Invariants of a General Branched Cover of P1

Bujokas, Gabriel; Patel, Anand

doi:10.1093/imrn/rnaa156

Cited by 4 publications

(4 citation statements)

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“…Similar results can be obtained by means of the resolvents with respect to A d−2 , S d−2 , S 2 ×A d−2 and S 2 ×S d−2 , whose scrollar invariants were determined in Corollary 42 and Theorem 4, using generic balancedness of the first syzygy bundle [11,Main Thm. ] in addition to Ballico's result.…”

Section: Proof This Follows From Indsupporting

confidence: 66%

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Scrollar invariants, syzygies and representations of the symmetric group

Castryck¹,

Vermeulen²,

Zhao³

2022

Preprint

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We give an explicit minimal graded free resolution of a certain Galois-theoretic configuration of d points in P d−2 , studied by Bhargava in the context of ring parametrizations, in terms of representations of the symmetric group S d . When applied to the generic fiber of a simply branched degree d cover of P 1 by a relatively canonically embedded curve C, our construction gives a new interpretation for the splitting types of the syzygy bundles appearing in its relative minimal resolution. Concretely, our work implies that all these splitting types consist of scrollar invariants of resolvent covers, up to a small shift. This vastly generalizes a prior observation due to Casnati, namely that the first syzygy bundle of a degree 4 cover splits according to the scrollar invariants of its cubic resolvent. Our work also shows that the splitting types of the syzygy bundles, together with the multi-set of scrollar invariants, belong to a much larger class of multi-sets of invariants that can be attached to C P 1 : one for each irreducible subrepresentation of S d , i.e., one for each non-trivial partition of d. We conjecture generic balancedness of all these multi-sets as soon as the genus of C is large enough when compared to d. * (1) 1 : . . . : α * (1) d−1 ], [α * (2) 1 : . . . : α * (2) d−1 ], . . . , [α * (d) 1 : . . . : α * (d) d−1 ].

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Section: Proof This Follows From Indsupporting

confidence: 66%

“…(1.8). Casnati-Ekedahl, building on earlier work of Schreyer [44], further showed that a minimal graded free resolution of the generic fiber can be completed to a minimal resolution of C relative to P(E), which takes the form (11) 0…”

mentioning

confidence: 97%

Scrollar invariants, syzygies and representations of the symmetric group

Castryck¹,

Vermeulen²,

Zhao³

2022

Preprint

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“…In general, we do not have a proof of this statement. However, the (un-)balancedness has been proved for the first bundle N 1 in some cases (see [BH15b] and [BP15]). For several cases our examples lead to the conjecture, that certain higher syzygy bundles in the relative canonical resolution are unbalanced.…”

Section: Database Of Experimentsmentioning

confidence: 99%

“…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 [BP15] 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 one. In [DP18] the authors give these syzygy divisor µ 1 , .…”

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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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Moduli of lattice polarized K3 surfaces via relative canonical resolutions

Bopp

Hoff

2022

Journal of Pure and Applied Algebra

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Invariants of a General Branched Cover of P1

Cited by 4 publications

References 5 publications

Scrollar invariants, syzygies and representations of the symmetric group

Scrollar invariants, syzygies and representations of the symmetric group

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

Moduli of lattice polarized K3 surfaces via relative canonical resolutions

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