The moduli space of Keum-Naie surfaces

TL;DR

This paper characterizes the moduli space of Keum-Naie surfaces, proving a homotopy equivalence implies isomorphism and describing the moduli space's geometric properties.

Contribution

It introduces a new description of Keum-Naie surfaces and their fundamental group, establishing the irreducibility and geometric structure of their moduli space.

Findings

Homotopy equivalence implies isomorphism for Keum-Naie surfaces

The moduli space component is irreducible, normal, and unirational

Dimension of the moduli space component is 6

Abstract

Using a new description of Keum Naie surfaces and their fundamental group, we prove the following main result: Let S be a smooth complex projective surface which is homotopically equivalent to a Keum - Naie surface. Then S is a Keum - Naie surface. The connected component of the Gieseker moduli space corresponding to Keum - Naie surfaces is irreducible, normal, unirational of dimension 6.