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(Probably some basic question, but I've never worked in the real world.) Let $X\subset\mathbb{P}^n_\mathbb{C}$ be a complex variety with the complex conjugation $\tau:X\to X$. So $\tau$ acts on $\mat …

Suppose X is a pure dimensional projective complex scheme, reducible and non-reduced but without embedded components of lower dimension. Let $X=\cup X_i$ be the decomposition such that $X_i$ is set-t …

Let $X\subset\Bbb P^n_{\Bbb C}$ be a connected, Cohen Macaulay sub-scheme. (Possibly singular, reducible or non-reduced.) For $k\gg0$ the numbers $h^0(\mathcal{O}_X(k))$ depend polynomially on $k$. De …

Let $Z\subseteq Sing(X)\subset M$, where all objects are complex reduced and projective, moreover $Z$ and $M$ are smooth. Consider the strict transform $\tilde{X}\subset Bl_Z M$. If all the singular …

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'm looking for a definition of Chern class (at least the first one) for a torsion-free sheaf $F$ (not necessarily locally free) on a singular curve (for simplicity can assume all the singularities …

Let $\nu:\tilde{X}\to X$ be the normalization of a projective variety with non-isolated singularity. The usual object to consider is $\nu_*\mathcal{O}_{\tilde{X}}/\mathcal{O}_X$. For example, one woul …

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 …

Let $V_X$ be a vector bundle of rank $r>1$ on a smooth (connected) projective variety of dimension $r$. Let s be the global holomorphic section, whose zero locus is a zero dimensional subscheme $Z\sub …

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?

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 $X\subset\mathbb{P}^n$ 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 …

(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 …

(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 …

(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 …