Is the closure of the image of $\mathbb C^n$ under vector-polynomial map is irreducible affine variety of dimension not higher than $n$?

Asked Jun 13 '13 at 16:42

Active Jun 13 '13 at 16:42

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I work with the subset $Z\subset\mathbb C^m$ which is the image of $\mathbb C^n$ under vector-polynomial map $f(x)=(p_1(x),...,p_m(x))$, that is $f$ sends $x\in\mathbb C^n$ to $f(x)\in Z\subset\mathbb C^m$.

Is the following true?

1) $Z$ is IRREDUCIBLE constructible subset of $\mathbb C^m$.

2) the dimension of $Z$ does not exceed $\min(m,n)$

3) the closure of $Z$ is the IRREDUCIBLE variety of dimension not high than $n$?

For 1) I know that the constructibility of $Z$ follows from the Chevalley theorem and the irreducibility follows from the fact that $f$ is a continuous map.

Statement 2) is intuitively clear, but I do not know how to prove it.

For 3) (if it is true) I have no any ideas.

asked Jun 13 '13 at 16:42

Ignat Domanov

1,123

2

For $3$, did you know that the closure of an irreducible topological space is again irreducible? – Jim Jun 13 '13 at 17:08
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For 2, see this related question: http://math.stackexchange.com/questions/95670/surjective-morphism-of-affine-varieties-and-dimension – Yuchen Liu Jun 14 '13 at 04:52
Thank you @Jim, I did not know the fact. I am reading this part of topology now. – Ignat Domanov Jun 14 '13 at 16:25
Thank you for the link @YuchenLiu, I will try to post my answer on all tree questions later. – Ignat Domanov Jun 14 '13 at 16:26

Is the closure of the image of $\mathbb C^n$ under vector-polynomial map is irreducible affine variety of dimension not higher than $n$?

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