Tschirnhausen bundles of covers of the projective line

Abstract: A degree $d$ genus $g$ cover of the complex projective line by a smooth curve $C$ yields a vector bundle on the projective line by pushforward of the structure sheaf. Which bundles are possible? Equivalently, which $\mathbb{P}^{d-2}$-bundles over $\mathbb{P}^1$ contain such covers? (In the language of many previous papers: what are the scrollar invariants of the cover?) We give a complete answer in degree $4$, which exhibits the expected pathologies. We describe a polytope (one per degree) which we propose gives the complete answer for primitive covers, i.e. covers that don't factor through a subcover. We show that all such bundles (for primitive covers) lie in this polytope, and that a ``positive proportion'' of the polytope arises from smooth covers. Moreover, we show the necessity of the primitivity assumption. Finally, we show that the map from the Hurwitz space of smooth covers to the space of bundles is not flat (for $d>3$ and $g \gg_d 0$).