Let $\pi$ be the blow up of $\mathbb{A}^n$ at an ideal $I_{p}$ which is supported at a close point $p$; let $E$ be the associated exceptional divisor. What are the possible varieties $Y$ that can appear as $E \cong Y$?

$\textbf{Partial Answers:}$

(1)If $I_p = (x_0, \ldots x_n)$, then $E \cong \mathbb{P}^n$. In this case $\pi$ is the blow up of $\mathbb{A}^n$ at a smoothing point.

(2) If $I_p = \left( x^{d/w_0}, \ldots, x_n^{d/w_n} \right)$ where $d=lcm(w_0, \ldots, w_n)$. Then $E \cong \mathbb{P}(w_0, \ldots w_n)$. In this case $\pi$ is the weighted blow up with respect the weights $(w_0, \ldots w_n)$.

I am wondering if there are similar picture for other varieties or other toric varieties? I am particularly interested in an explicit description! This means: To obtain the variety $E \cong Y$ we must blow up the ideal $I(Y)_{p}$ which generators are.....

Thanks!