Projective quadric cone with one singular vertex
= Projective quadric cone with one singular vertex
Over an algebraically closed field of characteristic other than two, the rank-$r+1$ quadric
$$
V(X_0^2+\cdots+X_r^2)\subseteq\mathbb P^{r+1}
$$
is an irreducible $r$-dimensional cone for $r\geq2$, and its singular locus is the omitted-coordinate vertex $[0:\cdots:0:1]$.