Standard stable Horikawa surfaces

Abstract

We consider the stable compactification $\bar{\mathfrak{H}}$ of the moduli space of Horikawa surfaces with $K_X^2=2p_g(X)-4$. When $K_X^2=8\ell$ we show that the closures of the two components ${\mathfrak{H}}^{\mathrm{I}}$ and ${\mathfrak{H}}^{\mathrm{II}}$ of the Gieseker moduli space intersect, for $\ell >2$ in a divisor parametrising explicitly described semi-smooth surfaces. With growing $K_X^2$ we find an increasing number of generically non-reduced irreducible components in the same connected component of the moduli space of stable surfaces.