Dimension of variety intersected with a hyperplane? I have an affine variety $V$ defined over $\mathbb{C}$, defined as a zero set of some polynomials in $\mathbb{C}[x_1, ..., x_n]$. 
Let
$$
W = V \cap \{ x_n = 0 \}.
$$ 
I have two questions.


*

*Does $W$  as an affine variety in $\mathbb{C}^n$ always have dimension $(\dim V) - 1$?

*Since we are restricting to $x_n = 0$, I suppose we could view $W$ as an affine variety in $\mathbb{C}^{n-1}$ also (I think... I might be confused about something here...). Is the dimension of $W$ as an affine variety in  $\mathbb{C}^n$ and  $\mathbb{C}^{n-1}$ always the same? 
 A: The answer for 1 is no: just take $V$ to be the line $x_2=0$ inside $\mathbb{C}^2$ (the horizontal line). Then $\dim V=1$ and $\dim W=1$, since $W=V$. The point here is (as it appears in the comments) that $V$ is contained in the hyperplane in question.
For the question 2, I think that it is easier to look at the dimension of the variety as the Krull dimension of its coordinate ring. Say that $V$ is defined by some ideal $I\subset \mathbb{C}[x_1,\ldots,x_n]$. In this case, $W$ can be viewed as the subvariety of $\mathbb{C}^n$ defined by $(I,x_n)\subset \mathbb{C}[x_1,\ldots,x_n]$ or as the subvariety of $\mathbb{C}^{n-1}$ defined by $$\overline{I}\subset \frac{\mathbb{C}[x_1,\ldots,x_n]}{(x_n)}\simeq \mathbb{C}[x_1,\ldots,x_{n-1}]$$ where the "bar" represents residual classes.
The point is that the coordinate rings $\frac{\mathbb{C}[x_1,\ldots,x_{n-1}]}{\overline{I}}$ and $\frac{\mathbb{C}[x_1,\ldots,x_n]}{(x_n,\,I)}$ are isomorphic as $\mathbb{C}$-algebras. It gives you that their Krull dimensions agree.
