TITLE: What is the zero subscheme of a section
QUESTION [16 upvotes]: 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.

REPLY [6 votes]: For any sheaf $F$ and $s\in \Gamma(F)$ a section, we regard it as a map $O\to F$ of sheaves. Taking duals, we obtain a map $F^*\to O$, its image is an ideal $I$, and the closed subscheme it defines is the zero section of the section $s$. 
This is a good definition as it is general, also it gives a scheme structure not just a closed set for each section. We recover the special definition where $F = O$ or a vector bundle. A regular section of a vector bundle is where the ideal $I$ defined above is a complete intersection. 
For a  reference, see Hartshorne's paper Stable Vector Bundles of Rank 2 on $P^3$ section 1.