Projective quadric cone with one singular vertex

Over an algebraically closed field of characteristic other than two, the rank-$r+1$ quadricis an irreducible $r$-dimensional cone for $r\ge 2$, and its singular locus is the omitted-coordinate vertex $[0:\cdots :0:1]$.

For $n>k\ge 0$, take an $(n-k)$-dimensional irreducible projective variety with one singular point and form its product with ${\mathbb{P}}^k$. The Segre embedding makes the product projective, and its singular locus is the singular point times ${\mathbb{P}}^k$, of dimension $k$. A quadric cone works in dimension at least two and a cuspidal cubic works in dimension one.