Covers of elliptic curves and slopes of effective divisors on the moduli space of curves

Consider genus g curves that admit degree d covers to elliptic curves only branched at one point with a fixed ramification type. The locus of such covers forms a one parameter family Y that naturally maps into the moduli space of stable genus g curves [Special characters omitted.] . We study the geometry of Y, and produce a combinatorial method by which to investigate its slope, irreducible components, genus and orbifold points. Moreover, a correspondence between our method and the viewpoint of square-tiled surfaces is established. We also use our results to study the lower bound for slopes of effective divisors on [Special characters omitted.] .