Questions tagged [affine-varieties]

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

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Criteria for empty vanishing set $V(J)$ for non-closed fields

Let $R = k[x_1,\dotsc,x_n]$ and $J = (f_1,\dotsc,f_m) \subseteq R$ be an ideal. I know that under the assumption that algebraically closed, $V(J)=\emptyset$ iff $J=(1)$. Namely, "$\Leftarrow$&...
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Locally isomorphic singularities = Locally isomorphic minimal blow-ups?

Let's say that one has two varieties $V_1, V_2$, with singularities at points $x_1$ and $x_2$ respectively, and that there exist open neighbourhoods $U_1$ and $U_2$ of these singularities such that $...
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9 votes
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Is a complex algebraic set with a Zariski dense subset of algebraic points already defined over the algebraic numbers?

Let $X$ be a complex algebraic set, i.e. the (not necessarily irreducible) vanishing set of some polynomials in $\mathbb{C}[X_1,\ldots,X_n]$. If $X$ contains a Zariski dense subset of points with ...
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Find the linear variety sum of $L_1$ and $L_2$(i.e., the smallest linear variety containing both $L_1$ and $L_2$)

Consider the linear varieties $L_1$ and $L_2$ in the Euclidean space $\mathbb{R}^3$, given by $$L_1: \pmatrix{x\\y\\z}=P_1+tV_1=\pmatrix{1\\1\\2}+t\pmatrix{2\\1\\1}$$ and $$L_2: \pmatrix{x\\y\\z}=P_2+...
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  • 571
3 votes
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Rank of limit points of a conjugacy class

Problem: Let $t \to P_t$ be a one parameter subgroup $\mathbb{C}^* \to \text{Gl}_{n}(\mathbb{C})$. Let $X$ be a $n \times n$ nilpotent matrix. I want to show that if $lim_{t \to 0} P_t^{-1}XP_t = Y$ ...
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1 answer
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Subfield of a ring of regular functions at a point

Consider $X\subseteq \mathbb A^n$ an affine variety, $p\in X$. Is it true that $\exists K_0\subseteq O_p(X)$ a subfield with $K_0\cong O_p(X)/m_p$ where $m_p$ is the maximal ideal of $O_p(X)$ ? We ...
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How to show $\operatorname{Pic}(X)=0$? Exercise $14.2$.Q Vakil's notes

I'm reading Vakil's notes and I'm struggling with the exercise $14.2$.Q. I've been able to prove everything except $\operatorname{Pic}(X)=0$ with $$ X=\operatorname{Spec}\frac{k[x,y,z]}{(xy-z^2)}. $$ ...
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transcendental dimension of a variety

I am trying to understand the definition of the dimension of a variety using the notion of a transcendental basis. Consider for an algebraically closed field $\mathbb{K}$ the variety $V=\left\{(x,y)\...
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Ring of regular functions

For $V\subseteq \mathbb A^n$ an affine variety set $$\Gamma (V)=\{f:V\to K| \exists F\in K[x_1,\dots,x_n]\text{ such that }f(c)=F(c)\forall c\in V\}=\{f:V\to K|f\text{ is regular}\}$$ $$K(V)=Frac(\...
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Extension theorem over reals

Is there an equivalent of the following theorem from Cox, Little & O'Shea over reals? Definition. Given $I=\left\langle f_{1}, \ldots, f_{s}\right\rangle \subseteq k\left[x_{1}, \ldots, x_{n}\...
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Showing that the set difference of two affine varieties need not be an affine variety [duplicate]

I was wondering what would be an example of two affine varieties $A, B$ over the field of complex numbers $\mathbb{C}$ such that $A\setminus B$ is not an affine variety? I was initially thinking about ...
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Question on the proofs of Hartshorne exercise I.3.20

