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Questions tagged [projective-varieties]

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Properties of base change of varieties

Let $X$ be an algebraic variety over a field $K$. Let's assume that $X$ is integral. Now let $L$ be any field extension of $K$ and let's construct the variety: $$X_L:= X\times_{\operatorname{Spec K}} ...
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Why Self-secant varieties are linear spaces

$\underline {Background}$:Let,$X$ be an irreducible ,reduced algebraic variety in $\mathbb{P}^{n}$.We denote the secant variety of $X$ by $\sigma_2(X)$ $\underline {Question}$: Given $\sigma_2(X)=X$,...
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Secant variety of a curve

$\underline {Background}$: Let,$X$ be a curve in $\mathbb{P}^{2}$ which is not a line.We denote secant variety of $X$ to be $\sigma_2(X)$ $\underline {Question}$: To prove $\sigma_2(X)=\...
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Clarification about a proof regarding a sum of polynomials being expressed as a linear combination of S-polynomials

I'm reading this proof from Ideals, Varieties, and Algorithms by David A. Cox, Donal O'Shea, and John Little. You can find an online version here. This is Lemma 5 of Chapter 2, page 85. From my ...
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Degree of the intersection of subvarieties from different branches

Let $k$ be an algebraically closed field. Let $X$ be an irreducible hypersurface of $\mathbb{P}_k^n$, where $n\geq 4$. Let $Y$ be an irreducible subvariety of $X$ of codimension one. Let $d$ be the ...
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On double points of a Projective space

$\underline {Background}$: Let, $ p\in Proj(K[x_0,....,x_n])$ . A double point at $ p\in Proj(K[x_0,....,x_n])$ is the scheme given by the square of the ideal (sheaf) of p. i.e we can consider it ...
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If a polynomial is reducible say $F=fg$($f \neq g$), does $V(F)$ still have codimension $1$?

If a polynomial is reducible say $F=fg$ ($f \neq g$), does $V(F)$ still have codimension $1$? This is a question to clear my conception, what I feel is no because we will have $V(F) \subsetneq V(f)$ ...
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Computing the structure sheaf $\mathcal{O}_X(U)$ for $U=X\setminus \mathbb{V}(x_0^2+x_1^2+x_2^2)$ and $X=\mathbb{P}^2$.

I was wondering how one computes $\mathcal{O}_X(U)$ for $U=X\setminus \mathbb{V}(x_0^2+x_1^2+x_2^2)$ and $X=\mathbb{P}^2$ by considering the usual cover of $\mathbb{P}^n$ by affine charts $U_i$. ...
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When is a morphism induced by a graph

Suppose $X$ and $Y$ are projective varieties and $\Gamma \subset X \times Y$ is a graph, that is for any $x \in X$ we require $\pi_{Y}(\pi^{-1}_{X}(x))$ to be a point in $Y$, call it $f(x)$. Does $\...
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example of algebraic variety with infinitely many singularities

Let $X$ ba an algebraic variety and $\mathrm{Sing}(X)$ be the set of all singular points. For a set $A$ , $|A|$ denotes the cardinality of $A$ . I konw examples of algebraic variety with finite ...
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Property of Unirational variety

Let $X$ be an algebraic variety over field $k$ snd $n=\mathrm{dim}(X)$ . We assume $X$ is unirational. There exists $m \in \mathbb{N}$ and a dominant rational map $\phi : \mathbb{P}_k^m \...
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Can an algebraic variety be embedded in projective space ???

Let $X$ be an algebraic variety over field $k$ . $X$ can be embedded in a complete variety by Nagata's compactification theorem. Moreover, can we embed $X$ in a projective space $\mathbb{P}_k^n$ $?...
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Set of Closed points of a Quasi-projective variety is dense

Let $k$ be an algebraically closed field and $V$ be a Qausi-projective variety in $\mathbb P^n_k$ i.e. $V$ is an open subset of an open subset of a Zariski-closed subset of $\mathbb P^n_k$, or in ...
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Dimension of product of varieties

I've got this exercise asking me to prove first that the product of quasi-projective varieties $X$ and $Y$ (henceforth just "varieties") is irreducible iff both $X$ and $Y$ are. I managed to solve ...
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1answer
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Existence of a hyper surface containing the singular locus of projective variety.

