Locally nilpotent derivations of graded integral domains and cylindricity

Let B be a commutative -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 = \bigoplus B_d 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 Ga-action on the affine cone over (Y,H).