Structure of geometrically non-reduced varieties

TL;DR

This paper investigates the structure of geometrically non-reduced varieties, providing new insights and applications to Fano varieties, especially in relation to Mori fibre spaces in positive characteristic.

Contribution

It establishes a structural theorem for geometrically non-reduced varieties and links their non-reducedness to the base of fibrations in certain Mori fibre spaces.

Findings

Non-reducedness of generic fibres originates from the base in characteristic p > 2n+1.

Structural characterization of geometrically non-reduced varieties.

Applications to the geometry of Fano varieties.

Abstract

We prove a structural result for geometrically non-reduced varieties and give applications to Fano varieties. For example, we show that if $X$ is the generic fibre of a Mori fibre space of relative dimension $n$, and the characteristic is $p>2n+1$, then any geometric non-reducedness of $X$ comes from the base of some fibration.