Characterization of realizable scrollar-invariant tuples

Characterize exactly which non-decreasing integer tuples can occur as the scrollar invariants of a finite cover of the projective line.

Background

For a finite degree-d map from a projective curve to the projective line, the scrollar invariants are the integers determined by the splitting type of the pushforward of the structure sheaf. Although the paper determines these invariants for broad classes of curves on toric surfaces, it emphasizes that their general realizability problem remains unresolved, even for smooth curves.

The problem asks for a complete intrinsic characterization of all tuples that arise, complementing the specific constructions and partial realizability results available in the literature.