Del Pezzo surfaces and Mori fiber spaces in positive characteristic

TL;DR

This paper proves that in positive characteristic, threefolds with terminal singularities cannot have Mori fibrations with geometrically non-normal fibers, by analyzing del Pezzo surfaces over imperfect fields.

Contribution

It establishes new structural results for del Pezzo surfaces over imperfect fields, extending Reid's classification to these settings.

Findings

No Mori fibrations with geometrically non-normal fibers on threefolds in positive characteristic.

Structural classification of del Pezzo surfaces over imperfect fields.

Analysis of geometrical non-reducedness in positive characteristic.

Abstract

We settle a question that originates from results and remarks by Koll\'ar on extremal ray in the minimal model program: In positive characteristics, there are no Mori fibrations on threefolds with only terminal singularities whose generic fibers are geometrically non-normal surfaces. To show this we establish some general structure results for del Pezzo surfaces over imperfect ground fields. This relies on Reid's classification of non-normal del Pezzo surfaces over algebraically closed fields, combined with a detailed analysis of geometrical non-reducedness over imperfect fields of p-degree one.