Moduli of non-standard Nikulin surfaces in low genus

TL;DR

This paper investigates the moduli space of non-standard Nikulin surfaces in low genus, revealing its rationality in specific cases and analyzing the family of such surfaces containing general Prym curves.

Contribution

It studies the birational geometry of non-standard Nikulin surfaces, proving rationality for genus 7 and 11, and constructing a rational double cover for genus 9, advancing understanding of their moduli.

Findings

Rationality of the moduli space for genus 7 and 11.

Existence of a rational double cover for genus 9.

Determination of the dimension of families containing general Prym curves.

Abstract

Primitively polarized genus $g$ Nikulin surfaces $(S,M,H)$ are of two types, that we call standard and non-standard depending on whether the lattice embedding $\mathbb{Z}[H] \oplus_{\perp} \mathbf{N} \subset \rm{Pic}(S)$ is primitive. Here $H$ is the genus $g$ polarization and $\mathbf{N}$ is the Nikulin lattice. We concentrate on the non-standard case, which only occurs in odd genus. In particular, we study the birational geometry of the moduli space of non-standard Nikulin surfaces of genus $g$ and prove its rationality for $g=7,11$ and the existence of a rational double cover of it when $g=9$. Furthermore, if $(S,M,H)$ is general in the above moduli space and $(C,M|_C)$ is a general Prym curve in $|H|$, we determine the dimension of the family of non-standard Nikulin surfaces of genus $g$ containing $(C, M|_C)$ for $3\leq g\leq 11$; this completes the study of the Prym-Nikulin map initiated in our previous work.