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 $p_{g}\geq 3$ 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 $6\leq K^{2}\leq 9$, 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 $K^{2}=8$.