${\mathbb{Z}}_3$-actions on Horikawa surfaces

Abstract

Minimal algebraic surfaces of general type $X$ such that $K_X^2=2\chi ({\mathcal{O}}_X)-6$ are called Horikawa surfaces. In this note ${\mathbb{Z}}_3$-actions on Horikawa surfaces are studied. The main result states that given an admissible pair $(K^2,\chi )$ such that $K^2=2\chi -6$, all the connected components of Gieseker's moduli space ${\mathfrak{M}}_{K^2,\chi }$ contain surfaces admitting a ${\mathbb{Z}}_3$-action. On the other hand, the examples considered allow to produce normal stable surfaces that do not admit a $\mathbb{Q}$-Gorenstein smoothing. This is illustrated by constructing non-smoothable normal surfaces in the KSBA-compactification ${\overline{\mathfrak{M}}}_{K^2,\chi }$ of Gieseker's moduli space ${\mathfrak{M}}_{K^2,\chi }$ for every admissible pair $(K^2,\chi )$ such that $K^2=2\chi -5$. Furthermore, the surfaces constructed belong to connected components of ${\overline{\mathfrak{M}}}_{K^2,\chi }$ without canonical models.

Theorem 1 of the first version had to be modified because of mistakes in its proof. The second version of the paper was a provisional version where the mistakes were removed. The third version of the paper does not contain the mistakes and includes a stronger version both of Theorem 1 and Theorem 2. 12 pages