New invariants of singularities in terms of higher Nash blow-up local algebras

Let $(V,0)$ be an isolated hypersurface singularity. In our previous work, we introduced a series of new local algebras called higher Nash blow-up local algebras associated with $(V,0)$. Thus many new invariants were introduced from these local algebras of $(V,0)$. We conjectured that singularities can be distinguished by a finite subset of these invariants. Furthermore, we proposed a generalized Halperin Conjecture. In this paper, we determine these invariants for simple curve singularities. As a result, we verify our conjecture for simple curve singularities. In the proof, we concretely compute the new invariants of simple curve singularities. Moreover, we verify the generalized Halperin Conjecture in some new cases.