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### Bezout's theorem for non-proper intersections?

Is there a version of Bézout's theorem for non-proper intersections?
For my specific problem, the setup is as follows: Let $P_1,P_2,P_3,P_4\in\mathbb{C}[z_1,z_2,z_3,z_4]$, and suppose that (as a ...

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### Irreducible components of reduced complete intersection

Let $Z$ be an irreducible and reduced scheme. Does there exist a reduced complete intersection $Y$ such that $Z$ is an irreducible component of $Y$?

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### Parameter space for complete intersections and their discriminant

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 ...

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### Factoriality of complete intersections

Let $X\subset\mathbb{P} _{\mathbb{C}}^N$ be a normal complete intersection.
$X$ is called factorial if every Weil divisor on it is Cartier;
equivalently if all local rings $\mathcal{O}_{X,x}$ are ...

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### what can be reached by flat degeneration of (globally) complete intersection?

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 ...

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### If $X$ is an affine variety, is $X$ one component of a complete intersection with two?

This is an idle question, but I give the example that motivated me below.
Say $X \subseteq {\mathbb A}^n_k$ is irreducible and $k$ is infinite. Then by picking a regular point of $X$ and picking ...

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### Sections of a fibration in intersections of quadrics

Suppose that we have a smooth variety $X$ of dimension $n$ that fibers (a flat morphism) over a curve $Y$, and s.t. the fibers of $X \to Y$ are all complete interesections of two quadric hypersurfaces ...

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### Looking for an inequality between Chern and Todd classes (something in style of Bogomolov-Miyaoka-Yau)

Consider a smooth projective surface $S\subset\Bbb P^N_{\Bbb C}$ which is a complete intersection of hypersurfaces of degrees $(d_1,..,d_{k\ge2})$ with $d_i\ge2$ for all i. Is it true that for such ...