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 P-1 -> P-2. The Rees algebra Rees (I) = R circle plus I circle plus I-2 circle plus... 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. Published by Elsevier B.V.