On surfaces with a canonical pencil

Abstract

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 \({\chi \gg 0,\,4\chi-10 \leq K^2 \leq 4\chi-8}\). All these surfaces are complete intersections in a toric 4-fold and bidouble covers of Hirzebruch surfaces. The surfaces with K 2 = 4χ − 8 were previously constructed by Catanese as bidouble covers of \({\mathbb{P}^1 \times \mathbb{P}^1}\).