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

TL;DR

This paper explores the geometric properties of Humbert-Edge's curves of type 5, relating them to Kummer surfaces and hyperelliptic curves of genus 2, and establishes moduli space isomorphisms.

Contribution

It characterizes Humbert-Edge's curves of type 5 via Kummer surfaces and constructs isomorphisms between their moduli spaces and those of hyperelliptic curves.

Findings

Construction of vanishing thetanulls on these curves

Isomorphism between moduli spaces of Humbert-Edge's curves and hyperelliptic curves of genus 2

Generalization of the isomorphism to higher genus hyperelliptic curves

Abstract

In this work we study the Humbert-Edge's 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's curves of type 5 and the moduli space of hyperelliptic curves of genus 2, and we let see 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's curves of type $n\geq 5$ where $n$ is an odd number.