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 $\f$ of $I$ has entries of degree $n$ and all of the other entries of $\f$ are linear. We identify an explicit generating set for the ideal $\Cal A$ which defines the Rees algebra $\Cal R=R[It]$; so $\Cal R=S/\Cal A$ for the polynomial ring $S=R[T_1,...,T_m]$. We resolve $\Cal R$ as an $S$-module and $I^s$ as an $R$-module, for all powers $s$. The proof uses the homogeneous coordinate ring, $A=S/H$, of a rational normal scroll, with $H\subseteq \Cal A$. The ideal $\Cal AA$ is isomorphic to the $n^{\text{th}}$ 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 $A/\Cal AA$ because the generators of $K^{(n)}$ are much less complicated then the generators of $\Cal AA$. 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 $\Cal C$ in projective $m-1$ space. The defining equations of the special fiber ring $\Cal R/(x,y)\Cal R$ yield a solution of the implicitization problem for $\Cal C$.