Abstract
Let \({{\rm C} \subset \mathbb{P}^{g-1}}\) be a general curve of genus g, and let k be a positive integer such that the Brill–Noether number \({\uprho(g,k,1)\geq 0}\) and \({g > k+1}\). The aim of this short note is to study the relative canonical resolution of \({{\rm C}}\) on a rational normal scroll swept out by a \({g^1_k=|{\rm L}|}\) with \({{\rm L} \in {\rm W}^1_k({\rm C})}\) general. We show that the bundle of quadrics appearing in the relative canonical resolution is unbalanced if and only if \({\uprho > 0}\) and \({\left(k-\uprho-\frac{7}{2}\right)^2-2k+\frac{23}{4} > 0}\).
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Bopp, C., Hoff, M. Resolutions of general canonical curves on rational normal scrolls. Arch. Math. 105, 239–249 (2015). https://doi.org/10.1007/s00013-015-0794-x
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DOI: https://doi.org/10.1007/s00013-015-0794-x