On surfaces with a canonical pencil

We classify the minimal surfaces of general type with $K^2\le 4\chi -8$ whose canonical map is composed with a pencil, up to a finite number of families. More precisely we prove that there is exactly one irreducible family for each value of $\chi \gg 0$, $4\chi -10\le K^2\le 4\chi -8$. All these surfaces are complete intersections in a toric $4-$fold and bidouble covers of Hirzebruch surfaces. The surfaces with $K^2=4\chi -8$ were previously constructed by Catanese as bidouble covers of ${\text{PP}}^1\times {\text{PP}}^1$.