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 which defines the Rees algebra ℛ = R[It]; so 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 . The ideal 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 because the generators of K(n) are much less complicated then the generators of . 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 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 .