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

TL;DR

This paper proves a conjecture by Tan and Tu that fibered surfaces of general type over the projective line have at least seven singular fibers, under specific geometric conditions.

Contribution

It confirms the conjecture in particular cases where the surface arises from blow-ups of minimal surfaces with certain pencils.

Findings

Proved the conjecture for surfaces from blowing-up minimal surfaces with transversal pencils.

Established lower bounds on the number of singular fibers in these cases.

Extended understanding of fibered surfaces of general type and their singular fibers.

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.