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A hypersurface of degree $d$ of $\mathbb{P}^{n}$ comes from a expression like $$ F=\displaystyle\sum_{i_{0}+\cdots+i_{n}=d}U_{i_{0},\ldots,i_{n}}X_{0}^{i_{0}}\cdots X_{n}^{i_{n}}. $$

The hypersurfaces of degree $d$ that passes through a point $p=(p_{0}:...:p_{n})\in\mathbb{P}^{n}$ are those whose coefficients satisfy the equation

$$ \displaystyle\sum_{i_{0}+\cdots+i_{n}=d}U_{i_{0},\ldots,i_{n}}p_{0}^{i_{0}}\cdots p_{n}^{i_{n}}=0, $$ which is a hyperplane of $\mathcal{L}^{n}_{d}$, the projective space of hypersurfaces of degree $d$ in $\mathbb{P}^{n}$.

Now, the hypersurfaces that passes through $p$ with multiplicity at least $m$ are those who satisfy the $\binom{m+n-1}{n}$ linear conditions given by the vanishing of the coefficients of the Taylor series expansion of the non-homogeneous polynomial asociated to $F$. My question is the next:

These linear conditions are supposed to be independent, but how do we know?

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Suppose we are in characteristic zero. The conditions are just that certain coefficients $U_{i_{0},\ldots,i_{n}}$ be zero, hence these conditions on the coordinates in the parameter space are linearly independent.

More explicitly, suppose that $p=(1:0:0:\dots:0)$.
If we write $F$ as $$F(X_0,\dots,X_n)=F_0(X_1,\dots,X_n)X_0^d+F_1(X_1,\dots,X_n)Z^{d-1}+\dots+F_d(X_1,\dots,X_n)X_0^d$$ (with $F_i$ homogeneous of degree $i$) the condition that the hypersurface $V(F)$ have multiplicity at least $m$ at $p$ is that $F_0=F_1=F_2=\dots=F_{m-1}=0$.
The vanishing of those homogeneous polynomials means that their coefficients are zero and since these coefficients are coefficients of $F$ the conditions are linearly independent coefficients in the coordinates of the projective space $\mathbb P^N$ of dimension $N= \binom {n+d}{n}-1$ parametrizing the hypersurfaces of degree $d$ of $\mathbb P^n$.
Hence there are $\binom {n+m-1}{n}$ linearly independent conditions and finally the hypersurfaces of degree $d$ having a singularity of order at least $m$ at $p$ form a projective space of dimension $N-1-\binom {n+m-1}{n}$

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  • $\begingroup$ Thank you, but I must be missing something. We may assume $p_{0}=1$. Then, if $f$ is the non-homogeneous polynomial asociated to $F$, we need (if I am not wrong): $$0=\dfrac{\partial^{k}f}{\partial X_{1}^{j_{1}}\cdots\partial X_{n}^{j_{n}}}(p)=$$ $$=\sum_{i_{0}+\cdots + i_{n}= d}U_{i_{0},\ldots,i_{n}}i_{1}\cdots(i_{1}-j_{1}+1)\cdots i_{n}\cdots(i_{n}-j_{n}+1)p_{1}^{i_{1}-j_{1}}\cdots p_{n}^{i_{n}-j_{n}}$$ for all $k\in\{0,...,m-1\}$ and $j_{1},...,j_{n}\geq 0$ such that $j_{1}+\cdots +j_{n}=k$. I do not see why these linear restrictions are equivalet to the vanishing of certain coefficients. $\endgroup$ Mar 12, 2015 at 21:52
  • $\begingroup$ Dear Gauloises, I have tried to make my answer more explicit by editing my post. $\endgroup$ Mar 13, 2015 at 9:25
  • $\begingroup$ @GeorgesElencwajg, why did you require characteristic zero? Where can your argument fail in positive characteristic? $\endgroup$
    – rmdmc89
    Sep 28, 2021 at 19:22
  • $\begingroup$ @GeorgesElencwajg, another question: in the last line, did you mean "a projective space of dimension $N-\binom{n+m-1}{n}$"? If not, why the $-1$? $\endgroup$
    – rmdmc89
    Sep 28, 2021 at 20:32

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