Things are in the complex algebraic setting.
Assume that a vector bundle $V$ of rank $n$ over a $\mathbb{P}^n$ has a global section $\sigma$. Is it true that the zero set of $\sigma$ is a $0$-dimensional subscheme of length $c_n(V)$? Thank you very much.
(collecting comments into an answer...)
As @Ariyan's comment points out, there is no reason in general that the zero set $Z(\sigma)$ should have the right dimension. For example, if $V = \mathcal{O}(1)^{\oplus n}$, and $f \in \Gamma \mathcal{O}(1)$ is a nonzero section, then the section $(f,\ldots,f) \in \Gamma(V)$ vanishes exactly along $Z(f)$, a subset of codimension 1 rather than $n$.
On the other hand, if you assume that $Z(\sigma)$ is zero-dimensional, then yes, it will be a subscheme of length $c_n(V)$. Fulton's Intersection Theory is the standard reference for these kinds of things; for this specific question there is surely a much simpler reference, but I don't know one offhand. (If anyone else does, please edit this answer to add it in!)