It is well-known that the versal deformations of nonsimple singularities depend on moduli. The first step in deeper understanding of this phenomenon is to determine the versal discriminant, which roughly speaking is an obstacle for analytic triviality of an unfolding or deformation along the moduli. The goal of this paper is to describe the versal discriminant of $Z_{k,0}$ and $Q_{k,0}$ singularities basing on the fact that the deformations of these singularities may be obtained as blowing ups of certain deformations...

Let $M^n$ be a germ at $0\in {ℂ}^m$ of an irreducible analytic set of dimension $n$, where $n\ge 2$ and $0$ is a singular point of $M$. We study the question: when does there exist a germ of holomorphic map $\phi :({ℂ}^n,0)\to (M,0)$ such that ${\phi }^{-1}\left(0\right)=\left\{0\right\}$ ? We prove essentialy three results. In Theorem 1 we consider the case where $M$ is a quasi-homogeneous complete intersection of $k$ polynomials $F=(F_1,...,F_k)$, that is there exists a linear holomorphic vector field $X$ on ${ℂ}^m$, with eigenvalues ${\lambda }_1,...,{\lambda }_m\in {\mathbb{Q}}_{+}$ such that $X\left(F^T\right)=U·F^T$, where $U$ is a $k\times k$ matrix with entries in ${𝒪}_m$. We prove that if there exists...

Given a germ $h$ of holomorphic function on $({ℂ}^n,0)$, we study the condition: “the ideal ${\text{Ann}}_{𝒟}1/h$ is generated by operators of order1”. We obtain here full characterizations in the particular cases of Koszul-free germs and unreduced germs of plane curves. Moreover, we prove that this condition holds for a special type of hyperplane arrangements. These results allow us to link this condition to the comparison of de Rham complexes associated with $h$.

It is well known that the versal deformations of nonsimple singularities depend on moduli. The first step in deeper understanding of this phenomenon is to determine the versal discriminant, which roughly speaking is an obstacle to analytic triviality of an unfolding or deformation along the moduli. The versal discriminant of the Pham singularity ($J_{3,0}$ in Arnold’s classification) was thoroughly investigated by J. Damon and A. Galligo [2], [3], [4]. The goal of this paper is to continue their work and...