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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 …

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 $X\subset\mathbb{P}^n$ 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 …

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 …

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 …

(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 …

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 $X$ 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 $(R,m)$ 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, $ …

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 …

A rational normal curve $C_d\subset\mathbb{P}^d$ is defined by quadrics. I guess, for the generic projection $\mathbb{P}^d\stackrel{\pi}{\to}\mathbb{P}^n$ the image $\pi(C_d)$ is still defined by quad …

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 $R\subset R'\subset \bar{R}$ …

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" …

(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 …

(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 …