Surfaces with pg=q=3

Abstract

 In this paper it is proved that a complete algebraic surface of general type with p _g =q=3, without irrational pencil of genus bigger than one is birationally equivalent to the two-symmetric product of a curve of genus 3. This result completes the classification of the surfaces with p _g =q=3. The main tools are the Lefschetz theorem and the use of the paracanonical system on the surface.