Toward a conjecture of Tan and Tu on fibered general type surfaces

TL;DR

The paper addresses the Tan–Tu conjecture, which posits that for a semistable non-isotrivial fibration $f:X\to \mathbb{P}^1$ with $X$ of general type, the number of singular fibers satisfies $s\ge 7$. It proves the conjecture in two concrete settings: (i) fibrations obtained from blowing up the base locus of a transversal pencil on an exceptional minimal surface $S$ (with $(K_S^2,p_g)=(2,3)$ or $(1,2)$), and (ii) fibrations from a transversal adjoint pencil, under appropriate $K_S^2$ bounds. The method combines Tan's inequality with detailed analysis of canonical or bi-canonical maps (e.g., $\phi_{K_S}$ and $\phi_{2K_S}$) and degree considerations on the images in projective planes to exclude configurations with too few singular fibers. These results demonstrate a viable strategy to extend Tan–Tu to broader classes by exploiting base-locus geometry and adjoint linear systems.

Abstract

Given a semistable non-isotrivial fibered surface $f:X\to \mathbb{P}^1$ it was conjectured by Tan and Tu that if $X$ is of general type, then $f$ admits at least $7$ singular fibers. In this paper we prove this conjecture in several particular cases, i.e. assuming $f$ is obtained from blowing-up the base locus of a transversal pencil on an exceptional minimal surface $S$ or assuming that $f$ is obtained as the blow-up of the base locus of a transversal and adjoint pencil on a minimal surface.

Theorems & Definitions (18)

• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Proposition 3.3
• proof
• Theorem 4.1
• proof
• Theorem 4.2
• proof