The classification of surfaces of general type with nonbirational bicanonical map

We classify surfaces of general type whose bicanonical map factors through a rational map of degree 2 to a rational or ruled surface. We prove that if such a surface has no pencil of genus 2 curves, then it is the smooth minimal model of a double plane branched along a reduced curve with certain singularities, a configuration already suggested by Du Val in the 1950s. This result allows us to complete the classification of surfaces of general type presenting the nonstandard case, with the possible exception of those with p g = q ≤ 1 p_g=q\le 1 .