Higher Nash blow-up local algebras of singularities and its derivation Lie algebras

Abstract

In this paper, we introduce new invariants to a singularity (V,0), i.e., the derivation Lie algebras Lk(V) of the higher Nash blow-up local algebra Mk(V). A new conjecture about the non-existence of negative weighted derivations of Lk(V) for weighted homogeneous isolated hypersurface singularities is proposed. We verify this conjecture partially. Moreover, we compute the Lie algebra L2(V) for binomial isolated singularities. We also formulate a sharp upper estimate conjecture for the dimension of Lk(V) for weighted homogeneous isolated hypersurface singularities and verify this conjecture for a large class of singularities.