Geography of minimal surfaces of general type with Z22$\mathbb {Z}_2^2$‐actions and the locus of Gorenstein stable surfaces

Abstract

In this note, the geography of minimal surfaces of general type admitting ℤ 2 2actions is studied. More precisely, it is shown that Gieseker's moduli space 𝔐 𝐾 2 ,𝜒 contains surfaces admitting a ℤ 2 2 -action for every admissible pair (𝐾 2 , 𝜒) such that 2𝜒 − 6 ≤ 𝐾 2 ≤ 8𝜒 − 8 or 𝐾 2 = 8𝜒. The examples considered allow to prove that the locus of Gorenstein stable surfaces is not closed in the KSBAcompactification 𝔐 𝐾 2 ,𝜒 of Gieseker's moduli space 𝔐 𝐾 2 ,𝜒 for every admissible pair (𝐾 2 , 𝜒) such that 2𝜒 − 6 ≤ 𝐾 2 ≤ 8𝜒 − 8.