two weights are not divisible by $p$. One says that a weight vector is *coprime* if $\gcd(\ell_0,\ldots,\ell_n)=1$; and it is *pairwise coprime* if $\gcd(\ell_i,\ell_j)=1$, for all $i\neq j$. For $n=1$, the only normalized weight vector is $(1,1)$; if $n=2$, a weight ${\underline}{\ell}$ is pairwise coprime if and only if it is normalized; if $n\geq 3$, every pairwise coprime weight vector is normalized but the converse is not true, e.g. $(1,1,\ell_2,\ldots,\ell_n)$ is normalized but not necessarily pairwise coprime.

Fixed a weight vector ${\underline}{\ell}=(\ell_0,\ldots,\ell_n)$, an action of $t\in {\mathbb{C}}^*$ on $z=(z_0,\ldots,z_n)\in {\mathbb{C}}^{n+1}\smallsetminus\{0\}$ is given by $z\mapsto (t^{\ell_0}z_0,\ldots,t^{\ell_n}z_n)$. If the action is restricted to the subgroup ${\mathrm{U}}(1)\subset  {\mathbb{C}}^*$, the unit sphere $S^{2n+1}\subset {\mathbb{C}}^{n+1}$ is an invariant submanifold.

The quotient ${\mathbb{C}}^{n+1}\smallsetminus\{0\} / {\mathbb{C}}^*$ yields the weighted projective space ${\mathbb{P}}({\underline}{\ell})={\mathbb{P}}(\ell_0,\ldots,\ell_n)$; when $\ell_0=\ldots=\ell_n=1$, this is the ordinary complex projective space ${\mathbb{C}\mathrm{P}}^n$. In general there is an embedding ${\mathbb{P}}({\underline}{\ell})\hookrightarrow{\mathbb{C}\mathrm{P}}^n$ realizing ${\mathbb{P}}({\underline}{\ell})$ as a complex projective (toric) variety. As a topological space, ${\mathbb{P}}({\underline}{\ell})$ is homeomorphic to the quotient $S^{2n+1}/{\mathrm{U}}(1)$ with respect to the weighted action defined above, a quotient which has a natural structure of orbifold.

It is known that two weighted projective spaces are isomorphic as projective varieties if and only if they are homeomorphic [@BFNR13]. Moreover, 