Geometry of genus 8 Nikulin surfaces and rationality of their moduli

TL;DR

This paper provides an explicit geometric construction of genus 8 Nikulin surfaces using Grassmannians and rational normal curves, and proves the rationality of their moduli space.

Contribution

It introduces a new geometric construction of genus 8 Nikulin surfaces and establishes the rationality of their moduli space.

Findings

Explicit construction of Nikulin surfaces via Grassmannian sections

Identification of a special family of sextic curves with bisecant lines

Proof of the rationality of the moduli space of genus 8 Nikulin surfaces

Abstract

Let S be a general complex Nikulin surface of genus 8, a geometric construction of S is given as follows. Consider a smooth 3-fold linear section T of the Grassmannian G(1,4) and the Hilbert scheme of rational normal sextic curves of T. In it consider the special family of sextics A which are also contained in the congruence of bisecant lines to a rational normal quartic curve of P^4. We show that S is biregular to a quadratic section of T containing a sextic A. In particular A admits a 1-dimensional family of bisecant lines contained in G(1,4) and 8 of them are in S. This explicit construction is then used to prove that the moduli space of genus 8 Nikulin surfaces is rational.