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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.

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Any irreducible hypersursurface in $X \subset \mathbb{P}^n$ of degree $d$ with an isolated singular point $p \in X$ of multiplicity $d$ is a cone over a hypersurface $Y$ of degree $d$ in $\mathbb{P}^{n-1}$.

In fact, take any point $q \in X$ different from $p$. Then the line $\overline{pq}$ has at least $d+1$ intersections with $X$ (counted with the right multiplicities), so by Bézout theorem it must be contained in $X$. But then $X$ is a cone of vertex $p$.

Coversely, if $Y \subset \mathbb{P}^{n-1} \subset \mathbb{P}^n$ is a hypersurface of degree $d$, then the cone $X$ of vertex a point $p \notin \mathbb{P}^{n-1}$ is a hypersurface $X \subset \mathbb P^n$ with a singular point of multiplicity $d$.

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Thanks! Could you point me to where i can find the definition of "a" cone? (i thought the only thing called like this was the quadratic cone) –  Joachim May 27 '13 at 13:49
    
Pfff i'm sorry this is in Hartshorne ex I.2.10 i knew i should have looked before asking.. Thanks for the answer Francesco! –  Joachim May 27 '13 at 13:53

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