Let $(0\in X)$ be a germ of a normal 3-fold with a singular point $0$ (over $\mathbb{C}$). We think of $X$ as a small neighborhood of $0$ (for studying singularity).
If we can think $X$ as a hypersurface in $\mathbb{C}^4$ defined by an equation $$ f(t,x,y,z)=t^2+g_4(x,y,z)\in \mathbb{C}[[t,x,y,z]], \ \ \ mult_0(g_4)\ge 4, $$ then what is the weighted blow-up $Y:=WBl_0X\to X$ of $X$ at $0$ with weights $w(t,x,y,z)=(2,1,1,1)$? What is a exceptional divisor? and what...? I don't know even basic things about weighted blow-up. So I want to know about weighted blow-up concretely. But I couldn't find a good reference.
Does anyone know a good reference for weighted blow-up?
