Questions tagged [complete-intersection]

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Sequence of Cohen-Macaulay subschemes and regular embedding

Let $X$ be a non-singular variety, $Y_1, Y_2$ be non-singular subvarieties of $X$ of the same dimension such that $Y_1 \cap Y_2$ is non-singular, irreducible and $Y_1 \cup Y_2$ is a local complete ...
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85 views

Local complete intersection on a smooth variety

Let $X$ be a smooth variety and let $x_{1},\ldots,x_{k}$ be general points. Let $T:=<T_{x_{1}}X,\ldots,T_{x_{k}}X>$ be the join of the tangent spaces at the points $x_{1},\ldots,x_{k}$. If we ...
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1answer
96 views

When is a graph morphism a regular embedding? [closed]

Let $f:X \to Y$ be a finite morphism, where $X$ is an affine, non-singular curve and $Y$ is an affine, non-singular surface over $\mathbb{C}$. Denote by $\Gamma_f \subset X \times Y$ the graph of the ...
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131 views

Structure of Complete Local Rings

Let $X$ be a proper $n$-dimensional $k$-scheme and $x \in X$ a closed point. Consider the stalk $\mathcal{O}_{X,x}$. We consider now it's completion $O_{X,x}^{\wedge}$ wrt it's maximal ideal $m_x$. ...
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1answer
145 views

Complete intersection subvariety of projective variety

I am not able to find any literature which studies complete intersection subvarieties of a projective variety, all good references consider CI in projective space. My guess is, a subvariety X of ...
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1answer
215 views

When does a subspace of the affine space form a regular sequence in a ring of regular functions?

Let $J$ be an ideal in the ring $\mathbb{C}[u] := \mathbb{C}[u_1,...,u_n]$ of regular functions on $\mathbb{C}^n$. Given a subspace $\mathbb{C}^k \subset \mathbb{C}^n$ and $I$ its ideal in $\mathbb{...
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253 views

Linear sections of Segre varieties and rational normal scrolls

In a projective space $\mathbb{P}^{k+2}$ consider two complementary subspaces $\mathbb{P}^1,\mathbb{P}^k$, and let $C\subset\mathbb{P}^k$ be a degree $k$ rational normal curve. Fixed an isomorphism $\...
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102 views

Bounding the dimension of the locus where a variety has larger than expected dimension

Disclaimer: I am a research mathematician, but not an algebraic geometer, and so I don't know if this is a good question. I welcome advice for improving it and/or better tags. Let $K$ be an ...
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2answers
557 views

How to define the intersection multiplicity of a projective variety and a complete intersection?

In the appendix of Algebaric Geometry by Hartshorne, he shows us that how Serre defines the intersection number in a more general case:$$i(X,Y;Z)=\sum(-1)^i\cdot\bigl(\operatorname{length} \...
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212 views

Find the generators of a complete intersection maximal ideal

Let $k$ be a field (not necessarily algebraically closed), define the $k$-algebra $$ B:=k[x_{0}, x_{1}, x_{2}, x_{3}]_{(F_{2}F_{3})} $$ (degree 0 part of the localization), it's the coordinate ring of ...
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1answer
312 views

ideals linked to an almost complete intersection

Is a grade $3$ type $3$ perfect ideal in a Noetherian ring linked to a grade $3$ almost complete intersection? I know that grade $3$ type $2$ perfect ideals are (by a work of Anne Brown).
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584 views

Automorphisms and infinitesimal deformations of a smooth complete intersection

Let $X\subset\mathbb{P}^{n+c}$ be a smooth complete intersection of dimension $n$. Is it known when $Aut(X)$ is finite ? Does there exist a formula for the dimension of the tangent space to the ...
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2answers
188 views

Sections of a sheaf of differentials on a weighted complete intersection

Let $X\subset\mathbb{P}(a_0,...,a_N)$ be a smooth $n$-dimensional weighted complete intersection in a weighted projective space $\mathbb{P}(a_0,...,a_N)$. Is it true that if $q\geq 1$ then $H^0(X,\...
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69 views

Rational connectedness of certain subvarieties of the linear series

Let $X$ be a smooth projective hypersurface in $\mathbb{P}^3$, $|\mathcal{O}_X(a)|$ be the complete linear system for some integer $a>0$. Ofcourse, a general element of the linear system is a ...
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307 views

Blow up along a section of a smooth morphism

Let $C$ be a ground locally notherian and quasiprojective scheme. Let $\pi:S\to B$ be a $C$-morphism of finite type $C$-schemes. We call $\pi$ a $C$-smooth morphism if the morphisms $S\to C$ and $B\to ...
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1answer
224 views

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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1answer
189 views

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 $B$...
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630 views

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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1answer
345 views

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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213 views

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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878 views

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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1answer
1k views

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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342 views

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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1answer
706 views

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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790 views

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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460 views

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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1answer
660 views

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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2answers
474 views

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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1answer
335 views

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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1answer
480 views

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 ...