On the classification of irregular surfaces of general type with nonbirational bicanonical map

The present paper is devoted to the classification of irregular surfaces of general type with p(g) equal to or greater than 3 and nonbirational bicanonical map. Our main result is that, if S is such a surface and if S is minimal with no pencil of curves of genus 2, then S is the symmetric product of a curve of genus 3: and therefore p(g) = q = 3 and K-2 = 6. Furthermore we obtain some results towards the classification of minimal surfaces with p(g) = q = 3. Such surfaces have 6 equal to or less than K-2 equal to or less than 9, and we show that K-2 = 6 if and only if S is the symmetric product of a curve of genus 3. We also classify the minimal surfaces with p(g) = q = 3 with a pencil of curves of genus 2, proving in particular that for those one has K-2 = 8.