rationality of weighted projective space

Question

A complex weighted projective is $\mathbb{P}(k_1, \cdots, k_{n+1})=Proj(\mathbb{C}[x_1, \cdots, x_{n+1}])$ with $x_i$ of degree $k_i$ (sometimes people ask for each $n$ of the weights being coprime). My first question is whether all weighted projective spaces are rational.

The rationality of $\mathbb{P}(k_1, \cdots, k_{n+1})$ is equivalent to the functional field $\mathbb{C}_0(x_1, \cdots, x_{n+1})$ being free generated by $n$ elements. When $n=1$, it is clear that $\mathbb{C}_0(x_1, x_2)=\mathbb{C}(\frac{x_1^{k_2}}{x_2^{k_1}})$, hence $\mathbb{P}(k_1, k_2)$ is always a rational curve. However, I guess in higher dimension this could not hold.

If the answer to the first question is no, then my second question is that whether some weaker rationality properties, like rational connectivity, stable rationality, etc hold for weighted projective space.

Thank you in advance!

Answer 1

The $n$-dimensional weighted projective space $X = \mathbb{P}(k_1,\dots,k_{n+1})$ is a toric variety, hence is automatically rational since it contains an isomorphic copy of $\mathbb{G}_m^n$ as a dense open subvariety. Indeed, let $T$ be the torus $\{(t_1,\dots,t_{n+1}) \mid t_i \in \mathbb{G}_m\}/\{(t,\dots,t)\mid t\in \mathbb{G}_m \}$. Then $T$ acts on $X$ via $$(t_1,\dots,t_{n+1}) \cdot [x_1 : \dots :x_{n+1}] = \left[t_1^{k_1}x_1 :\dots :t_{n+1}^{k_{n+1}}x_{n+1} \right]$$ and the orbit of $[1: \dots : 1]$ is isomorphic to $T$, since its stabilizer is a product of cyclic groups.