A connected component of the moduli space of surfaces of general type with $p_g=0$

Abstract

Let S be a minimal surface of general type with $p_g(S)=0$ and such that the bicanonical map $ϕ:S\to \pp^{K^2_S}$ is a morphism: then the degree of $\varphi$ is at most 4 and if it is equal to 4 then $K_S^2\le 6$. Here we prove that if $K_S^2=6$ and $°\varphi =4$ then S is a so-called {\em Burniat surface}. In addition we show that minimal surfaces with $p_g=0$, $K^2=6$ and bicanonical map of degree 4 form a 4-dimensional irreducible connected component of the moduli space of surfaces of general type.

LaTeX 2.09, 20 pages