Locally Nilpotent Derivations of Graded Integral Domains and Cylindricity

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