Questions tagged [affine-varieties]

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

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23 views

Proof that finite fields cannot be pseudo-algebraically closed

I'm studying pseudo-finite fields in particular on Chatzidakis notes. When dealing with the concept of pseudo-algebraically closed (PAC) fields, it is stated that they cannot be finite but without any ...
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107 views
+200

Isomorphism coordinate ring affine variety and regular functions

Hartshorne presents this result in Theorem I.3.2, but the proof is too much algebraic for me. I would like to really understand the reason of this isomorphism. So, suppose I consider an element $f\in\...
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1answer
112 views

Decomposition of the algebraic variety into irreducible components

I want to decompose the algebraic variety $v(\mathfrak{a}) \subset \mathbb{A}^3(\mathbb{C}),$ where $\mathfrak{a}$ is an ideal, $\mathfrak{a} = (xy+yz, x^3y^3 + x^2y^2),$ into irreducible components, ...
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38 views

Solving dimension of an affine algebraic variety.

By dimension, I used this definition: algebraic set $V$ has dimension d if maximum length of chains $V_0\subset V_1 \subset\cdots\subset V_d$ is $d$ where $V_i's$ are irreducible subvariety of $V$, ...
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0answers
79 views

Topologically unibranch in Mumfords book Complex Projective Varieties

David Mumford gives in his book Algebraic Geometry I, Complex Projective Varieties on page 43 the definition of topologically unibranch points of affine variety and I have a lot of problems to extract ...
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0answers
28 views

Maximal ideal in a polynomial field [duplicate]

I have been asked to prove that if $k$ is any field, then any maximal ideal in $R=k[x_1,x_2,\cdots,x_n]$ is generated by $n$ elements. I can easily see that the ideal generated by $x_1-a_1, x_2-a_2,\...
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1answer
60 views

Show a function is not regular on a variety

Let $V$ be the variety defined by the polynomial $x_1^2 + x_1^3 -x_2^2$ over an arbitrary field $F$, and consider the function $\phi = \frac{x_2}{x_1}$. I want to show that this function is not ...
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0answers
78 views

How can I prove that affine hypersurface $V(X^2 + Y^5+Z^5 + 1) \subset \mathbb{A}^3$ is not rational?

$(*)$ I would like to prove that $Spec(\mathbb{C}[X,Y,Z]/\langle X^2+Y^5+Z^5+1 \rangle)$ is not rational (or equivalently that $Proj (\mathbb{C}[X,Y,Z,T]/\langle X^2 T^3+Y^5+Z^5+T^5 \rangle)$ is ...
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1answer
36 views

Integral extension of a discrete valuation ring

$X$ and $Y$ are integral noether schemes over $\mathbb{C}$, and $F:X\rightarrow Y$ is a surjective morphism. Let $R$ be any discrete valuation ring over $\mathbb{C}$ with its fraction field $K$, and $...
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0answers
16 views

Reformulation of a point in the closure of an orbit

I'm trying to lemma 3.3 of this paper : https://web.northeastern.edu/iloseu/Barbara_14_complete.pdf I don't really know how to reformulate in a usable way the fact that we have $y \in X$ such that: \...
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76 views

Hartshorne's Ex. 3.15 (chap. 1) Product affine varieties

So I am trying to prove that the product of two affine varieties is the categorical product in the correspondent category. If $X$ and $Y$ are two affine varieties on $K^n$ and $K^m$ resp., then $X\...
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36 views

Proof of a simple statement about algebraic sets and their coordinate ring

Let $X\subset \mathbb A^n$ be an algebraic set, and let $A$ and $B$ be its two connected components. There is a theorem stating that $$\Gamma (X)\cong\Gamma(A)\times \Gamma(B).$$ I will indicate the ...
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0answers
38 views

Why tame automorphisms are important?

