#### Title

Syzygies and the Rees algebra

#### Document Type

Article

#### Publication Date

7-1-2008

#### Abstract

Let a, b, c be linearly independent homogeneous polynomials in the standard Z-graded ring R {colon equals} 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.

#### Publication Source (Journal or Book title)

Journal of Pure and Applied Algebra

#### First Page

1787

#### Last Page

1796

#### Recommended Citation

Cox, D., Hoffman, J., & Wang, H.
(2008). Syzygies and the Rees algebra.* Journal of Pure and Applied Algebra**, 212* (7), 1787-1796.
https://doi.org/10.1016/j.jpaa.2007.11.006