# Questions tagged [affine-varieties]

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

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### Criteria for empty vanishing set $V(J)$ for non-closed fields

Let $R = k[x_1,\dotsc,x_n]$ and $J = (f_1,\dotsc,f_m) \subseteq R$ be an ideal. I know that under the assumption that algebraically closed, $V(J)=\emptyset$ iff $J=(1)$. Namely, "$\Leftarrow$&...
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### Showing that the set difference of two affine varieties need not be an affine variety [duplicate]

I was wondering what would be an example of two affine varieties $A, B$ over the field of complex numbers $\mathbb{C}$ such that $A\setminus B$ is not an affine variety? I was initially thinking about ...
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• 2,443
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### Is a finitely generate $k$-algebra that has no nilpotent element ($k$ is a field) an integral domain?

In Hartshone section 1 exercise 1.5, he wants me to prove that If $B$ if a finitely generated $k$-algebra with no nilpotent elements, then $B$ is isomoprhic to the affine coordinate ring of some ...
• 4,926
1 vote
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### A Corollary on Estimation of Fiber Dimension from Kemper's Course on Commutative Algebra

I have a question about following proof from Kemper's Course in Commutative Algebra (page 142): Corollary 10.6. Let $f: X \to Y$ be a morphism of equidimensional affine varieties over an ...
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### Upper semicontinuity of fiber dimension for Affine Varieties (reference desired)

In This question following result is stated: If $f : X\to Y$ is a morphism between two irreducible affine varieties over an algebraically closed field $k$, then the function that assigns to each ...
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### When the ring of regular functions is Artinian?

Let $K$ be a field, $W$ an algebraic variety over $K$ and $A(W)=\{f:W\rightarrow K \mid f \text{ regular}\}$ the ring of regular functions. When is $A(W)$ an Artinian ring? I know that $A(W)$ is a ...
1 vote
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### Is there an easy example of a variety that is not rational? ${}$

I saw a lot of examples of varieties which are rational, now I was also wondering if there was a simple example of a variety that is not rational. Sadly I didn't find that much online, is it maybe ...
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### Confusion regarding dimension of variety and rank of Jacobian

I am getting the contradiction described below. I have been racking my brain since yesterday to figure out what the problem is here, but I just can't. I must be making some super silly mistake. Please ...
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### Showing that $\mathcal{V}(f,g) \subseteq \mathbb{A}_k^n$ has three irreducible components.

I was working on this following exercise and i would really like to see if i understood things correctly. Let $V:=\mathcal{V}(X^2-YZ,XZ-X) \subseteq \mathbb{A}_k^n$. Show that $V$ has three ...
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Suppose I have an affine variety $V \in k^n$ and $I := I(V)$. Let $f_1,\dots,f_t$ be generators of I. For a point $P \in V$, define the jacobian $J_P(I) = \begin{pmatrix} \frac{\partial f_1}{\partial ... • 57 1 vote 0 answers 152 views ### Generators of the ideal of the projective closure Let$X=\{(t,t^2,t^5): t\in k\}\subset \mathbb{A}^3$. Show that the projective closure$\bar{X}$is a projective variety of dimension 1, and say if it is isomorphic to$\mathbb{P}^1$. Compute$\mathbb{...
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Need to show that $\textbf{V}(y-x^2,xz-y^2)=\textbf{V}(y-x^2,xz-x^4)$ in $\mathbb{R}^3$. I was trying to use the fact that $\textbf{V}(f,g)=\textbf{V}(f,g_1)\cup\textbf{V}(f,g_2)$ when $g=g_1g_2$. ...