# Questions tagged [affine-varieties]

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

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### How to show that $U(f)$ is an affine variety [duplicate]

This question was asked in my assignment of Algebraic Geometry and I am struck on this. Question: Let $Y$ be an affine variety and $f\in A(Y)$. Let $U(f) = Y/Z(f)$. (a) Show that $U(f)$ is an affine ...
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### How can I prove $M+t$ is a hyperplane if $M$ is a maximal subspace

Let $M$ be a non-empty proper subset of a vector space $X$ over $\mathbb R$ and $t$ belongs to $X$, then $M$ is a maximal subspace if and only if $t+M$ is a hyperplane and $t$ belongs to $t+M$.
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### Chow's Lemma proof in Milne's Book: Cartesian diagram

I'm reading the proof of the following statement in Milne's book "Algebraic geometry". Chow's Lemma: Let $V$ be a complete irreducible variety. There exists a projective variety $V'$ and a ...
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### Chow's Lemma proof in Milne's Book: Locality of immersions.

I'm reading the proof of the following statement in Milne's book "Algebraic geometry". Chow's Lemma: Let $V$ be a complete irreducible variety. There exists a projective variety $V'$ and a ...
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### Does the completions $\hat{\mathscr{O}}_P(X)\simeq \hat{\mathscr{O}}_Q(Y)$ could deduce local rings $\mathscr{O}_P(X)\simeq \mathscr{O}_Q(Y)$?

In Hartshorne's algebraic geometry, he said the completion of local ring $\hat{\mathscr{O}}_P(X)$ takes much more 'local properties' than the local ring $\mathscr{O}_P(X)$. There are two natural ...
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### Is every affine variety an affine scheme? [duplicate]

I have seen that the notion of affine scheme is a generalization of the notion of affine varieties where the coordinate ring is replaced by any commutative unit ring, and the variety with the Zariski ...
1 vote
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### Is a morphism of varieties determined by its behaviour on an open subset

Let $\varphi:X\to Y$ be a morphism of varieties and let $X$ be irreducible. If $U\subseteq X$ is a non-empty open subset, is it true that $\varphi$ is the only extension to $X$ of $\varphi|_U$? I know ...
1 vote
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### Show that the equation of conic in $\mathbb{P}^2$ can we written as $ax^2+by^2+cz^2=0$.

Let $K$ be a field with characteristic not equal to $2$. We define a conic $C$ to be a subvariety of $\mathbb{P}^2_K$ defined by a homogeneous equation of degree $2$. I need to show that the equation ...
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### Checking smoothness of curves and finding multiplicites.

I am asked to check whether or not this curve is smooth (and if not provide singular points): $x_2^2x_0 = x_1^3 - x_1x_0^2$. The way I approached this was by to use the projective Jacobi criterion on ...
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### Dimension of an algebraic variety defined by two polynomials

This is a basic question, but the usual definitions of algebraic sets seem quite complicated for a non-specialist. I wanted to confirm if the following is correct: if we have two nontrivial ...
1 vote
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### Subvarieties of an abstract affine variety

I'm really confused on how to transfer constructions from "concrete" affine varieties (i.e. zeroes of polynomial equations in an affine space) and (abstract) affine varieties (i.e. ringed ...
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### How is the product of abstract affine varieties defined?

I'm studying Algebraic Geometry from Gathmann's Notes and I don't find really satisfying how the product of abstract affine varieties is defined. Page 34 (Gathmann's Notes) "[...] products ...
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### Definition of 'gluing two affine curves'

Let $C_0:$ be a projective closure of affine curve $y^{2}=x^4-7$. $C_0$ has singular point $[0:1:0]$ at infinity. Let another affine curve be $C_1: v^{2}=u^{4}(1/u^4-7)=1-7u^4$. To make smooth ...
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### Intuitive reason for why $\operatorname{Spec}(k[x,y,z]/(x-yz))$ is smooth at $O$ while $\operatorname{Spec}(k[x,y,z]/(x^2-yz))$ isn't?

Let $k$ be an algebraically closed field and consider the closed subschemes $X=\operatorname{Spec}(k[x,y,z]/(x-yz))$ and $Y=\operatorname{Spec}(k[x,y,z]/(x^2-yz))$ of $\mathbb{A}^3$. Then the gradient ...
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### Smoothness of affine variety consisting of finitely many points

Let $R$ be a commutative ring such that $X:=\mathrm{Spec}(R)=\{p\}$, where $p$ is a prime ideal. Then, obviously, the affine variety $X$ consists of one point only. However, can we decide if this ...
1 vote
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### Qing Liu Proposition 2.5.5, $\dim X=\sup\{\dim_x X \mid x \in X\}$

I've been trying to understand Liu's proof of the statement contained in Proposition 2.5.5 (d): "for X topological space, we have $\dim X=\sup\{\dim_x X \mid x \in X\}$." He basically ...
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### germ at origin: plane nocal cubic and x,y-axis

I am reading I.5 of Hartshorne's AG. In the example I.5.6.3, he explains how "completion" helps with analyzing the local structure of varieties by demonstrating two examples of completion: ...
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### rank of jacobian matrix of generators of ideal of affine variety is the dimension as a subspace of k^n

The question comes from the proof of Theorem I. 5.1 of Hartshorne's AG. Here I gave the theorem and the confusing part of the proof: Theorem 5.1 Let $Y\subseteq \mathbb{A}^n$ be an affine variety. Let ...
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### About the limit in affine line $\mathbb{A}^1$.

When I am reading the answer Example of a non extendable rational map, there is a point which made me confused. That is $\lim_{t\to 0}tx=(\lim_{t\to 0})x$, where $t,x\in\mathbb{A}^1=\mathbb{K}$, under ...
1 vote
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### The limit in affine (projective) space.

I have read an answer of Example of a non extendable rational map, which showed an example of non extendable rational map. I noticed that there is a limit $\lim_{t\to 0}(tx,ty,1)=(0,0,1)$ in affine ...
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### About the definition of the morphisms of varieties in Hartshorne's algebraic geometry.

I am reading Hartshorne's algebraic geometry. According to the definition about morphism of varieties in Hartshorne's book, a morphism $\varphi:X \to Y$ should not only be continuous but also preserve ...
1 vote