Geometry of 3-dimensional del Pezzo fibrations in positive characteristic

TL;DR

This paper surveys recent advances in understanding the geometry of 3-dimensional del Pezzo fibrations over imperfect fields in positive characteristic, especially focusing on cases with non-smooth general fibers.

Contribution

It compiles and discusses recent results on del Pezzo surfaces and fibrations in positive characteristic, highlighting cases with non-smooth fibers and their geometric implications.

Findings

Insights into the structure of del Pezzo surfaces over imperfect fields

Analysis of non-smooth general fibers in 3D fibrations

Applications to the classification of algebraic varieties in positive characteristic

Abstract

We survey some of the recent works on the geometry of del Pezzo surfaces over imperfect fields, with applications to 3-dimensional del Pezzo fibrations in positive characteristic. We place particular emphasis on cases where the general fibres of the fibrations are not smooth