  Let $A$ be the homogeneous coordinate ring of a rational normal scroll. The
ring $A$ is equal to the quotient of a polynomial ring $S$ by the ideal
generated by the two by two minors of a scroll matrix $\psi$ with two rows and
$\ell$ catalecticant blocks. The class group of $A$ is cyclic, and is infinite
provided $\ell$ is at least two. One generator of the class group is $[J]$,
where $J$ is the ideal of $A$ generated by the entries of the first column of
$\psi$. The positive powers of $J$ are well-understood, in the sense that the
$n^{\text{th}}$ ordinary power, the $n^{th}$ symmetric power, and the $n^{th}$
symbolic power all coincide and therefore all three $n^{th}$ powers are
resolved by a generalized Eagon-Northcott complex. The inverse of $[J]$ in the
class group of $A$ is $[K]$, where $K$ is the ideal generated by the entries of
the first row of $\psi$. We study the positive powers of $[K]$. We obtain a
minimal generating set and a Groebner basis for the preimage in $S$ of the
symbolic power $K^{(n)}$. We describe a filtration of $K^{(n)}$ in which all of
the factors are Cohen-Macaulay $S$-modules resolved by generalized
Eagon-Northcott complexes. We use this filtration to describe the modules in a
finely graded resolution of $K^{(n)}$ by free $S$-modules. We calculate the
regularity of the graded $S$-module $K^{(n)}$ and we show that the symbolic
Rees ring of $K$ is Noetherian.
