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Questions tagged [affine-varieties]

Use this tag for questions related to an affine variety over an algebraically-closed field.

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Let J be an ideal. Find a function in I(V(J)) such that the function f is not in J

Let $J$ be the ideal $\langle x^2+y^2-1,y-1\rangle$. Find $f \in \textbf{I}(\textbf{V}(J))$ such that $f \not \in J$. I'm confused on a number of aspects here. Firstly, how do I find $\mathbf{V}(J)$ ...
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Divisor class group 0 for affine and Z for projective nonsingular varieties given UFD [on hold]

The following exercise is from Shafarevich. Let $X$ be a nonsingular affine (resp. projective) variety over an algebraically closed field. Prove that $CL(X) = 0~(\text{resp. } \mathbb{Z})$ iff the ...
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The complement of a point in $\Bbb{A}^n$ is compact in the Zariski topology.

The book "Invitation to Algebraic Geometry" says the following: The complement of a point in $\Bbb{A}^n$ is compact in the Zariski topology. Why is this this the case? This is thing that is asked ...
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Quotient of Algebraic Variety

Let $\mathbb{A}^1$ be the affine line (considered as algebraic variety) endowed with action by $\mathbb{G}_m$ via $t \cdot a = ta$. If we form a set theoretical quotient $\mathbb{A}^1/\mathbb{G}_m$ ...
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Definition of Morphism between varieties

I have found the following definition of morphism. Let $X,Y$ be two varieties. A mapping $\phi:X\rightarrow Y$ is said to be morphism if $\phi$ is continuous. For each open set $U$ of $Y$ and $f\in ...
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Sheaf of a Closed Subset

I’ve been given the following definition: Let $(X,\mathcal{O}_X)$ be a ringed space which is locally isomorphic to an affine algebraic variety, and $Y\subseteq X$ be closed. Then for an open $V\...
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Tangent lines throught a point in an algebraic curve

In Fulton's Algebraic Curves, beginning of chapter 3, there is a quick discussion about multiple points and tangent lines. He gives many examples of affine curves, for instance: He then says that ...
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Prove that if k is a field, then any affine open subset of the affine n-space A_n^k is principal.

I think the question is clear and famous too but i didn' find an answer on this site, so : k : is a field not necessarily ( an algebraic closed field ). An open subset of the scheme is an open in ...
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Find a set of polynomials whose common zero set is $\{(1, 2), (0, 5)\}$.

Find a set of polynomials $\{P_1, \dots, P_n\}$, all of whose coefficients are real numbers, whose common zero set is the given set. I know what a zero set is, but I think my confusion comes from ...
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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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Is regular birational image of affine subset affine?

Let $X$ and $Y$ be projective varieties and $φ\colon X \to Y$ a regular birational map. If $U⊆X$ is an affine subset of $X$, is $φ(U)$ affine? This is related to my previous question.
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Sheaf of Regular Functions and Localisation

I’m trying to prove the following statement: Let $V$ be an affine algebraic set, $\Gamma(V)$ its coordinate ring, and $\Gamma(D(f),\mathcal{O}_V)$ the sheaf of regular functions of $D(f)=\{x\in V\...
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What is transition function of scheme theoretic vector bundle ??

Let X be a scheme X and $\mathscr{F}$ be a locally free sheaf of rank $n$ over $X$. I want to consider a vector bundle over an algebraic variety $X$ , that is , the relative spec over $X$ of ...
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Prove that $\mathcal {C}$ is an affine algebraic $K$-variety.

Let $K$ be a algebraically closed field. Consider the set $$\mathcal {C} = \left \{(l^p,l^q) \in K^2 \mid p,q\ \text {fixed in}\ \Bbb Z,p,q>0,l\ \text {varies over}\ K \right \}.$$ Prove that $\...
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Why is ${\varphi}^{-1} (A)$ closed in $V$?

Let $L$ be a field and $K$ be a subfield of $L.$ Let $V$ be an affine algebraic $K$-variety in $\Bbb A^n (L).$ Then the coordinate ring $K[V]$ is defined by $$K[V] = \frac {K[X_1,X_2, \dots ,X_n]} {\...
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What will be $\mathcal {I} (V)$ if $V = \varnothing$?

