Seminar of the Laboratory ATG "Rigidity and Rationality of Pham-Brieskorn Threefolds"

A ring B is said to be rigid if the only locally nilpotent derivation D: B -> B is the zero derivation. We prove that a 3-dimensional Pham-Brieskorn ring B_{a0, a1, a2, a3} = C[X₀,X₁,X₂,X₃] / (X₀^a₀ + X₁^a₁ + X₂^a₂ + X₃^a3) 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 (a₀, a₁, a₂, a₃) the ring B_{a0, a1, a2, a3} is rational. The content in the first part of the talk consists of joint work with Adrien Dubouloz.