Syzygies and the Rees algebra

Let a,b,c be linearly independent homogeneous polynomials in the standard image-graded ring Rcolon, equalsk[s,t] with the same degree d and no common divisors. This defines a morphism image. The Rees algebra image of the ideal I=left angle bracketa,b,cright-pointing angle bracket is the graded R-algebra which can be described as the image of an R-algebra homomorphism h: image. 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.