Rationality of an $S_6$-invariant quartic $3$-fold

TL;DR

This paper investigates the rationality of certain symmetric quartic threefolds in projective four-space, completing the classification for hypersurfaces invariant under the symmetric group S_6.

Contribution

It provides a complete analysis of the rationality problem for S_6-invariant quartic threefolds, a previously unresolved case.

Findings

Classified the rationality of S_6-invariant quartic threefolds

Identified conditions under which these hypersurfaces are rational or non-rational

Extended understanding of symmetry and rationality in algebraic geometry

Abstract

We complete the study of rationality problem for hypersurfaces $X_t\subset \mathbb{P}^4$ of degree $4$ invariant under the action of the symmetric group $S_6$.