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Questions tagged [affine-varieties]

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

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Coordinate ring of an open affine variety?

I have trouble understanding what the coordinate ring of an open affine variety, i.e. the ring of regular functions on that variety, is (in the classical sense). Let $k$ be algebraically closed. If $X ...
Gargantuar's user avatar
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Weight of a monomial

I have a question for the mathematicians in affine algebraic geometry: Given an algebraically closed field $k$, we define the projective $n$-space as the quotient space $\mathbb{P}^n = (k^{n+1} - \{0_{...
Mousa hamieh's user avatar
4 votes
1 answer
110 views

Making clear the definition of 'affine variety' in Mumford's book.

I am reading "The Red Book of Varieties and Schemes" by Mumford. In section 4 the author defines the term affine variety: An affine variety is a topological space $X$ plus a sheaf of $k$-...
Toni's user avatar
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1 answer
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Understanding why set theoretic intersection is not necceraly a complete intersection

If it is true that for projective varieties one can show that: $Z(f_1, f_2) = Z(f_1)\cap Z(f_2)$ for any homogenous polynomials, than why isn't true than any set theoretic complete intersection of ...
Joe's user avatar
  • 470
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rational map by F. Mangolte

I'm reading Real Algebraic Varieties by F. Mangolte. Definition 1.3.22 (in the book) If $X$ and $Y$ are algebraic varieties over a base field $K$ a rational map $\phi:X\dashrightarrow Y$ is an ...
isz's user avatar
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2 votes
1 answer
70 views

Remark in Atiyah, Macdonald: non-singularity $\Rightarrow$ analytic irreducibility

In Atiyah, Macdonald, Introduction to Commutative Algebra, Chapter 11, p. 124, after Proposition 11.24, there is a Remark. It follows from what we have said above that A is also an integral domain. ...
Elías Guisado Villalgordo's user avatar
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18 views

General way to determine whether a subset of a vector space is an affine space.

Given some implicit equations or a definition of a subset of a vector space, what is the requirement for it to define an affine subspace. I'm looking for something analogous to the test for vector ...
RealityIsSenseless's user avatar
1 vote
2 answers
74 views

Is $\textit{affine space}$ the same as $\textit{quotient space}$?

From the answer for this question, I understand that affine subspace is the same as affine subset, however (despite the somewhat misleading question's title), it doesn't say that affine space is the ...
Tran Khanh's user avatar
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28 views

Quasi-affine variety with non finitely-generated global section

Let $U\subseteq\mathbb{A}^{3}$ be $V\left(xy\right)\setminus V\left(x,z\right)$. I'm trying to show that $U$, which is quasi-affine, has a non finitely-generated global section $\mathcal{O}_{U}\left(U\...
Oria's user avatar
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1 answer
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Morphism of varieties is continuous between analytic varieties.

I'm seeking clarification on the significance of commutative diagrams in understanding the analytic topology of smooth varieties. In Aleksander Horawa's notes, a commutative diagram is used to ...
ben huni's user avatar
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Fact checking: are these inclusion relations regarding algebraic varieties of polynomial ideals correct?

I'm studying inclusion identities within polynomial ideals theory. More precisely, i'm interested in the correspondece of an ideal $I\subseteq \mathbb{F}[\vec{x}]$ and its associated affine variety $\...
Simón Flavio Ibañez's user avatar
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Dominant rational maps and dimension of affine varieties

Say we have $f : X \dashrightarrow Y$ as a dominant rational map between two affine varieties. Is it necessarily true that $\dim Y \leq \dim X$? $f$ being dominant means that we must have that $f(\...
Jeff's user avatar
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Existence of affine varieties and regular map solving functional equation for polynomials

I am wondering about the existence of projections from a multivariate polynomial $p\in\mathbb{R}[x_1,...x_n]$ to a polynomial $q\in\mathbb{R}[t]$ in the sense that there is a map $\phi:\mathbb{R}\to\...
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Zariski-density on almost diagonal embedding

It is not hard to see that the Gaussian integers $\mathbb{Z}[i]$ are Zariski-dense inside $\mathbb{C}$, seen as an affine space over $\mathbb{C}$. Consider now the set $$D = \{(z,\overline{z}) \in \...
Henrique Augusto Souza's user avatar
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What are the conditions for a function $f\colon X \to Y$ to be regular when $X$ and $Y$ are prevarieties?

