Let a , b , c be linearly independent homogeneous polynomials in the standard Z -graded ring R ≔ k [ s , t ] with the same degree d and no common divisors. This defines a morphism P 1 → P 2 . The Rees algebra Rees ( I ) = R ⊕ I ⊕ I 2 ⊕ ⋯ of the ideal I = 〈 a , b , c 〉 is the graded R -algebra which can be described as the image of an R -algebra homomorphism h : R [ x , y , z ] → Rees ( I ) . This paper discusses one result concerning the structure of the kernel of the map h and its relation to the problem of finding the implicit equation of the image of the map given by a , b , c . In particular, we prove a conjecture of Hong, Simis and Vasconcelos. We also relate our results to the theory of adjoint curves and prove a special case of a conjecture of Cox.