Questions tagged [complete-intersection]

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Are smooth irreducible affine varieties set theoretical complete intersection?

I'm trying to prove a lemma, so I wanted to use the next claim (which I do not know if it is true or if there is a counter example): every smooth irreducible affine variety is set theoretical complete ...
Leonardo Lanciano's user avatar
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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 ...
Bobby Grizzard's user avatar
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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 ...
Ron's user avatar
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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 ...
gio's user avatar
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Are there CM complete intersections of arbitrarily large degree and codimension?

For every $d, c$ does there exist a smooth complete intersection $X \subset \mathbb{P}^N$ of codimension $c$ and multidegrees $d_1, \dots, d_c \ge d$ such that the Mumford-Tate group of $X$ is abelian?...
Ben C's user avatar
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On a conjecture of Hartshorne

Hartshorne has a conjecture in his book Ample Subvarieties of Algebraic Varieties. It's in page 126 Conjecture 5.16 where he writes that if $B$ is a finitely generated flat module over a regular local ...
Jose Capco's user avatar
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Complete intersections in complex manifolds

Let $X$ be a complex manifold of dimension $n$ and $Y\subset X$ a closed submanifold of codimension $k$. a) Say that $Y$ is a complete intersection if the ideal $I(Y)\subset \mathcal O(X)$ of global ...
Georges Elencwajg's user avatar
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Lifting a Frobenius endomorphism under an étale morphism

Let $X$ be a smooth affine scheme over $\mathbb{Z}/{p^2}$ that is a complete intersection, say $X$ is the spectrum of $\mathbb{Z}/{p^2}[x_1,...x_n]/(f_1, ... f_r)$, where $n-r$ is the dimension of $X$....
user11235813's user avatar
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Noether intersection multiplicity for complete intersections

If I take two curves $C,D$ on a surface $M$ with isolated intersection point $p$, then Noether gives a formula equating the intersection multiplicity $i_p(C,D)$ of $C$ and $D$ at $p$ in terms of their ...
Stephen McKean's user avatar
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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 ...
Josh Zahl's user avatar
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Deformation of toric varieties to complete intersections

I am looking for some systematic study/examples of families of projective complete intersection varieties degenerating to a projective toric variety. In particular, given a projective toric variety, ...
user45397's user avatar
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Properties preserved in addition of ideals

If I and J are prime (radical) ideals then what are the conditions under which we can define a prime (radical) ideal from I+J?
Iqra Khan's user avatar
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128 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 ...
gigi's user avatar
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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 ...
MonLau's user avatar
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Can a surjective morphism between complete intersection rings be given by adding terms to a regular sequence?

Given a surjective morphism $$\frac{\mathcal{O}[[X_{1},\dots ,X_{n}]]}{I}\twoheadrightarrow \frac{\mathcal{O}[[X_{1},\dots ,X_{n}]]}{J}$$ where $I,J$ are genereated by regular sequences. Question Can ...
Marsault Chabat's user avatar
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Prescribed intersection of varieties

Every variety here is complex analytic, or complex algebraic if it solves anything. Given a germ of a (possibly singular, nor necessarily irreducible) hypersurface $(H,0)\subset(\mathbb{C}^{n+1},0)$ ...
MathBug's user avatar
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Check whether a closed point of a Noetherian affine scheme is a local complete intersection

Suppose that $k$ is an algebraically closed field and $A$ is the ring $k[a,b,c,d]/(ac-b^2,bd-c^2,ad-bc)$. Let $X$ be Spec$A$, and $m$ be the maximal ideal of $A$ generated by the quotient images of $a,...
Boris's user avatar
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How to compute the genus of the (singular) intersection of three quadratics in $\mathbb{C}P^4$?

Consider three quadratics in $\mathbb{C}P^4$: $$ x_0^2+4x_1^2+\frac{x^2_2}{4}=0,~ x_1x_4+x_2x_3=0, ~ x_0^2+x_3^2+x_4^2=0. $$ If there intersection was non-singular, then the intersection should be a ...
Zhaoting Wei's user avatar
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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 ...
user45397's user avatar
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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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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 ...
Lao-tzu's user avatar
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1 vote
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279 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 ...
Sivakanth Gopi's user avatar
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695 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 ...
Bil's user avatar
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