We show that for $5$-gonal curves of odd genus $g\geq 13$ and even genus $g\geq 28$ the $\lceil \frac{g-1}{2}\rceil$-th syzygy module of the curve is not determined by the syzygies of the scroll swept out by the special pencil of degree $5$.
Stability of genus five canonical curves
High-school algebra of the theory of dicritical divisors: atypical fibres for special pencils and polynomials
Special Linear Series and Syzygies of Canonical Curves of Genus 9
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