k = QQ m = 3; n = 4; R = k[x_(1,1)..x_(m,n),y_(1,1)..y_(m,n),t_1,t_2,z_(1,1)..z_(m,n)] Mx = genericMatrix(R,m,n) My = genericMatrix(R,y_(1,1),m,n) -- describe the variety V of matrices of rank 1 : Ix = minors(2,Mx) Iy = minors(2,My) zRelations = apply((1,1)..(m,n), ind->z_ind - t_1*x_ind - t_2*y_ind) I = Ix + Iy + ideal zRelations + ideal(t_1+t_2-1) Iz = eliminate(take(gens R, 2*m*n+2), I) Mz = genericMatrix(R,z_(1,1),m,n) -- the secant variety Sec_2(V) minors(3,Mz) == Iz