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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 $R\subset R'\subset \bar{R}$ …

Are there some standard ways to define the tangent space to a group $G$ at its unit element, $e$, when the group is not (pro)algebraic/(pro)Lie, not necessarily over a field, does not have the ``class …

Let $X$ be projective and Gorenstein (over $\mathbb{C}$), of dimension $n$, then $\chi(\mathcal{O}_X)=(-1)^n\chi(\omega_X)$. Hence a "generalization": $\chi(\omega^{\otimes k}_X)$. I'd like somethin …

Let $R$ be a commutative ring (zero characteristic). Take a skew-symmetric matrix $A\in Mat^{skew-sym}(n,R)$. If $n$ is even, then $\det(A)=Pf^2(A)$ and there exists the "Pfaffian adjugate/adjoint" …

Let $R$ be a local or graded ring. (If it helps, can assume the ring is "good", e.g. $R=k[[x_1,..,x_p]]$, where $k$ is a field of zero characteristic.) Let $M$ be a finitely generated $R$-module o …

(Fix some local ring $(R,\mathfrak{m})$ over a field of zero characteristic.) Suppose an ideal $J$ is defined by some complicated formula/procedure. And there is no hope of computing it/or writing do …

Let $R$ be a local (commutative, associative) ring over a field of zero characteristic. (My typical examples are $k[[x_1,..,x_p]]/I$, $k\{x_1,.,x_p\}/I$, $C^\infty(\Bbb{R}^p,0)$. If it helps one can …

(The question of the type "how to define?") Let $(R,\mathfrak{m})$ be a local ring over a field $k$ of zero characteristic. Consider the matrices over this ring, $Mat(m,R)$. I think of $Mat(m,R)$ as …

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?

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 …

I'd say the number can be arbitrary large if $X$ is not assumed smooth along $Z$. Take the simplest example: hypersurface singularity $\{f_p+f_{p+1}=0\}\subset\Bbb C^n$, where $f_p$ is totally reducib …

Let $(R,m)$ be a complete local ring over a field (of char=0). Consider a (not necessarily square) matrix $A$ over $R$. Consider its fitting ideal, $I_j(A)$. In general, a deformation of the matrix, $ …

(As Will Sawin has answered the question, in a comment above, I record here the detailed answer. Still, I'd like to know some references where all this is written.) \ (upd: there are several mistake …

Consider globally complete intersections in $\mathbb{P}^n$, of codimension $k$, of some fixed multi-degree $(d_1,\dots,d_k)$. Is there some nice (i.e. "explicit") parameter space for them? (even if …

Let $X\subset\mathbb{P}^n$ be a (globally) complete intersection, let $(X_t)_{t\in\mathbb{C}^1}$ be a flat family, with $X_1=X$. Which types of schemes can we get as $X_0$? Or, conversely, which (emb …