Scrollar invariants, syzygies and representations of the symmetric group

We give an explicit minimal graded free resolution of a certain Galois-theoretic configuration of $d$ points in $\mathbb{P}^{d-2}$, 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 $\mathbb{P}^1$ 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, 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 $C \to \mathbb{P}^1$: 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 $d$.