I have some questions about exercise I.3.20 of Robin Hartshorne's Algebraic Geometry: Let $Y$ be a variety of dimensions $\geq 2$, and $P\in Y$ be a normal point. Let $f$ be a regular function on $Y ...
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Bound on dimension of tangent space of an affine variety [duplicate]

I've been reading through my notes and the following fact is stated without any proof or justification: For an affine variety $X\subset\mathbb{A}^n$ and a point $p\in X$ we have $$dim_k T_pX\geq dim_p ...
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On a lemma used in proving a regular function is not a quotient of polynomials

Lemma: Consider algebraic set $X=V(wx-yz)\subset \mathbb{A}^{4}_{k}$, where $k$ is an algebraically closed field. Then there doesn't exist $h\in k[x,y,z,w]$ s.t. $h\mid _{X}$ is not a constant and $V(...
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Find the inverse image of the algebraic variety

Let $h: K^n \rightarrow K^m$ be a polynomial function and $Y= V_k(I)$ in $K^m$ the algebraic variety of an Ideal in $K[y_1,...,y_m]$. What would the inverse image $X=h^{-1}(Y)$ described as an ideal ...
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  • 777
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1 answer
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Linear subspace as an affine variety

I saw the following description in a book of Algebraic Geometry, as example of affine variety. Let $l_1,...,l_k $ be independent linear forms in $X_1,...,X_n$. Let $a_1,...,a_k \in \mathbb{C} $.Then $...
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2 votes
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Comparing the dimension of an algebraic variety and its tangent space [duplicate]

If $X\subseteq\mathbb{A}^N$ is a variety of dimension $n$, I would like to prove that $$\dim T_xX\geq\dim X=n$$ for any point $x\in X$. My first attempt was to use the Krull's Principal Ideal Theorem, ...
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Definition of affine toric variety

Sorry for my bad English. I'm trouble about a definition of an affine toric varieties. I often see a definition of affine toric varieties as follow; An affine toric variety is an irreducible affine ...
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3 votes
2 answers
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Nilradical in an algebra over a field

In general, if $K$ is a field, it could be that exists $f(x)\in K[x]$ such that $f(a)=0$ for all $a\in K$; for example, set $K:=\mathbb Z/(2)$ and $f(x):=x^2+x$. Now, if $f(x)$ vanishes on all $\...
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Is a finitely generate $k$-algebra that has no nilpotent element ($k$ is a field) an integral domain?

In Hartshone section 1 exercise 1.5, he wants me to prove that If $B$ if a finitely generated $k$-algebra with no nilpotent elements, then $B$ is isomoprhic to the affine coordinate ring of some ...
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1 vote
1 answer
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A Corollary on Estimation of Fiber Dimension from Kemper's Course on Commutative Algebra

I have a question about following proof from Kemper's Course in Commutative Algebra (page 142): Corollary 10.6. Let $f: X \to Y$ be a morphism of equidimensional affine varieties over an ...
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Upper semicontinuity of fiber dimension for Affine Varieties (reference desired)

In This question following result is stated: If $f : X\to Y$ is a morphism between two irreducible affine varieties over an algebraically closed field $k$, then the function that assigns to each ...
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When the ring of regular functions is Artinian?

Let $K$ be a field, $W$ an algebraic variety over $K$ and $A(W)=\{f:W\rightarrow K \mid f \text{ regular}\}$ the ring of regular functions. When is $A(W)$ an Artinian ring? I know that $A(W)$ is a ...
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1 vote
1 answer
144 views

Is there an easy example of a variety that is not rational? ${}$

I saw a lot of examples of varieties which are rational, now I was also wondering if there was a simple example of a variety that is not rational. Sadly I didn't find that much online, is it maybe ...
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1 vote
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Globalisation of varieties

I am trying to understand the way to globalize the notion of variety. Let $\mathcal A$ be a quasi-coherent sheaf locally of finite type over an affine variety $X$. We define a variety $Y=\mathrm{Specm}...
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Pullback of a rational curve