Let $L$ be a field and $K$ be a subfield of $L.$ Let $V$ be an affine algebraic $K$-variety in $\Bbb A^n (L).$ Consider the vanishing ideal of $V$ $$\begin{align} \mathcal {I} (V) & = \left \{f \...
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Proving $y-x^2, z-xy$ generate the ideal of the twisted cubic

This question comes from Hartshorne's exercise 1.2. He defines $Y:=\{(t,t^2,t^3)\in \mathbb{A}^3\mid t\in k\}$ and asks us, among other things, to find generators for the ideal $I(Y):=\{f\in k[x,y,z]\...
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Why is the algebraic torus an affine variety?

I am reading Toric Varieties by Cox, Little, and Schenck. I am stuck on the definition an algebraic torus, given in Part 1.1, page 10, which states: The affine variety $(\mathbb{C}^*)^n$ is a group ...
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Ideal generated by $yw-x^2$ and $z^2w-2xyz+y^3$ is not radical

How to prove that ideal generated by $yw-x^2$ and $z^2w-2xyz+y^3$ is not radical? This is one interpretation of the twisted cubic curve which is said to be not a radical ideal.
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Definition of codimension of variety

Let $X$ be an variety over field $k$. A Weil divisor on $X$ is an integral linear combination of irreducible subvarieties of $X$ of codimension $1$. So I want to know the definition of codimension of ...
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Finite Morphisms between Varieties have Finite Fibers

My question refers to following previous thread: Geometric interpretation of the Noether normalization lemma Obviously the core problem is: If $X= Spec(A), Y= Spec(R)$ are affine varieties and $\phi: ...
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Finite Dimensional $k$-Algebra

I have a question about an argument used in Tamas Szamuely's "Galois Groups and Fundamental Groups" in following excerpt (see page 102): We start with affine variety $X$ of dimension $n$. According ...
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About the tangent cone and tangent space of an affine variety

Let $X\subset \mathbb{A}^n$ be an affine variety then $X= Z(I)$ is the zero locus of the ideal $I$. In general the tangent cone at $0$ is define as $TC= Z(I^{in})$ where $I^{in}$ is the initial ideal ...
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Finite Dimensional $k$-Algebra has finitely many Maximal Ideals [duplicate]

I have a question about an argument used in Tamas Szamuely's "Galois Groups and Fundamental Groups" in following excerpt (see page 102): We start with affine variety $X$ of dimension $n$. According ...
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1answer
41 views

Uniform measure on an affine variety

I'm trying to find an algorithm that uniformly samples elements in the following variety: $A_p=\{(\lambda_1,...,\lambda_n) \in [0,1]^n / \sum \lambda_i = 1, \, \sum \lambda_i^2 = p\}$ I think it can ...
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1answer
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Prove that $\mathcal {I} (H) = (F_1 \cdot F_2 \cdots F_s).$

Let $\Bbb A^n (L)$ be an $n$-dimensional affine space over the field $L.$ Let $K (\subseteq L)$ be a subfield of $L.$ For any $V \subseteq \Bbb A^n(L)$ define $\mathcal I (V)$ as the set of all $F \in ...
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Notion of line on a complex affine space

$\newcommand{C}{\mathbb{C}} \newcommand{\A}{\mathbb{A}} \newcommand{\i}{\hspace{0.1em}\mathrm{i}} \newcommand{\R}{\mathbb{R}}$ Let $V = (\C, \C, +, \cdot)$ be the one-dimensional complex vector space ...
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Regular functions on an affine variety.

Let $X$ be a reducible affine variety (over an algebraically closed field $k$) with irreducible components $X_1,\ldots, X_r$. If you take $p_i\in k[X_i]$, then there exists $p\in k[X]$ such that $p=...
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affine variety from ideal

I have a problem with this excercise: *Take a field $F$, algebraic closed, $charF\neq 2$. 1)Consider the following ideal $I=(xy+yz+zx-\frac{1}{2}u^2 , x+y+z+u)$ in $F[x,y,z,u]$. Is $Z(I)$ affine ...
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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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About the image of an affine variety into a projective space.

