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!