I'm currently following A. Gathmann's Algebraic Geometry, chapter 5 - varieties. I've seen the concept of prevarieties (ringed spaces with finite covers of affine varieties) and I'm stuck with the ...
Lucas Henrique's user avatar
1 vote
1 answer
90 views

A question on isolated singularity involving analytic varieties.

Suppose $V\subset \mathbb{D}^2$ ($V$ is a subset of the unit bidisc) is a set. Also, suppose that the following conditions hold: If $\hat{V}$ is the polynomial convex hull of $V$, then $\hat{V}\cap \...
Anindya Biswas's user avatar
1 vote
3 answers
121 views

How to show that $(y-x^2, z-x^3) \in \mathbb{C}[x,y,z]$ is irreducible or radical?

In Fulton's introduction to Algebraic Geometry, there is the following exercise on page 11: I have been struggling for a bit longer than I care to admit on this problem, and have not been able to get ...
Nikolas Koutroulakis's user avatar
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0 answers
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Relation between $(A/I)^G$ and $A/I^G$ for a group action $G\curvearrowright A$

Let $A$ be a commutative ring and $I\subset A$ be an ideal. Moreover, for a group $G$, we are given group action $G\curvearrowright A$ such that $g(I)=I$ for any $g\in G.$ Then, naturally $G$ acts on $...
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classification of morphism between varieties

Let $K$ be an algebraically closed field. We have different important theorems to study morphism of variety, in particular there is a clear and constructive description of the morphism between affine ...
Mario's user avatar
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1 answer
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Hausdorff-ness of irreducible affine algebraic sets [closed]

In what case is $X$, an irreducible affine algebraic set not a Hausdorff topological space with respect to the Zariski topology? My understanding is that this would only occur if $X$ is infinite, ...
Jeff's user avatar
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1 answer
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Prove $S = \{ (a,b) \in \mathbb{A}^2 \ | \ a \overline{a} + b \overline{b} = 1\}$ is not Zariski closed

I'm working on a problem where I have to prove that Consider the subset $S$ of $\mathbb{A}^2$, the affine space over $\mathbb{C}$, defined by $$ S = \{ (a,b) \in \mathbb{A}^2 \mid a \overline{a} + ...
Oopsilon's user avatar
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At which points of $X=V(x^2+y^2-1)$ is the rational function $\frac{1-y}{x}$ regular?

Let $X \subseteq \mathbb{A}^2$ be the circle of equation $x^2+y^2=1$. I have to compute the points of $X$ such that $f=\frac{1-y}{x}$ is regular. (We can suppose $\operatorname{char}(K)\neq 2$) I ...
Mario's user avatar
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0 answers
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Exercise about fibre product of varieties

I am doing exercise $5.5.9$ of these lecture notes. I have produced my own solution but it just seems to trivial and there are possibly some mistakes in my reasoning. I would very much appreciate if ...
kubo's user avatar
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Regular functions on an algebraic variety

My question is related to Theorem 4.1.8 from these lecture notes. Let $X$ be an algebraic variety (you can check the definition of algebraic variety I am using in Definition 4.3.4). Let $x\in X$ be a ...
kubo's user avatar
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2 votes
0 answers
114 views

Miracle flatness on Wikipedia's "Cohen--Macaulay ring"

In Wikipedia's article on Cohen–Macaulay rings, the following geometric version of miracle flatness is stated, see this link: Let $X$ be a connected affine scheme of finite type over a field $K$ (for ...
Daniel W.'s user avatar
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1 vote
2 answers
67 views

Questions about singularity of a projective cubic in the projective plane.

