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

In this note, the geography of minimal surfaces of general type admitting Z22$\mathbb {Z}_2^2$‐actions is studied. More precisely, it is shown that Gieseker's moduli space MK2,χ$\mathfrak {M}_{K^2,\chi }$ contains surfaces admitting a Z22$\mathbb {Z}_2^2$‐action for every admissible pair (K2,χ)$(K^2, \chi )$ such that 2χ−6≤K2≤8χ−8$2\chi -6\le K^2\le 8\chi -8$ or K2=8χ$K^2=8\chi$. The examples considered allow to prove that the locus of Gorenstein stable surfaces is not closed in the KSBA‐compactification M¯K2,χ$\overline{\mathfrak {M}}_{K^2,\chi }$ of Gieseker's moduli space MK2,χ$\mathfrak {M}_{K^2,\chi }$ for every admissible pair (K2,χ)$(K^2, \chi )$ such that 2χ−6≤K2≤8χ−8$2\chi -6\le K^2\le 8\chi -8$.