On one stable birational invariant

TL;DR

This paper explores how fibrations of projective space by Abelian hypersurfaces serve as a stable birational invariant, potentially obstructing the stable rationality of algebraic varieties, with supporting evidence and corollaries.

Contribution

It introduces a new perspective on stable birational invariants via fibrations by Abelian hypersurfaces on projective spaces.

Findings

Fibrations by Abelian hypersurfaces can obstruct stable rationality.

Proposes a conjecture linking these fibrations to stable birational invariants.

Provides evidence and corollaries supporting this conjecture.

Abstract

This is an expository article, which contributes to the Proceedings of the conference "Groups of Automorphisms in Birational and Affine Geometry", held in Trento in 2012. We propose that (rational) fibrations on the projective space $\p^n$ by (birationally) Abelian hypersurfaces, for an arbitrary $n\ge 2$, provide an obstruction to stable rationality of algebraic varieties. We discuss the evidence for this proposition and derive some (almost straightforward) corollaries from it.