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 $\mathrm{deg}F\ge 2$.