On surfaces with a canonical pencil

We classify the minimal surfaces of general type with K ² ≤ 4χ ? 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 ${chi gg 0,,4chi-10 leq K^2 leq 4chi-8}$ . All these surfaces are complete intersections in a toric 4-fold and bidouble covers of Hirzebruch surfaces. The surfaces with K ² = 4χ ? 8 were previously constructed by Catanese as bidouble covers of ${mathbb{P}^1 times mathbb{P}^1}$ .