Let $F$ and $H$ be curves, and $G$ be a finite group acting faithfully on $F$ and $H$. Denote $\pi$ the canonical projection from $F\times H$ on $(F\times H)/G$, and suppose that this morphism $\pi$ ...
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2 votes
1 answer
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Theorem 3.2, Chapter 1, of Hartshorne's Algebraic Geometry

This is regarding a question about Theorem 3.2 in Chapter I of Hartshorne's Algebraic Geometry. Let $Y \subset \mathbb{A}^n$ be an affine variety with affine coordinate ring $A(Y)$. The final item (d) ...
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  • 439
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How to show that a function is not a regular map?

I'm proving that if $f:\mathbb{C}\to\mathbb{C}$ is a bijective map that is not a polynomial. So, $f$ is continuous in the Zariski topology but that it is not a regular map. I have proven that it is ...
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Definition and example of regular maps.

In our class we’ve just got to regular maps, but things are a bit abstract and I can’t relate them to the examples. I’d like to ask how the notions are relevant to the example below. First, this is ...
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2 votes
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Demonstrating how to turn the affine plane into a double-cone with a sheet of paper, is it possible?

I am currently learning a little bit about algebraic groups and quotients of varieties with J. Harris book "Algebraic Geometry" (http://userpage.fu-berlin.de/aconstant/Alg2/Bib/...
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1 vote
1 answer
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Rational functions on an affine variety are completely determined by their values where they are defined

Say $V$ be an irreducible affine variety contained in $\mathbb{A}^n_k$, where k is algebraically closed Since, the coordinate ring $\Gamma(V)=\frac{k[X_1,X_2,...,X_n]}{I(V)}$ of $V$ is an integral ...
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7 votes
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What schemes correspond to varieties in the sense of Weil?

Out of (perhaps morbid) curiosity I am trying to learn the basics of Weil's foundations of algebraic geometry. I tried to ask a question earlier but it turned out I had misunderstood some more basic ...
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Irreducibility of planar curve

Let $K$ be a field, $f(X,Y)=Y^2-X^3-X^2\in K[X,Y]$ and $C=\mathcal V_f(K)$ the vanishing set of $f(X,Y)$. I'd like to prove that $C$ is irreducible. I can do it under assumption that $C$ has ...
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1 answer
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Morphism between a projective and an affine varieties has finitely many points in its image

I was asked to show that if $X$ is a projective variety, $Y$ an affine variety and $\varphi:X\rightarrow Y$ is a morphism, then $\varphi(X)$ is finite. I think that I have to use this fact: Let $X$ ...
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  • 139
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1 answer
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Show that the projections $\mathbb{A}^2\to\mathbb{A}^1$ are morphisms

I am trying to show that the projections $\mathbb{A}^2\to\mathbb{A}^1$ exhibit $\mathbb{A}^2$ as $\mathbb{A}^1\times\mathbb{A}^1$ in the category of affine algebraic sets, for which I have to first ...
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0 answers
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The multiplication map $\mathrm{GL}(n,k)\times\mathrm{GL}(n,k)\to\mathrm{GL}(n,k)$ such that $A\times B\mapsto AB$ is a morphism.

I have to show that the multiplication map $\mathrm{GL}(n,k)\times\mathrm{GL}(n,k)\to\mathrm{GL}(n,k)$ such that $A\times B\mapsto AB$ is a morphism of affine algebraic sets. I know that given affine ...
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1 answer
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Projections between two affine algebraic sets

What are the two projections $\mathbb{A}^2\to\mathbb{A}^1$, that exhibits $\mathbb{A}^2$ as a product $\mathbb{A}^1\times\mathbb{A}^1$ in the category of affine algebraic sets?
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Ideals contained in a field not algebraically closed are not comaximal

My question is: why must $b$ be $1$? Also why can't there be $ax+y^2$ as polynomials?
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1 answer
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Codimension of the singular locus.

