Let C ⊂ P g −1 be a general curve of genus g and let k be a positive integer such that the Brill-Noether number ρ(g , k, 1) ≥ 0 and g > k + 1. The aim of this short note is to study the relative canonical resolution of C on a rational normal scroll swept out by a g 1 k = |L| with L ∈ W 1 k (C) general. We show that the bundle of quadrics appearing in the relative canonical resolution is unbalanced if and only if ρ > 0 and (k − ρ − 7 2 ) 2 − 2k + 23 4 > 0.
Based on computeralgebra experiments we formulate a refined version of Green's conjecture and a conjecture of Schicho-Schreyer-Weimann which conjecturally also holds in positive characteristic. The experiments are done by using our Macaulay2 package, which constructs random canonically embedded curves of genus g ≤ 15 over arbitrary small finite fields.
This short note provides a quick introduction to relative canonical resolutions of curves on rational normal scrolls. We present our Macaulay2 package that computes the relative canonical resolution associated to a curve and a pencil of divisors. We end with a list of conjectural shapes of relative canonical resolutions. In particular, for curves of genus g = n • k + 1 and pencils of degree k for n ≥ 1, we conjecture that the syzygy divisors on the Hurwitz scheme H g,k constructed by Deopurkar and Patel (Contemp. Math. 703 (2018) 209-222) all have the same support.
We show that for 5-gonal curves of odd genus g ≥ 13 and even genus g ≥ 28 the ⌈ g−1 2 ⌉-th syzygy module of the curve is not determined by the syzygies of the scroll swept out by the special pencil of degree 5.
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