Geometric aspects on Humbert-Edge curves of type 5, Kummer surfaces and hyperelliptic curves of genus 2

Abstract In this work, we study the Humbert-Edge curves of type 5, defined as a complete intersection of four diagonal quadrics in ${\mathbb{P}}^5$ . We characterize them using Kummer surfaces, and using the geometry of these surfaces, we construct some vanishing thetanulls on such curves. In addition, we describe an argument to give an isomorphism between the moduli space of Humbert-Edge curves of type 5 and the moduli space of hyperelliptic curves of genus 2, and we show how this argument can be generalized to state an isomorphism between the moduli space of hyperelliptic curves of genus $g=\frac{n-1}{2}$ and the moduli space of Humbert-Edge curves of type $n\geq 5$ where $n$ is an odd number.