It is known by Jung--van der Kulk theorem that any polynomial automorphism of $\mathbb{A} ^2$ is tame. It is known that in dimension 3 there are wild automorphisms. My question is what are the good ...
2
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1answer
46 views

Definition of variety in Silverman's AEC

In Silverman's AEC, p3, he defines a variety as the following: An affine algebraic set $V$ is called a variety if $I(V)$ is a prime ideal in $K’[X]$, where $K’$ denotes an algebraic closure of $K$. ...
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2answers
67 views

affin line $\mathbb A_{R}^{1}$ is not homeomorphic to real line $\mathbb{R}$

I want to show affin line $\mathbb A_{R}^{1}$ is not homeomorphic to real line $\mathbb{R}$(the topology is euclidean topology). I think the topology on $\mathbb A_{R}^{1}$ is just cofinite topology, ...
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1answer
33 views

On an inequality regarding fiber dimension of flat families

I'm trying to prove that if $\pi : V \to W$ is a flat familiy between affine algebraic varieties, with $V$ irreducible and $W$ of pure dimension, then $$\dim V_q = \dim V-\dim W$$ for any non-empty ...
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1answer
81 views

Vakil 3.2.10: Understanding the induced map $\operatorname{Spec} B\to \operatorname{Spec} A$

I'm having trouble understanding the following example from Vakil's "The Rising Sea": For example, consider a map from the parabola in $\mathbb{C}^2$ (with coordinates $a$ and $b$) given by ...
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1answer
52 views

Projection and dominant morphisms with finite general fibers

$\newcommand\C{\mathbb C} \newcommand\A{\mathbb A}$My base field throughout is $\C$. Let $m<n$ and suppose I have dominant morphisms $$f:\A^m \to \A^m \qquad g:\A^n \to \A^n$$ with finite general ...
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50 views

How do I check the primeness of ideals of a finitely generated algebra over a field?

In other words, how do I check whether the algebraic set given by a set of polynomial equations is irreducible? For example, I encountered the following example when working through Vakil's notes on ...
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0answers
27 views

Chart computation in weighted projective space [duplicate]

I'm confused about a relatively simple algebraic geometry computation. Let's take the affine plane curve $x^3-y^2=1$; this is an open subset of an elliptic curve. Indeed, if we embed affine space into ...
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1answer
63 views

Proving a multivariate polynomial is zero in $N$ easy steps. But what is $N$?

This question came about because multiplying polynomials in Magma is slow. Hence if I want to check that a matrix with linear coefficients in a (multivariate) polynomial ring cubes to zero, I just ...
2
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1answer
82 views

How can I understand a hypocycloid as an ideal in a polynomial ring?

Today I was reading On teaching mathematics by V. I. Arnol’d and came across the following quote. "Rephrasing the famous words on the electron and atom, it can be said that a hypocycloid is as ...
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2answers
120 views

The projective closure of the twisted cubic curve

I'm now reading Hartshorne's Algebraic Geometry and trying to solve Exercise 2.9(b). Let $Y$ be an affine variety in $\mathbb{A}^n$. Identifying $\mathbb{A}^{n}$ with the open subset $U_0$ of $\...
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1answer
26 views

Projective Closure of Morphism of Affine Varieties

Every affine variety $V$ has a unique projective closure $\overline{V}$, and there is an injective morphism $\iota : V \rightarrow \overline{V}$ given by something like $\iota(X_1,\ldots, X_n) = (X_1, ...
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1answer
128 views

How to understand group action especially Galois action on a scheme?

I read a lot of books and find none of them give explicit descriptions of group action on schemes. I am very confused now and have lots of questions. So I think these questions will be relatively long ...
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0answers
24 views

Exceptional Set vs Tangent Cone

Let $k$ be an algebraically closed field, $R=k[x_1,\ldots,x_r]/J$ for some ideal $J$, $X=Z(J)\subseteq\mathbb{A}^r$, and $I=(x_1,\ldots,x_r)$. I'm following Eisenbud's Commutative Algebra with a View ...
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1answer
45 views

Suppose that $X$ is an affine variety and $U$ is an open subset of $X$.Suppose that $p\in U.$ Show that $\mathcal O_{U,p} =\mathcal O_{X,p}$.

