# Varieties of sums of powers and moduli spaces of (1,7)-polarized abelian surfaces

## TL;DR

This paper explores the geometry of specific varieties of sums of powers linked to the Klein quartic, leading to a description of the birational geometry of certain moduli spaces of abelian surfaces, including their unirationality.

## Contribution

It provides a new geometric understanding of the moduli space of (1,7)-polarized abelian surfaces with additional structures, establishing its unirationality.

## Key findings

- The moduli space _2(1,7)^{-}_{sym} is unirational.
- A dominant morphism from a unirational conic bundle to the moduli space is constructed.
- The study connects varieties of sums of powers with the geometry of abelian surface moduli spaces.

## Abstract

We study the geometry of some varieties of sums of powers related to the Klein quartic. This allows us to describe the birational geometry of certain moduli spaces of abelian surfaces. In particular we show that the moduli space $\mathcal{A}_2(1,7)^{-}_{sym}$ of $(1,7)$-polarized abelian surfaces with a symmetric theta structure and an odd theta characteristic is unirational by showing that it admits a dominant morphism from a unirational conic bundle.