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