A connected component of the moduli space of surfaces with pg=0

Let S be a minimal surface of general type with p_g(S)=0 and K_s2⩾3 for which the bicanonical map ϕ : S→PK_S2 is a morphism. Then deg ϕ⩽4 by Mendes Lopes (Arch. Math. 69 (1997) 435–440) and if it is equal to 4 then K_S2⩽6 by Mendes Lopes and Pardini (A note on surfaces of general type with p_g=0 and K2⩾7, Pisa preprint, December 1999 (Eprint: math AG/9910074)). We prove that if K_S2=6 and deg ϕ=4 then S is a Burniat surface (see Peters (Nagoya Math. J. 166 (1977) 109–119)). We show moreover that minimal surfaces with p_g=0, K2=6 and bicanonical map of degree 4 form a four-dimensional irreducible connected component of the moduli space of surfaces of general type.