I was studying Algebraic Geometry and I found the following result: If $X$ is a normal variety, the set of singular points $Sing(X)$ has codimension $\geq 2$. I understand this result and its proof, ...
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  • 1,463
1 vote
1 answer
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Showing that $\mathcal V(Y^2-X^2+X)\subseteq \mathbb A_k^2$ has dimension $1$

I just want to confirm whether my understanding of this topic is generally correct. $k$ is an algebraically closed field. So basically, the dimension $\dim \mathcal V$ of $\mathcal V$ is the dimension ...
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  • 2,136
4 votes
2 answers
347 views

What is the meaning of the residue field of a point in scheme?

If I consider the analogy of local ring at a point to the space of function germs at the point, then the residue field can be seen as the values that functions can take at the point. But when I ...
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1 vote
0 answers
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Parametrization and Affine Varieties

This is an exercise (exercise 3.3) question from the text "Ideals, Varieties, and Algorithms" by Cox, Little, and O'Shea. I have question about part c. Using Macaulay2 (web) I got the second ...
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1 vote
1 answer
202 views

Smoothness implies a condition on the Jacobian in every affine open

In chapter 12 of FOAG, Ravi Vakil defines smoothness in the following way A k-scheme is k-smooth of dimension d, or smooth of dimension d over k, if it is of pure dimension d, and there exists a ...
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1 answer
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removing one polynomial equation from irreducible affine variety keeps irreducibility?

Suppose I have an irreducible affine variety $X = \{ \mathbf{x} \in \mathbb{A}^n_{\mathbb{C}}: f_i(\mathbf{x}) = 0, 1 \leq i \leq m \}$ for some integer $m$. Let us define $X_j = \{ \mathbf{x} \in \...
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  • 135
1 vote
1 answer
63 views

Confusion regarding dimension of variety and rank of Jacobian

I am getting the contradiction described below. I have been racking my brain since yesterday to figure out what the problem is here, but I just can't. I must be making some super silly mistake. Please ...
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  • 57
2 votes
1 answer
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Showing that $\mathcal{V}(f,g) \subseteq \mathbb{A}_k^n$ has three irreducible components.

I was working on this following exercise and i would really like to see if i understood things correctly. Let $V:=\mathcal{V}(X^2-YZ,XZ-X) \subseteq \mathbb{A}_k^n$. Show that $V$ has three ...
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  • 2,136
0 votes
1 answer
99 views

Relation between dimension of a variety and the jacobian

Suppose I have an affine variety $V \in k^n$ and $I := I(V)$. Let $f_1,\dots,f_t$ be generators of I. For a point $P \in V$, define the jacobian $J_P(I) = \begin{pmatrix} \frac{\partial f_1}{\partial ...
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  • 57
1 vote
0 answers
152 views

Generators of the ideal of the projective closure

Let $X=\{(t,t^2,t^5): t\in k\}\subset \mathbb{A}^3$. Show that the projective closure $\bar{X}$ is a projective variety of dimension 1, and say if it is isomorphic to $\mathbb{P}^1$. Compute $\mathbb{...
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  • 43
0 votes
1 answer
38 views

Proving two affine varieties are equal

Need to show that $\textbf{V}(y-x^2,xz-y^2)=\textbf{V}(y-x^2,xz-x^4)$ in $\mathbb{R}^3$. I was trying to use the fact that $\textbf{V}(f,g)=\textbf{V}(f,g_1)\cup\textbf{V}(f,g_2)$ when $g=g_1g_2$. ...
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0 votes
0 answers
16 views

Solving algebraic variety instead of homotopy continuation methods

Consider following ODE system with polynomial nonlinearity \begin{equation} \dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \begin{bmatrix} 0\\ \vdots\\ 0\\ ax_i + bx_i^3 + ax_i + bx_i^5 \\ 0\\ \vdots\\ 0 ...
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