The complete-intersection tag has no usage guidance.

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### A question about Complete Intersections

Suppose that $(A,\mathfrak{m})$ is a complete intersection and $\mathbf{x}$ is a minimal basis for $\mathfrak{m}$. Consider the Koszul homologies $H_\bullet(\mathbf{x},A)$. It is well-known that ...

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### When every module is a scalar extension?

Let $A \subseteq B$ be commutative noetherian domains.
Of course, if $M$ is an $A$-module, then $M \otimes_A B$ is a $B$-module.
I am curious to know if there exist additional conditions on $A$ and ...

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### Complete Intersection

Let $I$ be an ideal of the polynomial ring $P=K[x_{1},...,x_{n}]$ that is generated by degree two polynomials ${f_1,...,f_k}$.
The zero set $\mathcal{Z}(I)$ is isomorphic to an affine space of
...

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### A criterion for complete intersection in terms of the Hilbert series?

Let $(R, \mathfrak{m})$ be a complete local ring (of dimension $2$ if that makes a difference). I would like to be able to decide whether or not $R$ is a complete intersection (meaning, a quotient of ...

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### Residual Intersections of a complete intersection

Let $R$ be a Cohen-Macaulay local ring and $I=(b_1,\dots,b_s)$ be a complete intersection generated by a regular sequence $\underline{b}$. Let $\mathfrak{a}\subseteq I$ such that ...

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### Sufficient conditions to get complete intersection curves

Let $H_1,H_2\cdots,H_{d-1}$ be hypersurfaces in $\mathbb{P}^d$, if the intersection $B:=H_1\cap H_2\cap \cdots \cap H_{d-1}$ is $1$-dimensional then it is called a complete intersection curve.
What ...

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### Realizing algebraic curves as complete intersections

I have two related questions about smooth complete algebraic curves (over $\mathbb{C}$).
Does there exist a smooth complete algebraic curve $X$ that cannot be embedded as a complete intersection in ...

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### geometric interpretation and differences of Gorenstein rings, Complete intersections and regular rings

Let $R$ be a local Noetherian ring.
What is the geometric interpretation of:
1- Gorenstein rings
2- Complete intersections
3- Regular rings?
and how can I realize differences by geometric ...

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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 ...