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 ≥ 3 p_{g}\geq 3 and nonbirational bicanonical map. Our main result is that, if S S is such a surface and if S S is minimal with no pencil of curves of genus 2 2 , then S S is the symmetric product of a curve of genus 3 3 , and therefore p g = q = 3 p_{g}=q=3 and K 2 = 6 K^{2}=6 . Furthermore we obtain some results towards the classification of minimal surfaces with p g = q = 3 p_{g}=q=3 . Such surfaces have 6 ≤ K 2 ≤ 9 6\leq K^{2}\leq 9 , and we show that K 2 = 6 K^{2}=6 if and only if S S is the symmetric product of a curve of genus 3 3 . We also classify the minimal surfaces with p g = q = 3 p_{g}=q=3 with a pencil of curves of genus 2 2 , proving in particular that for those one has K 2 = 8 K^{2}=8 .