Relative linear extensions of sextic del Pezzo fibrations

TL;DR

This paper investigates sextic del Pezzo fibrations over curves, deriving invariants, establishing an embedding theorem, and classifying singular fibers, including non-normal ones, thus advancing understanding of their geometric structure.

Contribution

It introduces an embedding theorem showing these fibrations as relative linear sections of Mori fiber spaces, and classifies their singular fibers, addressing a question by T. Fujita.

Findings

Derived formulas for invariants of sextic del Pezzo fibrations.

Proved that such fibrations are relative linear sections of specific Mori fiber spaces.

Classified singular fibers, including non-normal fibers.

Abstract

In this paper, we study a sextic del Pezzo fibration over a curve comprehensively. We obtain certain formulae of several basic invariants of such a fibration. We also establish the embedding theorem of such a fibration which asserts that every such a fibration is a relative linear section of a Mori fiber space with the general fiber $(\mathbb{P}^{1})^{3}$ and that with the general fiber $(\mathbb{P}^{2})^{2}$. As an application of this embedding theorem, we classify singular fibers of such a fibrations and answer a question of T. Fujita about existence of non-normal fibers.