I am a beginner in algebraic geometry. I encountered such a question in the book which I read. Suppose that $X$ is an affine variety and $U$ is an open subset of $X$ (so that $U$ is a quasi-affine ...
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1answer
31 views

an irreducible affine curve is normal if and only if it is nonsingular

An is normal if and only if it is nonsingular. This statement comes from Kemper, A Course in Commutative Algebra. He says to use Proposition 8.10 and Theorem 14.1. Theorem 14.1. A Noetherian local ...
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0answers
51 views

The twisted cubic curve in $\mathbb P^3$ [duplicate]

I am trying to solve exercise 2.9 b from section 1.2 of Hartshorne algebraic geometry. It asks to find out the generator of $I(\bar Y)$ where $\bar Y$ is the projective closure of the twisted cubic ...
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1answer
63 views

Problem in proving a statement regarding projective closure of an affine variety.

In problem $2.9$ of Hartshorne section $1.2$, he defined projective closure of an affine variety. Let $Y\subset \mathbb A^n$ be an affine variety, let $\phi : U_0 \rightarrow \mathbb A^n$ be the ...
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1answer
32 views

Clarification about Ideal and zero sets of empty set in Varieties

While defining affine and projective varieties we consider Zariski topology on $\mathbb A^n$ and $\mathbb P^n$. In the process we define $ Z(T)$, zero set of $T $ where $T\subset A=k[x_1,...,x_n]$ and ...
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1answer
283 views

How to show that a prime ideal of height 2 can’t necessarily be generated by 2 elements? (Hartshorne exercise I.1.11) [duplicate]

In Hartshorne section 1.1 he gives a problem (ex 1.11) which says that, Let $Y \subset \mathbb A^3$ be the curve given parametrically by $x=t^3, y=t^4, z=t^5$. Show that $I(Y)\subset k[x,y,z]=A$ is ...
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1answer
30 views

Doubt of selecting only affine algebraic sets for defining affine coordinate ring

In Hartshorne section $1.1$ he defines affine coordinate ring to be $A/I(Y)$ where $Y\subset \mathbb {A}^n$ is an affine algebraic set and $A=k[x_1,...,x_n]$. My question is why it involves only ...
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0answers
24 views

Finding representative of a rational function on an affine variety to tell if given point is a pole

Suppose $X$ is an irreducible affine variety (i.e. embedded in some affine space) with coordinate ring $\Gamma(X)$. Suppose $\phi$ is a nonzero rational function on $X$. Let $p$ be a point on $X$. Is ...
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1answer
72 views

Show that $\sqrt{I}:J=\bigcap_{P\in M}P$

If $I,J$ are ideals in a ring $R$, the colon ideal is $$I:J=\{a\in R\mid ab\in I\text{ for all } b\in J\}.$$ (a) Set $M=\{P\in\mathrm{Spec}(R)\mid I\subset P\text{ and } J\not\subset P\}$. Show that ...
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1answer
60 views

Smooth morphism of relative dimension $>0$ and fiber bundle

Let $f:X\to Y$ be a smooth morphisms of relative dimension $>0$ of two smooth (affine) varieties over $\mathbb{C}$. I wonder if the corresponding holomorphic map of the corresponding complex ...
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1answer
85 views

Tangent space of a union of closed affine varieties

Suppose $X_{1}, X_{2} \subseteq \mathbb{A}^{n}$ are closed affine subvarieties such that $p:=(0,\dots,0) \in X_{1}\cap X_{2}$. Define $X := X_{1} \cup X_{2}$. I have shown the following inclusions ...
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1answer
29 views

Irreducible bivariate complex polynomial whose zero-locus contains two given points

Let $\alpha,\beta\in\mathbf{C}^2$ be distinct pairs of complex numbers. Is there an irreducible polynomial $f\in\mathbf{C}[x,y]$ vanishing at $\alpha$ and $\beta$?
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1answer
42 views

The function field of an affine part of a projective variety

Let $\phi:\mathbb{A}^n \to U_0\subseteq \mathbb{P}^n$ be given by $\phi(a_1,\ldots,a_n)=(1:a_1:\ldots:a_n).$ Let $X\subseteq \mathbb{P}^n$ be an irreducible Zariski-closed subspace (I call this a ...
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1answer
81 views

How to determine the local ring $\mathcal{O}_{X,p}$ of the variety $X=V(xy)\subset\mathbb{A}^2$ at $p=(0,0)$.

