Rational projective variety given by coprime homogeneous polynomials

Question

Let $K$ be algebraically closed field. Let $f_k, f_{k-1} \in K[x_1,...,x_n]$ coprime homogeneous polynomial of degree $k$ and $k-1$ respectively. I want to prove that:

The variety $$ X = \left\{ (x_0 : x_1 : ... : x_n) \in \mathbb{P}^n \mid f_k(x_1,...,x_n) + x_0 f_{k-1}(x_1,...,x_n) = 0 \right\}$$ is rational.

A projective variety is called rational if it is birational equivalent to $\mathbb{P}^m$.

I have an idea and it is to show that $K(V) \cong K(x_0,...,x_m)$ (Proposition 2.53 in Klaus Hulek, Elementary Algebraic Geometry). The fact that $f_k$ and $f_{k-1}$ are coprime means the following:

1. They are not zero simultaniously for any $(x_1,...,x_n) \ne (0,...,0)$.
2. There exists $a,b \in K[x_1,...,x_n]$ such that $a f_k + b f_{k-1} = 1$ so $\langle f_k , f_{k-1} \rangle = \langle 1 \rangle = K[x_1,...,x_n]$.

These two facts and Proposition 2.53 imply that I need to show somehow that the algebra of functions over $X$ is isomorphic to $K(x_1,...,x_n)$ but I don't how to it for this problem.

Answer 1

You need to show that there is some open subset of $X$ such that $X \simeq \mathbb{P}^{n-1}$.

HINT Consider the open subset $U$ where $f_{k-1}(x_1,\cdots,x_n) \neq 0$. And note that $X$ is precisely the set of points with $x_0=-f_k/f_{k-1}$.

Can you write up a bijection from $U$ to some open subset of $\mathbb{P}^{n-1}$?

Answer 2

Consider the open subset $W=\mathbb P^{n-1}\setminus V(f_{k-1}) $ defined by $f_{k-1}\neq 0$ and the open subset $U=X\setminus V(f_{k-1}) \subset X$ .
The two mutually inverse isomorphisms $$ W\to U: ( x_1 : \cdots : x_n) \mapsto(-\frac {f_k(x_1 : \cdots : x_n)}{f_{k-1}(x_1 : \cdots : x_n)} : x_1 : ... : x_n) \\ U\to W: (x_0 : x_1 : ... : x_n) \mapsto ( x_1 : \cdots : x_n) $$
show that $X$ is rational.

A riddle
Where have I used that $f_{k-1}$ and $f_k$ have no common factor?