# Questions tagged [affine-varieties]

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

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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $A=\{(... 1answer 29 views ### Does a closed point of a scheme have an affine open environment with the same dimension? Consider a scheme$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 ... 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. 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$x_1,\ldots,x_d$... 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 ... 0answers 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 ... 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 $are different. I'm ... 1answer 39 views ### Clarify on algebraic set contained in$\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)$cannot be generated by fewer than three elements. Call the three ... 0answers 37 views ### For$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))$. Clearly$\...
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 ...
### 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 ...