# Variation of stable birational types in positive characteristic

## TL;DR

This paper investigates how stable birational types vary among Fano hypersurfaces in positive characteristic, extending previous characteristic-zero results and showing that certain varieties are not stably birational to very general hypersurfaces.

## Contribution

It extends the study of stable birational types to positive characteristic, demonstrating that varieties without a diagonal decomposition are not stably birational to very general hypersurfaces.

## Key findings

- Varieties without diagonal decomposition are not stably birational to very general hypersurfaces.
- The method extends Shinder's characteristic-zero approach to arbitrary characteristic.
- Provides new insights into the birational geometry of Fano hypersurfaces in positive characteristic.

## Abstract

Let k be an uncountable algebraically closed field and let Y be a smooth projective k-variety which does not admit a decomposition of the diagonal. We prove that Y is not stably birational to a very general hypersurface of any given degree and dimension. We use this to study the variation of the stable birational types of Fano hypersurfaces over fields of arbitrary characteristic. This had been initiated by Shinder, whose method works in characteristic zero.