Complete families of generic covers of elliptic curves

Abstract We pose some conjectures about the existence of certain complete, one-dimensional families of degree $d$ , genus $g$ branched covers of an elliptic curve. The conjectures would imply that the slope of the corresponding Hurwitz space is precisely $5+6/d$ , and that the slope of the moduli space of stable genus $g$ curves is bounded below by $5$ . We provide evidence for the conjectures when $g=2$ or when $d \leq 5$ and $g \gg 0$ .