As in the title I am trying to solve the following execrcise: Compute the local ring $\mathcal{O}_{X,p}$ of the variety $X=V(xy)\subset\mathbb{A}^2$ at $p=(0,0)$. Show it is isomorphic to $A=\{(...
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1answer
29 views

Does a closed point of a scheme have an affine open environment with the same dimension?

Consider a scheme $X$, and an a closed point $x\in X$. I am wondering whether there is an affine open neighborhood $x\in U\subseteq X$ such that $$\dim \mathcal O_{X,x}=\dim U.$$ I tried the ...
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0answers
43 views

smooth morphisms and sheaf of differentials

Our course defined a smooth morphism in this way I'd like to know why this implies that the sheaf of differentials $\Omega_{X/S}^1$ is locally free of finite rank. Thanks in advance.
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1answer
80 views

Geometric Intuition of Systems of Parameters

I'm currently reading Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry, and he defines a system of parameters in Section 10.1. He says that, geometrically, if $x_1,\ldots,x_d$ ...
1
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1answer
77 views

$\mathcal{O}_X(U)=\mathcal{O}_X(X)=A(X)$ for any open subset $U$ that is the complement of an irreducible variety of codimension at least $2$

I am trying to solve Exercise 3.15 in Gathmann's 2014 notes. Given an irreducible affine variety $X$ such that $A(X)$ is UFD und $U$ like in the heading I want to show that the sheaf of regular ...
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41 views

Finding implicit equations for an affine subspace (“Ideals, Varieties and Algorithms” by Cox et al.)

I'm having trouble understanding the formulation of the following concept: in the second chapter, the question of implicitisation is posed, i.e. going from a system of parametric equations to implicit ...
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0answers
110 views

Zariski topology and product topology

I'm pretty new to algebraic geometry; I have not clear what does it mean that the Zariski topology and the product topology over the affine space $\mathbb A^m × \mathbb A^n $ are different. I'm ...
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1answer
39 views

Clarify on algebraic set contained in $\mathbb A^3$

Let $X ⊂ \mathbb {A^3}$ be the union of the three coordinate axes. Determine generators for the ideal $I(X)$, and show that $I(X)$ cannot be generated by fewer than three elements. Call the three ...
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37 views

For $Y_1,Y_2$ closed subsets of $\mathbb{A}_k^n$, $\mathcal{I}(Y_1 \cap Y_2)= \text{rad}(\mathcal{I}(Y_1)+\mathcal{I}(Y_2))$ [duplicate]

Let $Y_1,Y_2$ are two (Zariski) closed subsets of $\mathbb{A}_k^n$, for any field $k$. Then how do I show that $\mathcal{I}(Y_1 \cap Y_2)= \text{rad}(\mathcal{I}(Y_1)+\mathcal{I}(Y_2))$. Clearly $\...
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0answers
35 views

Field of definition of a polynomial ideal which is preserved under a Galois action

Let $k$ be a perfect field and $\bar{k}$ its algebraic closure. Let $H$ be a normal subgroup of $Gal(\bar{k}/k)$ of finite index. Let $L$ be the subfield of $\bar{k}$ of all elements which are fixed ...
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0answers
56 views

Degree of morphism between affine varieties $= \#$ preimages in general fiber

Let $f: V \dashrightarrow W$ a rational dominant map between two affine algebraic $k$-varieties $V \subset \mathbb{A}^t, W \subset \mathbb{A}^s$ of same dimension. By category equevalence this is ...

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