Counting vanishing conditions for a section of a line bundle I am reading Lazarsfeld's positivity 1 and there is a passage I'm not able to understand... In proposition 1.1.31 he determines the number of conditions for the vanishing to order $c$ of a global section $s$ of a line bundle on an $n$ dimensional variety as
$$\binom{n+c-1}{n}$$
Now, as far as I know this is the number of compositions of $n$ as sum of $c$ nonnegative integers. But why is this the total number of conditions
$$\frac{\partial^k s}{\partial x_{1}^{a_1}...\partial x_{n}^{a_n}}=0$$
for $k=0,...,c-1$?
Thanks!
 A: Think about the case of curves ($n=1$) first. So you have a line bundle $L$ on a curve $X$, and you want to know how many conditions describe the vanishing of $s,s',\dots s^{(c-1)}$ for a section $s\in H^0(X,L)$. The notion of derivative of a section of a line bundle is formulated in terms of jet bundles. If I am not misunderstanding your question, what you want is simply $$\textrm{the rank of the jet bundle }J^{c-1}L.$$ On a curve it is easy to see that $$\textrm{rank }J^{c-1}L=c.$$ Indeed, a section $s$ is represented on a local chart by a regular function $f$ on the curve, and the corresponding section $D^{c-1}s\in H^0(X,J^{c-1}L)$ is represented on that chart by the matrix $$\begin{pmatrix}
f\\
f'\\
\vdots\\
f^{(c-1)}
\end{pmatrix}.$$
So you see that there are $c=\binom{1+c-1}{1}$ vanishing conditions, as wanted.
For $n>1$, the "vanishing" becomes the vanishing of some minors. The number you are after is the number of minors that are supposed to vanish. For instance, for $n=2$ and local coordinates $x_1,x_2$, you want to describe the condition $$\begin{pmatrix}
f\\
\frac{\partial f}{\partial x_1}\\
\vdots\\
\frac{\partial^{c-1} f}{\partial x_1}
\end{pmatrix}\wedge \begin{pmatrix}
f\\
\frac{\partial f}{\partial x_2}\\
\vdots\\
\frac{\partial^{c-1} f}{\partial x_2}
\end{pmatrix}=0.$$
There are $\binom{c}{2}=\binom{2+c-1}{2}$ vanishing conditions here, as wanted. 
I leave it to you to generalize the above for arbitrary $n$. 
