Interaction between Picard splitting-type strata and Maroni loci

Determine the geometry of the interaction between the stratification of the Picard variety by splitting types of pushforwards of line bundles and the Maroni loci obtained by varying the curve while fixing the line bundle.

Background

The paper distinguishes two complementary stratifications: splitting-type strata in the Picard variety of a fixed curve, which underlie Hurwitz-Brill-Noether and Severi-Brill-Noether theories, and Maroni loci in Hurwitz spaces, where the curve varies with the line bundle fixed.

Beyond the divisorial case of Maroni loci and the degree-three case treated in prior work, the geometry of their interaction is not understood. The authors explicitly identify a more detailed study of this interaction as an open question.

References

A more detailed study of the interaction of these two stratifications remains an intriguing open question; as far as we know, only the degree $3$ case has been addressed in , in which case the scrollar invariants encode the same information as the more classical Maroni invariant.

— Scrollar invariants of singular curves on toric surfaces  (2608.14080 - Christ et al., 14 Aug 2026) in Section 1, Introduction