I'm having trouble understanding the definition of an ample line bundle as given in my lecture notes:
Definition: Let $X$ be a quasi-compact scheme, a line bundle $\mathcal{L}$ on $X$ is called ample if for every $x\in X$ there exists $m\ge 1$ and $s\in \mathcal{L}^{\otimes m}(X)$ a global section such that $D(s):= X\setminus V(s)$ is affine and $x\in D(s)$.
What is meant by $X\setminus V(s)$, for $s$ a global section? My first guess was that maybe this is defined via a cover of $X=\bigcup_{i\in I}U_i$ with $U_i=\operatorname{Spec} A_i$ open affine in such a way that $s$ locally (i.e. on $U_i$) becomes an element of $A_i$. (Does this work?$^1$) This would give us $U_i\setminus V(s)$ in a canonical way and $X\setminus V(s)=\bigcup_{i\in I}\left(U_i\setminus V(s)\right)$ would make sense.
However, I'm not sure if this is the correct way of looking at this.
Thank you very much in advance.
$^1$ The problem here might be that $s$ is a global section and thus not defined on $U_i$ but we could restrict it to each of the $U_i$.