The Rigid Pham-Brieskorn Threefolds

Abstract

We show that a $3$-dimensional Pham-Brieskorn hypersurface $\{X_0^{a_0}+X_1^{a_1}+X_2^{a_2}+X_3^{a_3}=0\}$ in ${\mathbb{A}}^4$ such that $\min \{a_0,a_1,a_2,a_3\}\ge 2$ and at most one element $i$ of $\{0,1,2,3\}$ satisfies $a_i=2$ does not admit a non-trivial action of the additive group ${\mathbb{G}}_a$.

Significantly shortened proofs of the (2,3,4,12) and (2,3,5,30) cases. Added proof of positive amplitude cases in all dimensions