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Suppose I have an affine variety defined by some polynomials $f_1(x_1, ..., x_n), ..., f_r(x_1, ..., x_n)$ in $\mathbb{C}^n$, and suppose this has dimension $L$. We can also view the polynomials as sitting inside $\mathbb{C}[x_1, ..., x_n, y_1, ..., y_r]$ so it produces an affine variety inside $\mathbb{C}^{n + r}$. I mean they are really the ``same", so does the two (the one sitting inside $\mathbb{C}^n$ and $\mathbb{C}^{n + r}$) have the same dimension?

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No. The variety corresponding to the ideal $I = (f_1, \dots, f_r) $ in $\mathbb C^{n+r}$ will be $X \times \mathbb C^r$, where $X$ is the zero set of $I$ in $\mathbb C^n$.

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  • $\begingroup$ I see. So the dimension will be $L+r$ then? $\endgroup$ – Johnny T. Jan 18 '17 at 17:23
  • $\begingroup$ Yes, this is correct. $\endgroup$ – user171326 Jan 18 '17 at 17:26

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