The relative canonical resolution: Macaulay2-package, experiments and conjectures

Abstract

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.