I just need one precision about the definition of zero locus of a general section of a vector bundle.
I know that if the $s$ is a section of a vector bundle $E$ on a scheme $X$, the zero locus of $s$ is the set $Z(s)$ which ideal sheaf is the image of the morphism $\mathcal E^{\vee}\rightarrow \mathcal O_X$, where $\mathcal E$ is the corresponding locally free sheaf to $E$.
So, I think that the zero locus of a general section is the zero locus of a section in a Zariski open subset in the space of global sections of $E$. Right?
Thanks for your answers in advance.
Best Answer
Yes, that would be my interpretation. Usually, you will have some Zariski closed condition on sections $s\in\Gamma(X,\mathcal{E})$. Then saying a "general" section of $\mathcal{E}$ has a property means that outside of a closed subset of $\Gamma(X,\mathcal{E})$ the property holds.
e.g. a general section of $\mathcal{O}(1)$ on $\Bbb{P}^1$ does not vanish at a given $P\in \Bbb{P}^1$.