Rational Normal Scrolls and the Defining Equations of Rees Algebras

Abstract. Consider a height two ideal, I, which is minimally generated by m homogeneous forms of degree d in the polynomial ring R = k[x, y]. Suppose that one column in the homogeneous presenting matrix φ of I has entries of degree n and all of the other entries of φ are linear. We identify an explicit generating set for the ideal A which defines the Rees algebra R = R[It]; so R = S/A for the polynomial ring S = R[T1, . . . , Tm]. We resolve R as an S-module and I as an R-module, for all powers s. The proof uses the homogeneous coordinate ring, A = S/H, of a rational normal scroll, with H ⊆ A. The ideal AA is isomorphic to the nth symbolic power of a height one prime ideal K of A. The ideal K(n) is generated by monomials. Whenever possible, we study A/K(n) in place of A/AA because the generators of K(n) are much less complicated then the generators of AA. We obtain a filtration of K(n) in which the factors are polynomial rings, hypersurface rings, or modules resolved by generalized Eagon-Northcott complexes. The generators of I parameterize an algebraic curve C in projective m− 1 space. The defining equations of the special fiber ring R/(x, y)R yield a solution of the implicitization problem for C.