurface in PP^5 to PP^4) i3 : time ideal phi -- used 0.00278176 seconds 2 2 o3 = ideal (x - x x , x x - x x + x x , x x - x + x x , x x - x x + 4 3 5 2 4 3 4 1 5 2 3 3 1 4 1 2 1 3 ------------------------------------------------------------------------ 2 x x , x - x x ) 0 4 1 0 3 QQ[x ..x ] 0 5 o3 : Ideal of ----------------------- 2 x - x x + x x - x x 2 2 3 1 4 0 5 i4 : assert(ideal phi == ideal matrix phi) i5 : phi' = last graph phi o5 = -- rational map -- source: subvariety of Proj(QQ[x , x , x , x , x , x ]) x Proj(QQ[y , y , y , y , y ]) defined by 0 1 2 3 4 5 0 1 2 3 4 { x y - x y + x y , 1 2 3 3 4 4 x y - x y - x y + x y , 0 2 1 3 2 4 3 4 x y - x y + x y - x y , 2 1 3 1 4 2 5 3 x y - x y + x y , 1 1 2 2 5 4 x y - x y + x y , 0 1 2 3 4 4 x y - x y + x y - x y , 2 0 3 0 4 1 5 2 x y - x y + x y , 1 0 3 1 4 2 x y - x y + x y , 0 0 2 2 4 3 2 x - x x + x x - x x 2 2 3 1 4 0 5 } target: Proj(QQ[y , y , y , y , y ]) 0 1 2 3 4 defining forms: { y , 0 y , 1 y , 2 y , 3 y 4 } o5 : MultihomogeneousRationalMap (rational map from 4-dimensional subvariety of PP^5 x PP^4 to PP^4) i6 : time ideal phi' -- used 0.141197 seconds o6 = ideal 1 QQ[x ..x , y ..y ] 0 5 0 4 o6 : Ideal of -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 2 (x y - x y + x y , x y - x y - x y + x y , x y - x y + x y - x y , x y - x y + x y , x y - x y + x y , x y - x y + x y - x y , x y - x y + x y , x y - x y + x y , x - x x + x x - x x ) 1 2 3 3 4 4 0 2 1 3 2 4 3 4 2 1 3 1 4 2 5 3 1 1 2 2 5 4 0 1 2 3 4 4 2 0 3 0 4 1 5 2 1 0 3 1 4 2 0 0 2 2 4 3 2 2 3 1 4 0 5 i7 : assert(ideal phi' != ideal matrix phi')