On the embedded versions of degree-2 tropical covers of an elliptic curve

Abstract

We study an equivalent problem to the Hurwitz existence problem in the context of tropical algebraic geometry. For this, we introduced the idea of an algebraic realization of a tropical cover c and homeomorphically faithfulness of the embeddings. In this study, our approach is constructive and we constructed algebraic realizations which are homeomorphically faithful for an arbitrary tropical cover of degree 2 and genus 2 of an abstract elliptic curve. On the basis of length conditions, we divide this into two cases: when the lengths in a tropical cover are equal and when the lengths are unequal. To achieve these results, we also progressed in unfolding and generalized a existing technique to unfold a cycle of certain length under certain conditions.