Let $\mathbb{WP}^1(1,a)$ be the weighted projective line where $(x_0:x_1)$ is identified with $(\lambda x_0:\lambda^a x_1)$. Some questions:
1) Is $\mathbb{WP}^1(1,a)$ smooth for all a?
2) When is $\mathbb{WP}^1(1,a)$ isomorphic to $\mathbb{WP}^1(1,b)$?
what about the projective plane? Is there an easy way to classify $\mathbb{WP}^2(1,a,b)$?