I would like to understand the calculation of higher Ext groups of a skyscraper sheaf $\mathcal{O}_p$ at $p$.
The calculation I have seen does this using a Koszul resolution. It starts out like this
"Assuming $X$ is affine, local coordinates near $p$ define a section $s$ of $\mathcal{O}^n$ ($n = \dim X$) vanishing transversely at $p$. "
I know that for such a section we get a koszul resolution of $\mathcal{O}_p$ that will lead to the Ext groups. However, I fail to understand the quotation. More specifically: how is this section obtained and why is $p$ the zero locus?
Thanks in advance for your answers.
Carsten
ps: the calculation is from http://math.mit.edu/~auroux/18.969-S09/mirrorsymm-lect16.pdf