Upper bounds on the genus of hyperelliptic Albanese fibrations

TL;DR

This work establishes a universal quadratic upper bound on the genus g of hyperelliptic Albanese fibrations for minimal irregular surfaces of general type, tying g to χ($𝒪_S$) and, more generally, to χ_f via relative invariants. The authors develop and apply Xiao’s hyperelliptic singularity indices to derive explicit quadratic estimates and prove sharpness by constructing families attaining the bound; in the special case p_g=q=1 they obtain a uniform cap g ≤ 14, with only two exceptional possibilities when K_S^2 = 8. The methods integrate slope theory, Noether-type relations for fibrations, and explicit double-cover constructions, yielding both quantitative bounds and structural restrictions. The results illuminate how hyperelliptic fibrations constrain the geometry of irregular surfaces and demonstrate that the quadratic bound is essentially optimal in families realizing large genus.

Abstract

Let $S$ be a minimal irregular surface of general type, whose Albanese map induces a hyperelliptic fibration $f:\,S \to B$ of genus $g$.We prove a quadratic upper bound on the genus $g$, i.e., $g\leq h\big(χ(\mathcal{O}_S)\big)$, where $h$ is a quadratic function. We also construct examples showing that the quadratic upper bounds can not be improved to the linear ones. In the special case when $p_g(S)=q(S)=1$, we show that $g\leq 14$.

Theorems & Definitions (46)

• Theorem 1.1
• Remark 1
• Theorem 1.2
• Lemma 1: Xiao91Xiao92
• Definition 1
• Example 1
• Lemma 2: Xiao91Xiao92
• Theorem 2.1
• Corollary 1: Xiao92lt13
• Theorem 3.1