Scrollar invariants, syzygies and representations of the symmetric group

Abstract

AbstractWe give an explicit minimal graded free resolution, in terms of representations of the symmetric groupSd{S_{d}}, of a Galois-theoretic configuration ofdpoints in𝐏d-2{\mathbf{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 degreedcover of𝐏1{\mathbf{P}^{1}}by a relatively canonically embedded curveC, 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 toC→𝐏1{C\to\mathbf{P}^{1}}: one for each irreducible representation ofSd{S_{d}}, i.e., one for each partition ofd.