Rigidity and Rationality of Pham-Brieskorn Threefolds

A ring $B$ is said to be rigid if the only locally nilpotent derivation $D: B \rightarrow B$ is the zero derivation. We prove that a $3$-dimensional Pham-Brieskorn ring $B_{a_0, a_1, a_2, a_3} = C[X_0, X_1, X_2, X_3] / ( X_0^{a_0} + X_1^{a_1} + X_2^{a_2} + X_3^{a_3} )$ is not rigid if and only if $a_i = 1$ for some $i$ or $ a_i = a_j = 2$ for distinct $i$ and $j$. We also determine for which $(a_0, a_1, a_2, a_3)$ the ring $B_{a_0, a_1, a_2, a_3} $ is rational. The content in the first part of the talk consists of joint work with Adrien Dubouloz.