I'm studying algebraic geometry, in particular the singularity of certain affine and projective varieties and differential forms in said sets. In particular I have, in $P^2$ (over complex numbers) ...
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1 vote
1 answer
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Extension of a morphism of affine varieties $f: X \to Y$ to $\overline{f}: \mathbb{A}^m \to \mathbb{A}^n$

I am attempting Exercise 5.5.7 from this lecture notes. Let $X \subset \mathbb{A}^m$ and $Y \subset \mathbb{A}^n$ be closed subsets and a morphism of varieties $f: X \to Y$, extend $f$ to a morphism ...
Mystery girl's user avatar
1 vote
1 answer
116 views

$ (\mathbb{P}^n \times \mathbb{P}^n) \setminus \Gamma$ is isomorphic to an affine variety

We denote $\mathbb{P}^n$ to be the standard projective space over $\mathbb{C}$. Define $$\Gamma = \{ ([p_0 : ... : p_n] , [q_0 : ... : q_n]) \in \mathbb{P}^n \times \mathbb{P}^n | \sum_{i = 0}^n p_i ...
Steve's user avatar
  • 184
0 votes
1 answer
71 views

compute Matrix relative to monomial Basis

I have the radical Ideal of $\mathbb{C}$ [x,y,z] generated by $\{x - 3 y - z + 9, z^2 -3z + 2, yz -2y - 3z + 6, y^2 - 5y + 6\}$ which form a Gröbner basis with TO deglex (x>y>z) and I want to ...
Heidehexer's user avatar
1 vote
1 answer
79 views

If $Y$ is a projective variety and $Y \cap \mathbb{V}(g) = \emptyset$, then $Y$ is affine

In this question, I denote $\mathbb{P}^n$ for the standard $n$-dimensional projective space over $\mathbb{C}$. Also, with a variety I always mean a Zariski closed subset. I am currently following an ...
Steve's user avatar
  • 184
0 votes
1 answer
110 views

Show product of projective spaces minus a hypersurface is affine

Let $L=\{([a_{0}:a_{1}:....:a_{n}],[b_{0}:b_{1}:...:b_{n}]) \in P^{n}* P^{n}: \sum_{i}a_{i}b_{i}=0\}$ Show $P^{n}*P^{n}-L$ is affine. I don't know where to start. I only know that if we use a Veronese ...
Kevin's user avatar
  • 1
0 votes
0 answers
25 views

Existence of function in coordinate ring vanishing at a point

I am dealing with the following problem. Let $X$ be an affine variety of positive dimension. For a point $x \in X$, there exists a nonzero function $f \in A(X)$, where $A(X)$ is the coordinate ring ...
GendoTendoLendo's user avatar
2 votes
1 answer
82 views

Confusion about $k$-algebra homomorphisms from $k[W] \to k[V]$.

I am currently reading chapter $15.1$ in Dummit & Foote:s "Abstract Algebra", 3rd edition (the passage below is from page $662$). Before I get to why I am perplexed, note that $V \subset ...
Ben123's user avatar
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4 votes
1 answer
295 views

Why is the noetherian ring property not definable in first-order logic?

I am reading this paper on the connection between model theory and algebraic geometry. https://math.uchicago.edu/~may/REU2015/REUPapers/Zhang,Victor.pdf On page 9, I have trouble understanding Example ...
Y.X.'s user avatar
  • 4,203
2 votes
1 answer
51 views

Simpler way to express the affine variety V(f, g, h) in terms of union and intersection of varieties

Is it correct that we can write the affine variety $V(xy, xz, yz)$ in $\mathbb{R}^3$ as $$[V(x) \cup V(y, z)] \cup [V(x, y) \cup V(z)]?$$ Is there another way to write this in order to better imagine ...
user avatar
0 votes
0 answers
39 views

Maximal ideal of a point is principal

Let $X$ be an affine variety defined by a smooth curve $F=0$ on $\mathbb{A}^2$. Let $p=(a,b)$ be a point on $X$. Now $m_p$ be the maximal ideal generated by $x-a,y-b$. Then why $$F_X(p)(x-a)+F_Y(p)(y-...
Soham Chatterjee's user avatar
1 vote
0 answers
46 views

Is open always Zariski open in the context of Algebraic geometry, unless otherwise mentioned?

Going through Smith et al.'s Invitation to Algebraic Geometry I sometimes find myself wondering whenever they use the word open, do they mean open in the usual sense on $\mathbb{A}^1$ for example, or ...
Neckverse Herdman's user avatar
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0 answers
24 views

How to prove that $\phi (x,\frac{1}{x}) = x$ and $ \psi( x) = (x,\frac{1}{x}) $ are morphisms of varieties?

