Locally nilpotent derivations of graded integral domains and cylindricity

TL;DR

This paper explores the connection between cylindrical elements and locally nilpotent derivations in graded integral domains, providing conditions for derivation extension and generalizing results on cylindricity and G_a-actions.

Contribution

It establishes new links between cylindrical elements and locally nilpotent derivations, and extends previous results on cylindricity of polarized projective varieties.

Findings

Characterization of cylindrical elements in graded domains.

Conditions for extending derivations from subrings to the entire domain.

Generalization of results relating cylindricity to G_a-actions.

Abstract

Let B be a commutative $\mathbb{Z}$-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 > 0, we give sufficient conditions that guarantee that every derivation of $B^{(d)} = \oplus_i B_{di}$ 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).