Syzygies and the Rees algebra

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 P1→P2. The Rees algebra Rees(I)=R⊕I⊕I2⊕⋯ 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.