Affine cone over a projective hypersurface
= Affine cone over a projective hypersurface
If $X=V_+(F)\subseteq\mathbb P^n$, its affine cone is $Y=V(F)\subseteq\mathbb A^{n+1}$. If $X$ is smooth, every nonzero point of $Y$ is smooth; hence the vertex is the only possible singular point. The vertex is smooth for linear $F$ and singular whenever $\deg F\geq2$.