# Variation of Stable Birational Types of Hypersurfaces

## TL;DR

This paper investigates how stable birational types of hypersurfaces vary in smooth proper families, showing that high-degree Fano hypersurfaces are generally not stably birational to each other, with supporting results from Voisin.

## Contribution

It extends the understanding of stable birational types in hypersurfaces, demonstrating their variation in families and providing new examples of non-stably birational hypersurfaces.

## Key findings

- High-degree Fano hypersurfaces are generally not stably birational.
- Stable birational types can vary within smooth proper families.
- Voisin's appendix confirms similar results using Chow decomposition.

## Abstract

We introduce and study the question how can stable birational types vary in a smooth proper family. Our starting point is the specialization for stable birational types of Nicaise and the author and our emphasis is on stable birational types of hypersurfaces. Building up on the work of Totaro and Schreieder on stable irrationality of hypersurfaces of high degree, we show that smooth Fano hypersurfaces of large degree over a field of characteristic zero are in general not stably birational to each other. In the appendix Claire Voisin proves a similar result in a different setting using the Chow decomposition of diagonal and unramified cohomology.