All varieties will be projective and over $\mathbb{C}$.

If $S$ is any surface in $\mathbb{P}^3$ of degree 2 that posseses an ordinary double point, it follows easily that $S$ is projectively isomorphic to the cone: $$ x^2 + y^2 + z^2 = 0. $$ Is there a similar standard form for quartics that possess an ordinary quadruple point, or is there a family of such standard forms? More generally what can we say about a degree $d$ surface in $\mathbb{P}^3$ that contain an ordinary $d$-uple point?

Thanks.