Scrollar invariants, syzygies and representations of the symmetric group II

2023/06/08 by Vermeulen, Floris
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2306.05269

Abstract

Let φ: C→ ℙ¹ be a degree d cover of curves. In work by Castryck, Zhao and the author, we showed how one can attach to each partition λ of d a multi-set of scrollar invariants of λ with respect to φ. We studied these invariants when φ is simply branched, and related these new scrollar invariants to known geometric data. In this article we show how one can remove this simple branching condition, using the notion of S_d-closure as developed by Bhargava and Satriano. With this new framework, we are able to generalize all results from this work to arbitrary covers C→ ℙ¹. In particular, we obtain a syzygy-free interpretation for the splitting types of the syzygy bundles in the Casnati--Ekedahl resolution of an arbitrary cover φ: C→ ℙ¹. By using the Maroni bound, we are able to give new general bounds on these splitting types.

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• Scrollar invariants, syzygies and representations of the symmetric group
• Invariants of a general branched cover of \bf P¹
• Secant varieties and Hirschowitz bound on vector bundles over a curve
• Projecting Syzygies of Curves
• Syzygies of canonical ribbons on higher genus curves