# Questions tagged [affine-varieties]

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

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### 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 $k$-...
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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 ...
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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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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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 ...
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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 ...
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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 ...
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### 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 ...
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### 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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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $V$....
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### 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 ...
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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 ...