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 $(w_0, \ldots w_n)$.

(3) Kawakita article [K] proves that a divisorial contraction in dimension $n=3$ which contracts its exceptional divisor to a smooth point is a suitable weighted blow-up.

I am wondering if there are similar picture for other varieties? ( The toric case is clarified below by Lev Borisov's answer). 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...

Particularly, is it possible to obtain a grassmannian this way?

Thanks!

[K] Divisorial contractions in dimension three which contract divisors to smooth points,Kawakita, Masayuki, Inventiones mathematicae, 145, 1, 105--119,2001