Hyperplane in $\mathbb{P}^{n}$

Question

If we let $(x_{1},\ldots, x_{n}, y)$ be coordinates on $\mathbb{C}^{n+1}$ such that $y \neq 0$, then $\big[\tfrac{x_{1}}{y}, \ldots, \tfrac{x_{n}}{y}\big]$ are local coordinates on $\mathbb{P}^{n}$. Specifically, they are local coordinates on the patch defined by $y \neq 0$. The polynomial defined by

$$f(x_{1}, \ldots, x_{n}, y) = x_{1} + \cdots + x_{n} -y$$

is linear and homogeneous, so I believe its vanishing defines a projective variety $X \subseteq \mathbb{P}^{n}$, specifically a $(n-1)$-dimensional hyperplane, since it is linear.

I'm wondering if everything I've said so far is correct? I'm undertaking my own translation of a physics paper which isn't clear about the mathematics they're using, so I'm not very confident. I think a point in $X$ is simply a point in the hyperplane satisfying

$$\frac{x_{1}}{y} + \ldots + \frac{x_{n}}{y} =1.$$

Am I correct to say that varying $y$ leaves this point in $X$ invariant, but parameterizes a complex line in $\mathbb{C}^{n+1}$? I'm also curious what happens if $y=0$. I'm guessing this is related to a line at infinity.

Answer 1

I'm wondering if everything I've said so far is correct?

Yes.

I think a point in $X$ is simply a point in the hyperplane satisfying ...

Well, $X$ also contains points for which $y=0$. The equation you write with $y$ in the denominator is how the projective variety looks in that particular affine chart, but that's not the whole thing.

Probably going through the example $n=2$ would be helpful ($n=1$ has only one point as solution).

Am I correct to say that varying $y$ leaves this point in $X$ invariant, but parameterizes a complex line in $\mathbb{C}^{n+1}$?

I don't understand what you mean by this. Can you elaborate? The homogeneity of the polynomial ensures that multiplying all the variables by a constant doesn't change the zero locus, yet if you multiply only one of them by a constant you obtain a different solution to the equation.