Connectedness of Prym Eigenform Loci in Genus 5

Abstract

The moduli space of holomorphic differentials on curves of genus g admits a natural action of the group \(G{{L}_{2}}(\mathbb{R})\). The study of orbits of this action and their closures has attracted the interest of a wide range of researchers in the last few decades. In the 2000s, C. McMullen described an infinite family of orbifolds that are closures of such orbits in the space of holomorphic differentials on curves of genus 2. In spaces of holomorphic differentials on curves of higher genera, well-known examples of orbifolds that are unions of \(G{{L}_{2}}(\mathbb{R})\)-orbit closures are Prym eigenform loci. They are nonempty for surfaces of genus at most 5. This paper presents the first nontrivial calculations of the number of connected components in Prym eigenform loci for surfaces of maximum possible genus.