Prym varieties and moduli of polarized Nikulin surfaces

TL;DR

This paper provides a detailed description of the moduli space of Prym curves of genus 7, demonstrating its structure as a projective bundle, and explores the implications for related Nikulin surface moduli spaces, including their unirationality and rationality.

Contribution

It introduces a structure theorem for the moduli space R_7 of Prym curves of genus 7 as a projective bundle, establishing new unirationality and rationality results for associated Nikulin surfaces.

Findings

R_7 is a projective bundle over 7-nodal rational curves

Unirationality of R_7 and Nikulin surfaces of genus 7

R_8 is uniruled

Abstract

We present a structure theorem for the moduli space R_7 of Prym curves of genus 7 as a projective bundle over the moduli space of 7-nodal rational curves. The existence of this parametrization implies the unirationality of R_7 and that of the moduli space of Nikulin surfaces of genus 7, as well as the rationality of the moduli space of Nikulin surfaces of genus 7 with a distinguished line. Using the results in genus 7, we then establish that R_8 is uniruled.