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If I understand correctly, you ask what can be the results of the collision of two singular points, of $A_4$ and $A_1$ types. In general there does not seem to exist an ultimate effective method to tr …

Not clear why do you want only the hyperplane sections (and not all the smooth hypersurface sections of the initial singularity). For example, the plane curve singularity $\{y^2=x^k\}$, $k>2$ give th …

Let $(X,0)\subset(\mathbb{C}^N,0)$ be the (formal) germ of a singular space (isolated singularity). Let $\mathbb{P}T_{(X,0)}\subset\mathbb{P}^{N-1}$ be its projectivized tangent cone (considered as a …

Well, for the hypersurface $X_d\subset\Bbb{P}^n$ the "most degenerate" isolated singularity is of the type: $\{x^d_1+\cdots+x^d_n=0\}$. Thus, $\mu_{max}=(d-1)^n$. Is this what was meant?

(Perhaps a not very well defined question) Let $(S_t)_t$ be a (flat) family of compact complex surfaces. Assume the generic member is smooth while $S_0$ has isolated singularities. As the simplest c …

Let $R$ be the local ring of a complex curve singularity. (Can assume the singularity planar, the ring locally analytic or formal.) Let $\bar{R}$ be the normalization, let $R\subset R'\subset \bar{R}$ …

(this can be considered as some ad) Consider the system of equations $F(x,y)=0$. (Here $x$, $y$ are multi-variables. The equations are over a local ring. e.g. polynomial/analytic/formal/$C^\infty$ et …