every isomorphism class can be represented by a normalized weight, and two such spaces are isomorphic if and only if they have the same normalized weights, up to order. As a corollary, for $n=1$ every such space is isomorphic to ${\mathbb{C}\mathrm{P}}^1$. Let us stress that, despite this, ${\mathbb{P}}(\ell_0,\ell_1)$ has orbifold singularities in every case except $\ell_0=\ell_1=1$; the spaces ${\mathbb{P}}(\ell_0,\ell_1)$ are all homeomorphic to ${\mathbb{C}\mathrm{P}}^1$, but not isomorphic as orbifolds.

It is useful to have in mind few basic examples.

The map $S^3\to{\mathbb{R}}^3$, $(z_1,z_2)\mapsto (x_1,x_2,x_3):=\big({\mathrm{Re}}(z_1^*z_2^3),{\mathrm{Im}}(z_1^*z_2^3),z_1z_1^*\big)$, factors to a homeomorphism between ${\mathbb{P}}(3,1)$ and the variety in ${\mathbb{R}}^3$ defined by the equation $x_1^2+x_2^2=x_3(1-x_3)^3$. The latter has an orbifold singularity (a cusp) at $x=(0,0,1)$. As a toric variety, the map is an isomorphism ${\mathbb{P}}(3,1)\to{\mathbb{C}\mathrm{P}}^1$.

As the ${\mathbb{P}}(3,1)$ example shows, ${\mathbb{P}}({\underline}{\ell})$ may be isomorphic to ${\mathbb{C}\mathrm{P}}^n$ as complex algebraic variety and, at the same time, have orbifold singularities.

There is an isomorphism of algebraic varieties between ${\mathbb{P}}(1,1,2)$ and the cone in ${\mathbb{C}\mathrm{P}}^3$ with equation $y_0y_2-y_1^2=0$; the isomorphism is given by $$[z_0{\hspace{-1pt}:\hspace{-1pt}}z_1{\hspace{-1pt}:\hspace{-1pt}}z_2]\mapsto [y_0{\hspace{-1pt}:\hspace{-1pt}}y_1{\hspace{-1pt}:\hspace{-1pt}}y_2{\hspace{-1pt}:\hspace{-1pt}}y_3]{\hspace{-1pt}:\hspace{-1pt}}=[z_0^2{\hspace{-1pt}:\hspace{-1pt}}z_0z_1{\hspace{-1pt}:\hspace{-1pt}}z_1^2{\hspace{-1pt}:\hspace{-1pt}}z_2] \;,$$

As a preliminary step, we investigate the conditions on the weight vector ${\underline}{\ell}$ for which . Firstly, the observation that ${\mathbb{P}}({\underline}{\ell})\simeq{\mathbb{P}}(m{\underline}{\ell})$ for all $m\geq 1$ allows one to consider coprime weight vectors 