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

TL;DR

This paper introduces a Macaulay2 package for computing relative canonical resolutions of curves on rational normal scrolls, presents experimental data, and proposes conjectures about the structure of these resolutions.

Contribution

It provides a new computational tool for relative canonical resolutions and formulates conjectures on their shapes and properties in specific cases.

Findings

Experimental data on relative canonical resolutions

Conjecture on syzygy divisors support in Hurwitz space

Introduction of a Macaulay2 package for computations

Abstract

This short note provides a quick introduction to relative canonical resolutions of curves on rational normal scrolls. We present our Macaulay2-package which computes the relative canonical resolution associated to a curve and a pencil of divisors. Most of our experimental data can be found on a dedicated webpage. We end with a list of conjectural shapes of relative canonical resolutions. In particular, for curves of genus $g=n\cdot k +1$ and pencils of degree $k$ for $n\ge 1$, we conjecture that the syzygy divisors on the Hurwitz space $\mathscr{H}_{g,k}$ constructed by Deopurkar and Patel all have the same support.