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The classical criterion of Castelnuovo for the rationality of a complex algebraic surface S states that S is rational (i.e. birationally equivalent to IP 2) if and only if P2 =q =0. It is therefore natural to ask for the existence of non-rational surfaces with pg=q=0. Perhaps the oldest example is due to Enriques and this surface still bears his name. See Kodaira, [-8] for more details. Here we only remark that c~ is zero for this surface. Campedelli seems to be the first to give examples of non-rational surfaces with pg = q = 0, but c 2 > 0,

On two types of surfaces of general type with vanishing geometric genus