A bound on embedding dimensions of geometric generic fibers

TL;DR

This paper establishes bounds on the embedding dimensions of geometric generic fibers in positive characteristic, impacting the minimal model program by excluding certain fiber structures.

Contribution

It introduces a new bound on embedding dimensions of geometric generic fibers in positive characteristic and applies it to the minimal model program.

Findings

Bound on embedding dimensions of geometric generic fibers.

Exclusion of certain non-normal del Pezzo surface fibrations.

Application to the minimal model program in positive characteristic.

Abstract

We limit the singularities that arise in geometric generic fibers of morphisms between smooth varieties of positive characteristic by studying changes in embedding dimension under inseparable field extensions. We then use this result in the context of the minimal model program to rule out the existence of smooth varieties fibered by certain non-normal del Pezzo surfaces over bases of small dimension.