Weighted projective plane ($\mathbb{P}(a,b,c)$)

This is a two-dimensional weighted projective space. The weighted projective plane $\mathbb{P}(a,b,c)$ is the quotient of ${\mathbb{A}}^3\setminus \{0\}$ by $\lambda \cdot (x,y,z)=({\lambda }^{a}x,{\lambda }^{b}y,{\lambda }^{c}z)$. It is generally singular but is a proper toric surface.

For coprime positive integers $a,b$,and the natural map identifies the Picard group with $ab\mathbb{Z}$ inside the class group. The plane is smooth exactly when $a=b=1$.