Assume to have a morphism $f: \mathbb{A}^n \to \mathbb{P}^m$, I want to compute the ideal of the the projective closure of $ f(\mathbb{A}^n)$. I think that in general the image of an affine variety ...
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1answer
29 views

Computing Coordinate Rings of Varieites

I am a complex analyst who was been screwed over by fate and now has to work with elliptic curves for my doctoral dissertation. This entails learning about (non-category-theoretic) algebraic geometry. ...
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Affine Varieties over separably closed fields

Let $E=E^{sep}$ be a separably closed field, i.e. any separable polynomial over $E$ has a solution in $E$, and let $Var_E$ be the category of smooth, affine schemes of finite type over $E$. On the ...
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Irreducible components of the subvariety of matrices s.t. $M^n=\mathbb{I}$

Fixed $n\in\mathbb{N}$, describe the irreducible components, and prove that there are ${n+1\choose{2}}$ of them, of the following algebraic variety: $$V=\{A\in M_{2}(\mathbb{C})\mid A^n=\mathbb{I}\}$$ ...
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Affine variety of an algebraic set in ghathmann notebook

In the book "Algebraic geometry" by Andreas Gathmann, in page 26 Lemma 2.3.16 , why the ideal $J$ is prime?. i mean why $Z(J)$ is affine variety?.it says" we claim that the ringed space ... is ...
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1answer
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How to show that $\phi: \mathbb{A}^1(\mathbb{C}) \to V(Y^2 - X^3)$ is no isomorphism.

Consider the polynomial function $$\phi: \mathbb{A}^1(\mathbb{C})\to V(Y^2 - X^3): t \mapsto (t^2, t^3)$$ Show that $\phi$ is a bijective polynomial function, but no isomorphism (= $\phi^{-1}$ is ...
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Singular points on a variety. $V: 4x^2y^2=(x^2+y^2)^3$

So I was studying some stuff about projective varieties from the book "The Arithmetic of Elliptic Curves" from Silverman, and there is this exercise at the very beginning, about determine the singular ...
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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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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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Morphism from $\mathbb{C}P^1$ to affine space

I am interesting the morphisms from $\mathbb{C}P^1$ to $\mathbb{C}^n$. I intuitively think that there should be only one nontrivial map, which is just embedding when $n>1$, and there is no ...
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Showing that an affine variety is not isomorphic to the affine line.

Let $k$ be algebraically closed. Show that $\mathbb{A}_k^1$ is not isomorphic to $C:= \{(x,y) \in \mathbb{A}_k^2 :x^2-y^2=1\}$. I know that this is equivalent to showing that $k[t]$ and $k[x,y]/(x^2-...
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No zeros or poles implies regular

The following is Theorem 8.14 of Milne's notes (page 176): Let $V$ be a normal variety, let $ f $ be a rational function on $V$. If $f$ has no zeros or poles on an open subset $U$ of $V$, then $f$ is ...
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Invertible polynomial maps on algebraic sets

Two affine varieties (defined as irreducible algebraic sets) are isomorphic if there are mutualy inverse polynomial maps between them. Is it true that if I have an affine variety $V$ and an ...
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1answer
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Kernel of a morphism of $k$-algebras

Assume you have two affine varieties $X$ and $Y$. A morphism $\phi$ between them induces a morphism between the k-algebras of regular functions (functions on them that are locally the quotients of ...
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Fact regarding the product of Affine varities

I am reading Andreas Gathmann's Algebraic Geometry notes where following statement is given regarding the product of affine varities: Let $X\subset \mathbb{A}_{k}^{n}$ and $Y\subset \mathbb{A}_{k}^{...
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1answer
55 views

Proposition about regular functions on an affine variety

I have the following basic issue: Let $X$ be an affine variety in $\mathbb{A}_{k}^ {n}$. Co-ordinate ring of $X$ be $A(X):= k[x_1,\ldots ,x_n]/I(X)$ whereas $K(X)$ be its quotient field. $$ \mathcal{...
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1answer
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det-1 is irreducible

I'm interesting about $V(det-1) \subset \mathbb{A}^{n^2}$ with the determinant seen as a polynomial. I know that $det$ is irreducible. But I want to show that $det-1$ is irreducible. In a paper, it ...
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1answer
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Some Questions on Algebraic Varieties

I'm doing some research on algebraic varieties for an engineering application. I am totally new to the subject and have a general (B.S.-level) background in math. I am currently using Ideals, ...
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Find the irreducible components of $V(x - yz, xz - y^2)$.

I think I have an answer but I'm looking for some verification. Since $x - yz = 0$, then I can rewrite my problem as $V(x - yz, yz^2 - y^2)$. This is equal to $V(x - yz, y) \cup V(x - yz, z^2 - y)$...
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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.