Let $V_n$ be an irreducible algebraic variety in projective space $\mathbb{C}P^n$. Denote by $V_{n_1}$ the subvariety cut out on $V_n$ by a hyper surface $W$ containing the singular locus of $V_n$ and ...
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Proving that Projective Varieties are Compactifications of Affine Varieties

I am working through Karen Smith's Invitation to Algebraic Geometry and one of the problems is: Prove that every projective variety is the compactification of an affine variety in the Zariski topology ...
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Find the dimension of an (explicit) projective variety

I have to compute the dimension of : $$ X=V(xz-y^2,xz-xw,yz-yw,z^2-zw) \subset \mathbb{P}^3(\mathbb{C}) $$ My idea was to compute irreducible components of this algebraic sets and intersect with some ...
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Projecting a projective variety away from a linear subspace

I read in Harris' book at page 148, proposition 11.37 and got slightly confused regarding the argument. Harris mentions a projective space $\mathbb{P}^{2n+1}$ and a linear subspace $\mathbb{P}^{n}$ ...
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Automorphism Group of a Variety acts on Local Sections

The motivation/background of my question arises from following thread: Galois morphism - group acting on the variety The original setting is that we have a finite Galois morphism $f: X \to S$, where ...
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Group Acting on Variety

My question refers to some comments occured in following thread: Galois morphism - group acting on the variety The setting is that we have a finite Galois morphism $f: X \to S$, where $X$ and $S$ ...
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Galois Action on Coherent Sheaves Exact Functor

Let $X$ be a non-singular, connected projective variety and $G$ be a finite automorphism group of $X$ such that the quotient $X/G$ is well defined as variety. (especially there is a well defined ...
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Degree and codimension of nondegenerate varieties

A projective variety $X \subset \mathbb{P}^r$ (i.e. reduced, irreducible closed subscheme) is called nonegenerate, if it is not contained in any hypersurface $H \subset \mathbb{P}^r$. Equivalently, ...
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Intersection number for projective plane curves

Context: In our lecture about algebraic geometry we defined the intersection number of two algebraic curves $F,G$ to be $dim_k (\mathcal{O}_P (\mathbb{A}^2)/(F,G))$. Then we proved that it satisfies ...
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Differential 1-forms on an irreducible projective variety

Let $k$ be an algebraically closed field and let $X$ be an irreducible projective variety over $k$. I am wondering what the module of differential 1-forms on $X$ is. Since $X$ is a projective variety,...
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Shafarevich, locally regular $\Rightarrow$ globally regular

So I am confused with this argument in 3rd Ed, pg 47, Basic Alg. Geo. 1. Definition 1: if $X \subseteq \Bbb P^n$ is a quasiprojective variety, $x \in X$, and $f=P/Q$ is a homogenous function of ...
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1answer
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Is the Zariski topology on a variety $V$ a maximal Noetherian topology?

Let $K$ be an algebraically closed field. By a variety $V$ definable over $K$, I mean a quasi-projective or an algebraic variety in sense of Weil. It is the set of points in an affine or a projective ...
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Image of the Segre embedding of an hypersurface

Suppose that my initial envoirenment space is $\mathbb{P}^{4}$ with homogeneous coordinates $[x_{0}:x_{1}:x_{2}:x_{3}:x_{4}]$ and I define an hypersurface with equation (for example) $h:x_{0}x_{3}+2x_{...
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2answers
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Showing the join of two disjoint projective varieties is a projective variety.

I'm looking at the following proposition in Harris' Algebraic Geometry: a first course: Proposition: Let $X,Y\subset\mathbb P^n$ be disjoint projective varieties. Then the join $J(X,Y)$ of $X$ and $...
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1answer
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Why can a projective variety of dimension $n$ be covered by $n+1$ affine open subsets?

My question: Why can a projective variety of dimension $n$ be covered by $n+1$ affine open subsets? I can see the result holds when the variety is $\mathbb P^n$ or a hypersurface $X$ in $\mathbb P^{...
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30 views

Degree of some type of projective varieties

Let $X\subseteq\mathbb{P}^{N+1}$ be a projective variety lying in some plane $\mathbb{P}^{N}\subset\mathbb P^{N+1}$ and take a point $p\in\mathbb P^{N+1}-\mathbb P^N$. Now consider the projective ...
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Smoothness of an algebraic subvariety

Is every subvariety of a smooth algebraic variety a smooth variety ? Thanks in advance for your help.
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Degree of images of projective varieties

Let $X\subseteq\mathbb{P}^n\times\mathbb{P}^m$ be a projective variety of dimension $p$ and degree $d$ defined over an algebraically closed field $k$. Let $X'\subseteq \mathbb{P}^n$ be the projection ...