I've often read that the first chern class can be seen as "the number of zeroes a section must have".
How precise can this statement be made?
I'm only interested in Line bundles. I actually know how a generic section looks like on my manifold. From this I can even tell what the "lowest" number of zeros is.
Is it safe to say that
minimum number of zeros $> 0$ $\to$ $c_1 \neq 0$?