Tropical degenerations and stable rationality

TL;DR

This paper introduces tropical degeneration techniques to demonstrate that certain complex hypersurfaces and complete intersections are stably irrational, expanding the understanding of stable rationality in algebraic geometry.

Contribution

It combines motivic obstructions with tropical degeneration methods to identify new classes of stably irrational hypersurfaces and complete intersections.

Findings

Very general quartic fivefolds are stably irrational.

Complete intersections of a quadric and a cubic in P^6 are stably irrational.

Tropical degeneration techniques are effective in studying stable rationality.

Abstract

We use the motivic obstruction to stable rationality introduced by Shinder and the first-named author to establish several new classes of stably irrational hypersurfaces and complete intersections. In particular, we show that very general quartic fivefolds and complete intersections of a quadric and a cubic in $\mathbb P^6$ are stably irrational. An important new ingredient is the use of tropical degeneration techniques.