This paper, joint with A. Deopurkar, establishes an important result in the moduli theory of branched coverings. A branched covering X/Y of a smooth projective curve Y determines a vector bundle E on Y, sometimes called the Tschirnhausen bundle. The main result proves a formal smoothness-type statement about the functor which assigns to a branched covering its Tschirnhausen bundle. In particular, when Y is the projective line, the main result implies an "asymptotic" existence statement for scrollar invariants.