On surfaces with a canonical pencil

TL;DR

This paper classifies minimal surfaces of general type with specific invariants whose canonical maps are composed with a pencil, identifying unique families and their geometric constructions, extending previous work by Catanese.

Contribution

It provides a classification of such surfaces with $K^2 \,\leq\, 4\chi - 8$, describing their geometric structures and establishing the uniqueness of families for large $\\chi$.

Findings

Exactly one irreducible family for each large $\\chi$ in specified range.

All surfaces are complete intersections in toric 4-folds and bidouble covers of Hirzebruch surfaces.

Surfaces with $K^2=4\chi-8$ previously constructed by Catanese.

Abstract

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