Let $f(x) \in \mathbb{C}[X_1,...,X_n]$ be a homogeneous polynomial in $n$ variables such that the zero locus $V$ of $f$ in $\mathbb{C}^n$ is singular only at the origin. Denote by $\pi:\widetilde{V} \to V$ be the blow-up of $V$ at the origin. Then,
1) Is there any known condition on $f$ under which $\widetilde{V}$ is non-singular?
2) Is there any known condition on $f$ under which the fiber of $\pi$ over the origin is isomorphic to the hypersurface in $\mathbb{P}^{n-1}$ defined by $f$?
Any hint/reference will be most welcome.
The base of the cone is a hypersurface $$ \bar{V} \subset \mathbb{P}^{n-1} $$ with equation $f$. By assumption it is smooth. The blow up is isomorphic to the total space of the line bundle $\mathcal{O}(-1)\vert_{\bar{V}}$, in particular it is smooth. Furthermore, the fiber over the origin is the zero section of the total space, in particular it is isomorphic to $\bar{V}$.