arXiv:math/9910074v3 [math.AG] 18 Sep 2000 The bicanonical map of surfaces with p g = 0 and K 2 ≥ 7 ∗ Margarida Mendes Lopes Rita Pardini Abstract A minimal surface of general type with p g (S ) = 0 satisfies 1 ≤ K 2 ≤ 9 and it is known that the image of the bicanonical map ϕ is a surface for K 2 S ≥ 2, whilst for K 2 S ≥ 5, the bicanonical map is always a morphism. In this paper it is shown that ϕ is birational if K 2 S =9 and that the degree of ϕ is at most 2 if K 2 S = 7 or K 2 S = 8. By presenting two examples of surfaces S with K 2 S = 7 and 8 and bicanonical map of degree 2, it is also shown that this result is sharp. The example with K 2 S = 8 is, to our knowledge, a new example of a surface of general type with p g = 0. The degree of ϕ is also calculated for two other known surfaces of general type with p g = 0, K 2 S = 8. In both cases the bicanonical map turns out to be birational. 1 Introduction Many examples of complex surfaces of general type with p g = q = 0 are known, but a detailed classification is still lacking, despite much progress in the theory of algebraic surfaces. Surfaces of general type are often studied using properties of their canonical curves. If a surface has p g = 0, then there are of course no such curves, and it seems natural to look instead at the bicanonical system, which is not empty. Minimal surfaces S of general type with p g (S ) = 0 satisfy 1 ≤ K 2 S ≤ 9. By a result of Xiao Gang [15], the image of the bicanonical map is a surface * 2000 Mathematics Subject Classification: 14J29, 14E05 1