Kernel of locally nilpotent 𝑅-derivations of 𝑅[𝑋,𝑌

Abstract

In this paper we study the kernel of a non-zero locally nilpotent R R -derivation of the polynomial ring R [ X , Y ] R[X,Y] over a noetherian integral domain R R containing a field of characteristic zero. We show that if R R is normal then the kernel has a graded R R -algebra structure isomorphic to the symbolic Rees algebra of an unmixed ideal of height one in R R , and, conversely, the symbolic Rees algebra of any unmixed height one ideal in R R can be embedded in R [ X , Y ] R[X,Y] as the kernel of a locally nilpotent R R -derivation of R [ X , Y ] R[X,Y] . We also give a necessary and sufficient criterion for the kernel to be a polynomial ring in general.