Algebraic analysis of Siersma's non-isolated hypersurface singularities

Abstract

We examine isolated line singularities, transversal $A_1$-type non-isolated hypersurface singularities studied by D. Siersma, in the context of algebraic analysis. By using Poincaré-Birkhoff-Witt algebra, we explicitly compute a Gröbner basis of the annihilator $\text{Ann}_{D_X[s]}f^s$ in a non-commutative ring associated with these singularities. We compute local cohomology solutions of the associated holonomic $D$-modules by utilizing the Gröbner basis of $\text{Ann}_{D_X[s]}f^s$ and determine in particular the monodromy structure of the local cohomology solutions along a singular stratum of hypersurfaces. As a byproduct, we obtain micro-local $b$-functions of isolated line singularities in an explicit manner.