Numerical properties of isotrivial fibrations

TL;DR

This paper explores the numerical characteristics of isotrivial fibrations on surfaces, establishing sharp inequalities relating the canonical divisor squared and the Euler characteristic, with implications for minimal surfaces of general type.

Contribution

It proves new sharp bounds on the invariants of isotrivial fibrations, extending previous results and characterizing cases of equality with specific surface singularities.

Findings

Established the inequality K_X^2 8 \u03a7(1_X)-2 for certain surfaces.

Proved the inequality K_X^2 8 8 1_X-5 under ampleness of K_X.

Characterized the surfaces attaining equality as minimal surfaces with two ordinary double points.

Abstract

In this paper we investigate the numerical properties of relatively minimal isotrivial fibrations $\varphi \colon X \lr C$, where $X$ is a smooth, projective surface and $C$ is a curve. In particular we prove that, if $g(C) \geq 1$ and $X$ is neither ruled nor isomorphic to a quasi-bundle, then $K_X^2 \leq 8 \chi(\mO_X)-2$; this inequality is sharp and if equality holds then $X$ is a minimal surface of general type whose canonical model has precisely two ordinary double points as singularities. Under the further assumption that $K_X$ is ample, we obtain $K_X^2 \leq 8 \chi(\mO_X)-5$ and the inequality is also sharp. This improves previous results of Serrano and Tan.