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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What is the dimension and nature of this variety?
Let $1 < N \in \mathbb{N}$ and $x, a \in \mathbb{C}^N$ with $a$ fixed; also, let $b \in \mathbb{R}_{\ge 0}^N$ be fixed (this last bit can be weakened to the extent it makes no difference). For $n \...
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Proof of Chevalley's theorem in Milne's book
I'm facing some difficulties into understanding the proof of the famous "Chevalley Theorem" in Milne's "Algebraic Geometry" (page 196)
A regular morphism $\varphi:W\to V$ between ...
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Application of ZMT to birational maps in Milne's Algebraic Geometry
I'm studying Algebraic Geometry from this book. At page 187 the author proves that
If $\varphi:W\to V$
is a regular birational quasi-finite map between irreducible varieties and $V$ is normal, then $\...
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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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Is Every Closed Algebraic Set of Dimension $n$ Contained in a Closed Variety of Dimension $n+1$
Let $V$ be an algebraic variety of dimension $m$ over an algebraically-closed field of characteristic $0$, and let $n<m$ and $U\subset V$ be a closed subset of $V$. Must there exist a subvariety $U\...
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Constructing intersection of two varieties modulo $p$.
Suppose I have a sequence $a_n$ given by $s_1^m+s_2^m$ modulo $p$. This takes on some values which are a subset of $\mathbb{Z}_p$. Suppose I also have a second sequence $b_n$ given by $t_1^m + t_2^m$.
...
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Dimension of product variety using finite morphisms
I've read the following proof for $\dim(X\times Y)=\dim(X)+\dim(Y)$ with $X,Y$ algebraic varieties. Because dimension is a birrational property we can suppose that $X,Y$ are affine. Now, Noether ...
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Potential typo in Fulton's "Algebraic Curves"
Could anyone verify whether the following is a typo? I'm studying William Fulton's "Algebraic Curves" and he's in the process of studying the relationship between affine space and ...
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Confusion about the definition of local ring of an affine variety at a point and its relation with localization
I am reading Gathmann notes on Algebraic Geometry. He defines the local ring of an affine variety $X \subset \mathbb{A}^n$ (affine variety means an irreducible algebraic set here) as follows
$$
\...
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Geometrical interpretation of $\operatorname{Spec}(\mathbb{R}[x,y]/(x^2+y^2))$
$\def\Spec{\operatorname{Spec}}$If am not mistaken, the prime spectrum of $\Spec(\mathbb{R}[x,y]/(x^2+y^2))$ consists of the points $(\overline{x},\overline{y})$, $(\overline{x}-a)$, $(\overline{y}-b)$...
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Dimension of open subset of affine variety [duplicate]
If $X$ is an irreducible affine variety and $U \subseteq X$ a non-empty and open (in $X$), then $\dim U = \dim X$.
The part $\dim U \le \dim X$ is clear, but I do not see how to prove the other part. ...
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Finding $I(V)$ for $V = V(x^2 + y^2 - z^2, 2z^2 - y)$
I'm trying to find the ideal of the affine variety $V(x^2 + y^2 - z^2, 2z^2 - y) \subseteq \mathbb{A}_{\mathbb{C}}^3$ in order to calculate its singularities. Is this some well-known variety or is ...
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Is left t-exactness of $Rf_\ast$ for $f : X\longrightarrow Y$ affine just Andreotti-Frenkel
Sorry I'm a little lost in those perverse sheaves. I think this left t-exactness property is called Artin-Grothendieck vanishing, which I take to be the relative version of $H^n(X,\mathcal{P})=0$ for $...
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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 ...
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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 ...
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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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Trouble understanding Hartshorne's Algebraic Geometry Exercise 2.2 (Chapter 1).
I am trying to solve the following exercise:
Let $\mathcal{a}$ be a homogeneous ideal such that $\mathcal a \subset
S = K[x_0,\dots,x_n].$ Show that the following affirmations are
equivalent:
$Z(a) =...
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Show that $\mathcal{I}(\bigcup_iX_i)=\bigcap_i\mathcal{I}(X_i)$.
Let $\mathcal{I}$ be the vanishing ideal. I understand that $\mathcal{I}(A\cup B)=\mathcal{I}(A)\cap\mathcal{I}(B)$ where $A,B\subset \mathbb{A}^n$. For any collection of subsets $X_i$ of $\mathbb{A}^...
