Complete Families of Generic Covers of Elliptic Curves

TL;DR

The paper discusses a conjecture about the existence of complete families of covers of elliptic curves, which could impact the understanding of the moduli space of curves and its slope bounds.

Contribution

It introduces a conjecture on the existence of complete 1-dimensional families of elliptic curve covers and explores its implications.

Findings

Conjecture relates to the existence of certain families of covers.

Implication of the conjecture could establish a slope lower bound of 5.

Failure of the conjecture presents an interesting phenomenon to investigate.

Abstract

We introduce and motivate a conjecture about the existence of complete, 1-dimensional families of covers of an elliptic curve. If the conjecture holds, then it would imply a uniform lower bound of 5 for slope of the moduli space of curves. If the conjecture fails, this would itself be an interesting phenomenon worthy of explanation.