The author considers the minimal free resolutions for the coordinate ring of 5-gonal canonically embedded curves \(C\) in \(\mathbb P^{g-1}\); in particular, the interest is focused on the relation between the Betti numbers of \(C\) and those of the scroll \(X\) defined by a minimal-degree pencil on \(C\).
More precisely, the author proves that given a 5-gonal canonical curve \(C\), s.t. its genus is g and taking \(n=\overline{(g-1)/2}\), the following relation holds for Betti numbers: \(\beta_{n,n+1}(C)>\beta_{n,n+1}(X)\) for odd genus \(g\geq 13\) and even genus \(g\geq 28\).