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 which computes the relative canonical resolution associated to a curve and a pencil of divisors. Most of our experimental data can be found on the following webpage [BBH18]. 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 in [DP18] all have the same support. Relative Canonical ResolutionsThe relative canonical resolution is the minimal free resolution of a canonically embedded curve C inside a rational normal scroll. Every such scroll is swept out by linear spaces parametized by pencils of divisors on C.Studying divisors on moduli spaces reveals certain aspects of the global geometry of these spaces. A famous example for curves of odd genus g is the Koszul-divisor on the moduli space of curves M g . It is induced by the minimal free resolution of C ⊂ P g −1 . Set-theoretically the Koszul-divisor consists of curves, such that the minimal free resolution of the canonical model has extra-syzygies at a certain step. In [BP15],[DP15] and [DP18], the relative canonical resolution was used to define similar syzygy divisors on Hurwitz spaces H g ,k , parametizing pairs of curves of genus g and pencils of divisors of degree k (equivalently, covers of P 1 of degree k by curves of genus g ).We will briefly summarize the connections between pencils of divisors on canonical curves and rational normal scrolls in order to define the relative canonical resolution.