# RationalMap * RationalMap -- composition of rational maps

## Synopsis

• Operator: *
• Usage:
phi * psi
compose(phi,psi)
• Inputs:
• phi, , $X \dashrightarrow Y$
• psi, , $Y \dashrightarrow Z$
• Outputs:
• , $X \dashrightarrow Z$, the composition of phi and psi

## Description

 i1 : R = QQ[x_0..x_3]; S = QQ[y_0..y_4]; T = QQ[z_0..z_4]; i4 : phi = rationalMap(R,S,{x_0*x_2,x_0*x_3,x_1*x_2,x_1*x_3,x_2*x_3}) o4 = -- rational map -- source: Proj(QQ[x , x , x , x ]) 0 1 2 3 target: Proj(QQ[y , y , y , y , y ]) 0 1 2 3 4 defining forms: { x x , 0 2 x x , 0 3 x x , 1 2 x x , 1 3 x x 2 3 } o4 : RationalMap (quadratic rational map from PP^3 to PP^4) i5 : psi = rationalMap(S,T,{y_0*y_3,-y_2*y_3,y_1*y_2,y_2*y_4,-y_3*y_4}) o5 = -- rational map -- source: Proj(QQ[y , y , y , y , y ]) 0 1 2 3 4 target: Proj(QQ[z , z , z , z , z ]) 0 1 2 3 4 defining forms: { y y , 0 3 -y y , 2 3 y y , 1 2 y y , 2 4 -y y 3 4 } o5 : RationalMap (quadratic rational map from PP^4 to PP^4) i6 : phi * psi o6 = -- rational map -- source: Proj(QQ[x , x , x , x ]) 0 1 2 3 target: Proj(QQ[z , z , z , z , z ]) 0 1 2 3 4 defining forms: { x , 0 -x , 1 x , 0 x , 2 -x 3 } o6 : RationalMap (linear rational map from PP^3 to PP^4) i7 : (map phi) * (map psi) 2 2 2 o7 = map (R, T, {x x x x , -x x x , x x x x , x x x , -x x x }) 0 1 2 3 1 2 3 0 1 2 3 1 2 3 1 2 3 o7 : RingMap R <--- T