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 $Y$ ?



Multiplicity of a variety along a subvariety
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 $[1:0:0:0]$.


