Albanese fibrations of surfaces with low slope

TL;DR

This paper establishes a sharp linear upper bound on the genus of Albanese fibrations for certain surfaces with low slope, characterizes cases reaching this bound, and constructs examples with quadratic growth in fiber genus.

Contribution

It provides a new linear upper bound on the genus of Albanese fibrations for surfaces with $K_S^2 \\leq 4\\chi(\\mathcal{O}_S)$ and characterizes the extremal cases, along with constructing sequences exceeding this bound.

Findings

Linear upper bound on genus g for surfaces with $K_S^2 \\leq 4\\chi(\\mathcal{O}_S)$

Examples showing the bound is sharp

Construction of surfaces with quadratic growth in fiber genus

Abstract

Let $S$ be a minimal irregular surface of general type, whose Albanese map induces a fibration $f:\,S \to C$ of genus $g$.We prove a linear upper bound on the genus $g$ if $K_S^2\leq 4\chi(\mathcal{O}_S)$. Examples are constructed showing that the above linear upper bound is sharp. We also give a characterization of the Albanese fibrations reaching the above upper bound when $\chi(\mathcal{O}_S)\geq 5$.On the other hand, we will construct a sequence of surfaces $S_n$ of general type with $K_{S_n}^2/\chi(\mathcal{O}_{S_n})>4$ and with an Albanese fibration $f_n$, such that the genus $g_n$ of a general fiber of $f_n$ increases quadratically with $\chi(\mathcal{O}_{S_n})$,and that $K_{S_n}^2/\chi(\mathcal{O}_{S_n})$ can be arbitrarily close to $4$.