I would like to understand what's a weighted blow-up in a very simple case: $\mathbb{C}^2$ blown-up in the origin with weights $(a,b)$.
In found some notes online saying that this is the surface $X$ defined in $\mathbb{C}^2\times \mathbb{P}(a,b)$ by $x^{a}v-y^{b}u=0$. Is this correct?
I am asking because I think that the equation should be invariant under the action of $\mathbb{C}^{*}$ on $\mathbb{C}^2\setminus\{0\}$ whose quotient is $\mathbb{P}(a,b)$ but this is not.
Does anyone know a description of $X$ as a quotient and a way to compute what kind of singularities may appear on the exceptional divisor depending on the weights?
