How to compute the generic point of $\{ \mathbf{x} \in \mathbb{P}^{n}: x_0 + \cdots + x_{n-1} + x_n = 0 \}$?

Question

I am considering an irreducible projective variety $\{ \mathbf{x} \in \mathbb{P}^{n}: x_0 + \cdots + x_{n-1} + x_n = 0 \}$. How does one compute the generic point of this variety? Any hints/comments are appreciated. Thank you.

Answer 1

The generic point can be obtained by working affine-locally. In the affine patch where $x_0 \neq 0$, you have that the variety is cut out of $\mathbb{A}^n = \operatorname{Spec} \mathbb{C}[x_{1/0}, \dots, x_{n/0}]$ by the equation $x_{1/0} + \cdots + x_{n/1} = 0$, where $x_{i/1} = x_i/x_0$. Thus, the coordinate ring of the variety is given by $\mathbb{C}[x_{1/0}, \dots, x_{n/0}]/(x_{1/0} + \cdots + x_{n/0})$, and the generic point is then simply the $0$-ideal of this ring.