Flenner–Zaidenberg rigidity conjecture for affine Pham–Brieskorn hypersurfaces

Let $n\geq 2$, let ${{\rm \bf k}}$ be an algebraically closed field of characteristic zero, and let $a_0,\dots,a_n$ be positive integers. The affine Pham–Brieskorn hypersurface $X_{a_0,\dots,a_n}$ is the affine hypersurface

$$X_{a_0,\dots,a_n}=\operatorname{Spec}\left({{\rm \bf k}}[X_0,\dots,X_n]/\langle X_0^{a_0}+\cdots+X_n^{a_n}\rangle\right).$$

A variety is rigid if it admits no non-trivial action of the additive group $\mathbb{G}_a$, equivalently if its coordinate ring admits no non-zero locally nilpotent derivation.

Flenner–Zaidenberg rigidity conjecture. The affine Pham–Brieskorn hypersurface $X_{a_0,\dots,a_n}$ is rigid if and only if $\min\{a_0,\dots,a_n\}\geq 2$ and at most one element $i$ of $\{0,\dots,n\}$ satisfies $a_i=2$.

This conjecture gives a proposed complete numerical characterization of rigidity for affine Pham–Brieskorn hypersurfaces. The paper proves the stated criterion for $3$-dimensional hypersurfaces under the corresponding exponent assumptions, while the general conjecture remains open.