Let $M$ be smooth and purely $r$-dimensional, $E$ be a vector bundle of rank $r$ over $M$, $s$ be a regular section of $E$ and $Z$ the zero scheme of $s$. Then $[Z]$ is dual to the top Chern class $c_r(E)$, which is set-theoretically $\mu$ points, where $\mu$ the top Chern number. When (under what conditions) can we say $Z$ consists of $\mu$ points? Or is it always true when $s$ is general enough? Thanks!
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3$\begingroup$ Depends. $E$ may have too few sections and then $Z$ may not be even zero dimensional for any section. If $E$ is globally generated, then general section will have the property you expect. $\endgroup$– MohanCommented Jun 24, 2014 at 16:28
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1$\begingroup$ Since you tagged this "algebraic topology", probably you are working over $\mathbb{C}$. I just want to point out, this type of thing can fail for smooth schemes over a finite field. If you assume that the field is infinite, then you should be able to apply Bertini theorems (probably this is what Mohan is alluding to as well). $\endgroup$– Jason StarrCommented Jun 24, 2014 at 17:38
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$\begingroup$ Thanks a lot, Mohan and Jason. Another question, say we are working over $\mathbb{C}$, under what conditions is $s$ general enough? $s$ intersects transversally with the zero section? I think this is not enough. $\endgroup$– JYQCommented Jun 24, 2014 at 18:37
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