Syzygy divisors on Hurwitz spaces

Deopurkar, Anand; Patel, Anand

doi:10.1090/conm/703/14139

Cited by 4 publications

(9 citation statements)

References 18 publications

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“…The same phenomenon appears for classes of divisorial Brill-Noether loci in the moduli space M g (see [Eisenbud and Harris 1987a] and [Harris and Mumford 1982]). For the divisorial Brill-Noether classes it is known that these classes are supported on different sets, and in [Deopurkar and Patel 2018] the authors conjecture that this also happens for the syzygy divisors on H g,k .…”

Section: Experiments and Conjecturesmentioning

confidence: 92%

“…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. This yields interesting subvarieties which also turn out to be divisors in some cases (see [Deopurkar and Patel 2018]). Similar to Koszul divisors on the moduli space M g , the study of the divisors obtained from the relative canonical resolution sheds light on the global geometry of the Hurwitz space.…”

Section: Relative Canonical Resolutionsmentioning

confidence: 99%

“…Hence, we can deduce the degrees of the bundles in this relative resolution of C ⊂ ‫(ސ‬E T ) from Proposition 1.4. These degrees have also been computed directly in [Deopurkar and Patel 2018]. Definition 1.6.…”

Section: Relative Canonical Resolutionsmentioning

confidence: 99%

“…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 , . .…”

Section: Experiments and Conjecturesmentioning

confidence: 99%

See 3 more Smart Citations

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

Bopp

Hoff

2021

J. Softw. Alg. Geom.

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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.

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Section: Experiments and Conjecturesmentioning

confidence: 92%

Section: Relative Canonical Resolutionsmentioning

confidence: 99%

Section: Relative Canonical Resolutionsmentioning

confidence: 99%

Section: Experiments and Conjecturesmentioning

confidence: 99%

See 2 more Smart Citations

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

Bopp

Hoff

2021

J. Softw. Alg. Geom.

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“…Hence, we can deduce the degrees of the bundles in this relative resolution of C ⊂ P(E T ) from Proposition 1.4. These degrees have also been computed directly in [DP18].…”

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confidence: 99%

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

Bopp,

Hoff

2018

Preprint

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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.

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The integral Picard groups of low-degree Hurwitz spaces

Samir

Larson

2023

Math. Z.

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Syzygy divisors on Hurwitz spaces

Cited by 4 publications

References 18 publications

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

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

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

The integral Picard groups of low-degree Hurwitz spaces

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