## 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*[*T*
_{1}, . . . , *T _{m}*]. We resolve ℛ as an

*S*-module and

*I*as an

^{s}*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

*n*

^{th}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 .

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