Burniat surfaces II: secondary Burniat surfaces form three connected components of the moduli space

TL;DR

This paper proves that four families of Burniat surfaces with specific invariants are each connected components of the moduli space, and establishes their rationality, revealing new phenomena in the structure of these moduli spaces.

Contribution

It demonstrates that each Burniat surface family forms a distinct connected component of the moduli space and proves their rationality, also uncovering non-reduced structures in the moduli spaces.

Findings

Each Burniat surface family is a connected component of the moduli space.

The moduli components are rational.

Non-reduced structures occur in the moduli spaces, especially in the nodal case.

Abstract

We prove in one go that each of the 4 families of Burniat surfaces with K^2 = 6,5,4, is a connected component of the moduli space of surfaces of general type. We prove also the rationality of each component. In the nodal case (one of the two families for K^2_S = 4) a very surprising and new phenomenon occurs. Both the moduli space for the minimal models and the Gieseker moduli space for canonical models are everywhere non reduced. But the nilpotence order is higher for the first.