A Step in Castelnuovo theory via Grobner bases

We establish the first previously unknown case of the Eisenbud-Harris conjecture in Castelnuovo theory concerning algebraic curves of high genus in ${\bf P}^n$. The problem is reduced to a question about zero-dimensional schemes $\Gamma \subset {\bf P}^{n-1}$ in symmetric position with certain constrains on the Hilbert function. The method of Gr\"obner bases is then applied to study the homogeneous ideal of $\Gamma$.