The most common formulation of the weighted projective space is perhaps the global quotient
$$
(\mathbb{C}^{n+1} \setminus \{(0,\ldots,0\}) / \mathbb{C}^\ast
$$
with the $\mathbb{C}^\ast$ group action given by
$$
\lambda \cdot (x_0,\ldots,x_n) = (\lambda^{w_0} x_0, \ldots, \lambda^{w_n} x_n),
$$
using *positive integer* weights $(w_0, \ldots, w_n)$. Of course, it can also be, equivalently, defined as a toric variety using an appropriate lattice.

My question is: (1) is there any classification on weighted projective space with rational weights? (i.e. $w_0, \ldots, w_n$ are nonzero rational numbers). It seems to me that many of the results should remain valid. In particular, all the singularities should still be of the cyclic quotient singularity type.

(2) How about using real weights? (let $w_0, \ldots, w_n$ be real numbers) With a properly chosen branch of logarithm, powers like $\lambda^{w_0}$ are still meaningful. Is there any classification results on such weighted projective spaces?