Hartshorne say that a $ \phi : V \to W$ is a morphism of variety if it is continuous and for every regular function $f$ on $W$, the map $ f \circ \phi^{-1} $ is also regular function. Using this ...
LearningMath's user avatar
1 vote
1 answer
59 views

$\tilde f_i = f_i + g_i$, then $\dim \mathbb C[t_1,\dots, t_d]/\langle f_i \rangle \leq \dim \mathbb C[t_1,\dots,t_d]/ \langle \tilde f_i \rangle$

Let $\mathbb C[t_1,\dots, t_d]$ be the multivariate polynomial ring with a certain monomial order $\leq$. I will denote monomials by $x^\alpha$, where $\alpha \in \mathbb N^d$. For $f = \sum_\alpha a_\...
user2345678's user avatar
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0 votes
1 answer
54 views

The plane in the quadric 3-fold is not a (set-theoretic) hypersurface

I think my question has a top-bottom answer, but as of yet I am not familiar enough with divisors and class groups to be sure of what I am claiming. I also include an "elementary answer" to ...
Andrei.B's user avatar
  • 1,005
1 vote
0 answers
110 views

Tangent space of a quotient of an algebraic variety

A word of warning: I have no background in algebraic geometry, so please excuse my ignorance. References welcome (but please refrain from saying things like "read Hartshorne's Algebraic Geometry, ...
Margaret's user avatar
  • 1,769
0 votes
1 answer
98 views

how to prove that a set is not algebraic

Consider the Zariski topology on $\mathbb{C}^2$. How can I prove that a set is not an algebraic variety? For example let $Y=\{(z_1,z_2)\in \mathbb{C}^2| \text{Im}(z_1) \cdot \text{Im}(z_2)>0\}$ be ...
Mario's user avatar
  • 739
0 votes
0 answers
37 views

Varieties with parameters

This seems like a very basic question to me and I am certain that people studied it a lot. It is for sure related to deformation theory and families of varieties, but I am not sure how these fields ...
Daniel W.'s user avatar
  • 1,780
2 votes
0 answers
62 views

Find $\dim_k(m/m^2)$ in a local ring without geometric interpretation.

I have the following question: Let $V$ be an irreducible affine variety over an algebraically closed field $K$. For any point $p \in V$, the ring $P(V) := ${$f \in K(V) | f$ is regular at $p$} is ...
mad_scientist's user avatar
0 votes
0 answers
48 views

Algebraic tangent space of image of a morphism

I'm getting lost trying to figure out what seems to be a simple question. I've looked in Shafarevich and Görtz & Wedhorn but the points are usually dealt with using schemes, which I haven't gotten ...
Absent mind's user avatar
0 votes
0 answers
90 views

Tangent space at a point of product of algebraic varieties

Let $X$ and $Y$ be algebraic varieties over an algebraically closed field $k$. Consider two points $a\in X$ and $b\in X$. I want to prove that the natural projection maps $p_X:X\times Y\to X$ and $p_Y:...
Albert's user avatar
  • 3,052
0 votes
0 answers
21 views

Can every chain of distinct, irreducible, Zariski closed subspaces of an affine algebraic set be extended to a chain of maximal length?(dimension+1) [duplicate]

We shall be working in affine $n$-space $\mathbb{A}^n$ Fix an algebraic set, say, $V$ Now, the dimension of $V$ is defined as the supremum of all distinct, irreducible, Zariski-closed subspaces of $V$....
Academic's user avatar
  • 307
1 vote
0 answers
53 views

Have I figure out the correct set of singular points?

I am working with the following problem. Let $X = V \left( X_1^2 - X_2X_3, X_1X_3 - X_1 \right) \subset \mathbb{C}^3$. Determine the set of all singular points of $X$. I have proved that the ...
GendoTendoLendo's user avatar
1 vote
0 answers
109 views

The cardinality of the affine algebraic variety $V_{\mathbb{C}}(I)$ and the dimension of $\mathbb{R}[\mathbf{x}]/I$

I am reading Laurent's notes Sums of squares, moment matrices and optimization over polynomials and, since I am not really experienced in algebraic geometry, I have some questions. This one concerns ...
math_cpt's user avatar
  • 184
1 vote
0 answers
40 views

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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