Covers of Elliptic Curves and the Lower Bound for Slopes of Effective Divisors on ${\bar{\mathcal{M}}}_g$

Abstract

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 ${\bar{\mathcal{M}}}_g$. We study the geometry of $Y$, and produce a combinatorial method by which to investigate its slope, irreducible components, genus and orbifold points. As a by-product of our approach, we find some equalities from classical number theory. 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 ${\bar{\mathcal{M}}}_g$.

41 pages, 19 figures