The Rigid Pham-Brieskorn Threefolds

TL;DR

This work proves that a 3-dimensional Pham-Brieskorn hypersurface $X_{a_0,a_1,a_2,a_3}$ with $\min\{a_i\}\ge2$ and at most one $a_i=2$ is rigid, i.e., admits no nontrivial $\mathbb{G}_a$-action. The authors reduce the Main Conjecture to well-formed cases via the cotype-0 condition and to the non-existence of anti-canonical polar cylinders, then proceed by dimension induction. They solve eight simplifying cases and treat the two remaining families, $B_{2,3,5,30}$ and $B_{2,3,4,12}$, using detailed del Pezzo surface geometry, Demazure’s construction, and polar-cylinder obstructions to prove rigidity. The paper also discusses extensions to higher dimensions, clarifying reductions to well-formed hypersurfaces and the role of polar cylinders in singular Fano varieties, and provides a self-contained treatment for the $n=3$ case. Overall, it closes a long-standing gap in the dimension-3 case of the rigidity conjecture for Pham-Brieskorn hypersurfaces while outlining the path for future higher-dimensional investigations.

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 $\min\{a_0, a_1, a_2, a_3 \} \geq 2$ 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 $\mathbb{G}_a$.

Theorems & Definitions (78)

• Corollary
• Theorem 1.1.3
• Proposition 1.1.5
• Definition 1.2.2
• Theorem 1.3.4: Nakai-Moishezon
• Theorem 1.4.4
• Example 1.5.2
• Definition 1.5.3
• Lemma 1.5.4
• Definition 1.6.1