--Worksheet 5, Mathematics 535, Fall 2008 --Imagge of rational maps --To find the image of a rational map from X to P^n --Let I be the ideal of a ring R describing the source X of the rational map --Let f_i be polynomials in variables of R describing the coordinate functions --Then the kernel of the mapfrom K[x_0...x_n] to R/I given by sending x_i to f_i --is the ideal describing the image of the rational map --Example: Projection of the rational normal quartic to P^3 --rational normal quartic in P^4 R=QQ[x_0..x_4] S=QQ[s,t] quart=ker map (S,R,{t^4,t^3*s,t^2*s^2,t*s^3,s^4}) --projection from the point [0,0,0,0,1] T=QQ[y_0..y_3] par={x_0,x_1,x_2,x_3} --find ideal of image of projection ker map(R/quart,T,par) --note that it is a twisted cubic --Equation of the cubic scroll in P^4 T3=QQ[z_0..z_2] ker map(T3,R,{z_0^2,z_1^2,z_0*z_1,z_0*z_2,z_1*z_2}) ---Problems --1. Find the equation of the image of the rational map of P^2 to P^6 given by -- {z_0*z_1^2,z_0*z_2^2,z_1*z_0^2,z_1*z_2^2,z_2*z_0^2,z_2*z_1^2,z_0*z_1*z_2}