The Picard rank conjecture for the Hurwitz spaces of degree up to five

Abstract

We prove that the rational Picard group of the simple Hurwitz space ${\mathcal H}_{d,g}$ is trivial for $d$ up to five. We also relate the rational Picard groups of the Hurwitz spaces to the rational Picard groups of the Severi varieties of nodal curves on Hirzebruch surfaces.Comment: Added a figure. Fixed typos and numerical mistake

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

“…By o15, we see that the second syzygy bundle N 2 is unbalanced in our example. Although this single example does not show that the generic relative canonical resolution is unbalanced for this case, one can show that this is indeed the generic form (see [Bopp and Hoff 2017]).…”

“…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 [Bujokas and Patel 2015;Deopurkar and Patel 2015;, the relative canonical resolution was used to define similar syzygy divisors on Hurwitz spaces H g,k , parametrizing pairs of curves of genus g and pencils of divisors of degree k (equivalently, covers of ‫ސ‬ 1 of degree k by curves of genus g). We also refer to [Farkas 2018] for divisors on Hurwitz spaces.…”

“…The sublocus inside H g,k parametrizing covers such that the associated scroll is unbalanced defines a divisor on H g,k precisely if g is a multiple of (k − 1). This divisor is called the Maroni divisor (for more details on the Maroni divisor see, e.g., [van der Geer and Kouvidakis 2017] and [Deopurkar and Patel 2015]). On the other hand, knowing the splitting type of the syzygy bundles in the relative canonical resolution for generic elements in H g,k one can study the sublocus inside H g,k consisting set-theoretically of curves for which a certain syzygy bundle has nongeneric splitting type.…”

“…Deopurkar and Patel used the relative canonical resolution to describe new effective divisors on the Hurwitz scheme H g,k . If the degree k divides g − 1, it is shown in [Bujokas and Patel 2015] that the relative canonical resolution for a generic element in H g,k is totally balanced, and hence, the locus µ i , corresponding set-theoretically to covers in H g,k for which the i-th syzygy bundle N i is unbalanced, has expected codimension 1. Deopurkar and Patel [2018] give these syzygy divisors µ 1 , .…”

A Note on Planarity Stratification of Hurwitz Spaces

“…Proposition 4 shows that there is a connection of the planarity defect with the slope invariants of meromorphic functions and with the Maroni strata, comp. [6] and [18]. In fact, the following statement is true.…”