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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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Irreducible components, dimension and degree of projective varieties
I have this problem given to me in my review session for my algebraic geometry final:
Describe the irreducible components and compute the degree and dimension of $V_p(x_0x_2-x_1^2, x_0x_3-x_1x_2)\...
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$X\times Y$ and the product topology [duplicate]
I'm studing Andreas Gathmann's notes on algebraic geometry (pdf here: https://agag-gathmann.math.rptu.de/de/alggeom.php). In chapter 4 (about Morphisms) he was using the universal property of products ...
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Birational morphism of the affine line
Let $k$ be an algebraically closed field. Suppose $K$ has characteristic nonzero, can we characterize the birational morphism of $A^1$ the way we characterize isomorphisms(all linear maps)? I know it ...
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Extending a map $\mathbb{A}^1-\{0\}\to \mathbb{P}^1$
I know this question has been asked a couple of times on this site, but both of the times $\mathbb{P}^1$ was regarded as a space of homogeneous coordinates and not as $\mathbb{A}^1\cup \infty$ and I ...
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$\mathbb Z$-graded coordinate ring $\iff$ $\mathbb C^*$ action on affine variety
Let $X$ be an affine variety with coordinate ring $\mathcal O$. I want to show that a $\mathbb C^*$ action on $X$ is equivalent to a $\mathbb Z$-grading on $\mathcal O$. I have problems with both ...
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$\mathscr{O}_X(X-Y)=\mathbb{K}[X]$ if and only if $\text{codim}_X(Y)\geq 2$ [duplicate]
I'm having an hard time in solving exercise 3.12 of Gathmann's algebraic geometry book:
Let $X$ be an affine variety such that $\mathbb{K}[X]$ is an UFD and let $Y\subseteq X$ be a non-empty ...
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regular functions on distinguished open sets in affine space
I read the algebraic geometry notes of Prof.dr.Andreas Gathmann. In his notes I find the following proposition but fail to understand.
I am confused with the paragraph which is below the equation $...
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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 ...
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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 ...
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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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For $f \in K[x_0,\ldots,x_n]$ then, $(\phi_i^{-1})^*(\phi_i^*(f)) | f$
For $f \in K[x_0,\ldots,x_n]$ then, $(\phi_i^{-1})^*(\phi_i^*(f)) | f$
Lets consider the dehomogenisation defined by:
$ \phi_i^*: K[x_0,\ldots,x_n] \rightarrow K[x_1,\ldots,x_n] $, such that, $\phi_i^*...
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Confusion with multiple definition of curves
I'm seeing more and more definitions of curves that I can't seem to match together. First define a curve as an algebraic variety of dimension 1. Now define a curve to be the zero set of some $f\in K[x,...
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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 ...
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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 ...
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What is the relation between an affine variety and an affine space?
I was learning about algebraic varieties and the wikipedia page presents affine varieties as the "conceptually easiest" type. Having read about affine spaces and affine varieties, I am ...
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On the height of the Jacobian ideal of the determinant of a square matrix of variables
Let $k$ be a field of characteristic $0$, let $\mathbf X=[X_{ij}]_{1\le i,j\le n} $ be a square matrix of indeterminates where $n\ge 2$. Consider the polynomial $f(\mathbf X)=\text{det}(\mathbf X)\in ...
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The definition for the morphism of varieties in Hartshorne's book.
I am reading the Hartshorne's algebraic geometry now, and by his definition for the morphism of varieties:
A continuous map $\varphi:X\to Y$, such that for every open set $V\subseteq Y$ and every ...
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Regular functions on distinguished open sets
I’ve been reviewing algebraic geometry by reading through A. Gathmann’s notes. Somehow I don’t understand few steps included in the following proof. I appreciate any help :) \
How does $V(f_a)=V(h_a)$...
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Whether a parameterized subset is affine algebraic
I am wondering whether a parameterized subset of an affine space is closed and is there a way to determine the ideal that generates it? For a simple example, let $k$ be algebraically closed and ...
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Examples of empty affine varieties
The nullstellensatz says that a system $S$ of polynomial equations $f_1=0,f_2=0,…f_n=0$, where $f_i$ are elements of a polynomial ring over a field $K[x_0,x_1,…]$, will have a non-empty affine variety ...
