It is well known that the exceptional divisor associated to the blow up of a smooth point at $\mathbb{A}^n$ is $\mathbb{P}^n$ and the related ideal in this case is $(x_0, \ldots x_n)$. More generally, if we take a weighted blow up with respect the weights $(w_0, \ldots w_n)$, then our associated exceptional divisor is $\mathbb{P}(w_0, \ldots w_n)$. In this case, the related ideal is: $$ \left( x^{d/w_0}, \ldots, x_n^{d/w_n} \right) $$ where $d=lcm(w_0, \ldots, w_n)$.

Is there are similar picture for other varieties or other toric varieties? I am particularly interested in an explicit description :)

Thanks!