Homogeneous locally nilpotent derivations having slices and embeddings of affine spaces

Abstract

Let m and n be positive integers such that n ⩾ m and let B be a polynomial ring in m + n + 1 variables over a field k of characteristic 0. We give a bijective correspondence between the equivalence classes of embeddings A m → A n and the equivalence classes of sequences of mutually commuting locally nilpotent derivations δ i ( 1 ⩽ i ⩽ m ) on B in some form, which are homogeneous with respect to a Z -grading on B and have slices. The intersection A of the kernels of δ i for 1 ⩽ i ⩽ m inherits the Z -grading on B. We show that A is a polynomial ring with homogeneous coordinates if and only if the corresponding embedding is rectifiable.