Rational Normal Scrolls and the Defining Equations of Rees Algebras

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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 of I has entries of degree n and all of the other entries of 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 Iˢ 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.