Let $X$ be a scheme, $\mathcal F$ a locally free sheaf of rank $r$ and $s \in \Gamma(X, \mathcal F)$ a global section of $\mathcal F$.
Question: What is the zero subscheme of $s$?
I can't believe that pouring through Hartshorne hasn't turned up a definition of this. It should be some subscheme of $X$. The only thing I can think of is the set of points $x \in X$ where $s$ goes to $0$ in the stalk $\mathcal F_x$, i.e., the complement of the support of $s$. But that would make the zero subscheme of $s$ open and that doesn't make sense because in what I'm reading there is a hypothesis that $s$ is a regular section, and that this has something to do with the codimension of the zero scheme in $X$ (which would always be $0$ if the zero scheme were open). Which leads me to question $2$:
Question 2: What is a regular section? Is it a section whose zero subscheme is regular? Cause that would be great if it were true.