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Fix a field k of characteristic zero. If a 1 , . . . , a n (n ≥ 3) are positive integers, the integral domain B a1,...,an = k[X 1 , . . . , X n ]/ X a1 1 + · · · + X an n is called a Pham-Brieskorn ring. It is conjectured that if a i ≥ 2 for all i and a i = 2 for at most one i, then B a1,...,an is rigid. (A ring B is said to be rigid if the only locally nilpotent derivation D : B → B is the zero derivation.) We give partial results towards the conjecture.

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Let B be a commutative Z-graded domain of characteristic zero. An element f of B is said to be cylindrical if it is nonzero, homogeneous of nonzero degree, and such that B (f ) is a polynomial ring in one variable over a subring. We study the relation between the existence of a cylindrical element of B and the existence of a nonzero locally nilpotent derivation of B. Also, given d ≥ 1, we give sufficient conditions that guarantee that every derivation of B (d) = i∈Z B di can be extended to a derivation of B. We generalize some results of Kishimoto, Prokhorov and Zaidenberg that relate the cylindricity of a polarized projective variety (Y, H) to the existence of a nontrivial G a -action on the affine cone over (Y, H).