Scrollar invariants, syzygies and representations of the symmetric group

TL;DR

This paper provides an explicit minimal graded free resolution of certain Galois configurations using symmetric group representations, revealing new geometric insights into syzygy bundles and scrollar invariants of algebraic curves.

Contribution

It introduces a novel interpretation of syzygy bundle splitting types via symmetric group representations and generalizes prior results on scrollar invariants for degree 4 covers.

Findings

Explicit minimal graded free resolution in terms of $S_d$ representations.

All splitting types of syzygy bundles are scrollar invariants of resolvent covers.

Splitting types and scrollar invariants form a larger class of invariants associated with $C o P^1$.

Abstract

We give an explicit minimal graded free resolution, in terms of representations of the symmetric group $S_d$, of a Galois-theoretic configuration of $d$ points in $\mathbb{P}^{d-2}$ that was studied by Bhargava in the context of ring parametrizations. When applied to the geometric generic fiber of a simply branched degree $d$ cover of $\mathbb{P}^1$ 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. 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 $C \to \mathbb{P}^1$: one for each irreducible representation of $S_d$, i.e., one for each partition of $d$.