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Let $X\subset\mathbb{P}^n$ be an hypersurface given by the vanishing of a polynomial $F\in k[x_0,...,x_n]_d$. Let $Y\subset X$ be a subvariety. Then $X$ has multiplicity $m$ along $Y$ if all the partial derivatives of order $m-1$ of $F$ vanish on $Y$ but there is at least one partial derivative of $F$ of order $m$ not vanishing along $Y$.

I am wondering if there is a criterion like this for higher codimension varieties.

More precisely, assume that $X\subset\mathbb{P}^n$ has codimension greater that one, and let $I = (F_1,...,F_r)$ be its homogeneous ideal. Is it true that $X$ has multiplicity $m$ along $Y$ if all the partial derivatives of order $m-1$ of $F_1,...,F_r$ vanish on $Y$ but there is at least one partial derivative of $F_i$, for some $i$, of order $m$ not vanishing along $Y$ ?

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    $\begingroup$ Even if $X$ is a complete intersection, wouldn't you need to consider something like the order of vanishing of the $r$-by-$r$ minors of the Jacobian matrix of $F_1,\ldots,F_r$? $\endgroup$ – Joe Silverman Sep 6 '14 at 14:38
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The comment of Joe Silverman is correct. You have to consider the order of vanishing of $r\times r$ minors of $Jac(F_1,...,F_r)$. For instance, consider $F_1 = xw-yz$ and $F_2 = w$ the intersection of a quadric with its tangent plane in $[1:0:0:0]\in\mathbb{P}^3$. Note that $\frac{\partial F_2}{\partial w} \neq 0$. However, the Jacobian matrix is $$Jac(F_1,F_2)=\left(\begin{matrix} w & -z & -y & x\\ 0 & 0 & 0 & 1 \end{matrix}\right)$$ We see that all the $2\times 2$ minors of $Jac(F_1,F_2)$ vanish with order one in $[1,0,0,0]$. Therefore, $[1:0:0:0]$ is a point of multiplicity two of $C = \{xw-yz =w = 0\}$. Indeed $C$ is a degenerate conic which is the union of two lines intersecting in $[1:0:0:0]$.

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