Non-rationality of the S_6-symmetric quartic threefolds

TL;DR

This paper proves that a general quartic threefold in P^4, symmetric under the S_6 group, is not rational, highlighting the non-rationality of a specific family of symmetric algebraic varieties.

Contribution

It establishes the non-rationality of a general member of the S_6-symmetric quartic threefolds family, a previously unresolved case in algebraic geometry.

Findings

A general S_6-symmetric quartic threefold is not rational.

The proof involves analyzing the symmetry and geometric properties of the family.

This result contributes to the classification of rationality in algebraic varieties.

Abstract

The quartic hypersurfaces in P^4 invariant under the standard representation of S_6 form a linear pencil. We prove that a general member of this pencil is not rational.