What does Ravi Vakil mean with a "geometric point" of a divisor? In this article Ravi Vakil talks about "geometric points" of a divisor on a projective variety (specifically theorem 1.13). I have not seen this notion used anywhere else, so I'm not quite sure what he means.
I imagine that if $D=\sum_{i=1}^nk_iY_i$ is a (Weil) divisor, the geometric points of $D$ are the closed points contained in $\cup Y_i$. Is that correct?
As a follow up question (assuming my interpretation above is correct): what does it mean that $D$ has exactly one singular geometric point? I see two possible meaning:
there is a single closed point $p\in \cup Y_i$ and:
1) a single $Y_i$ that is singular at $p$.
2) at least one, but possibly multiple $Y_i$ that are singular at $p$.
which of these is correct?
 A: For a scheme $X$ defined over a field $k$ (ie equipped with a morphism to $\operatorname{Spec} k$), geometric points are morphisms from $\operatorname{Spec}\overline{k}$ to $X$ for $\overline{k}$ an algebraic closure of $k$ (occasionally you want the separable closure here, but I don't know enough about that sort of situation to say much about it). One can think of these points as the $\overline{k}$ points of the base-change of $X$ to $\overline{k}$ via the natural map $\operatorname{Spec} \overline{k} \to \operatorname{Spec} k$. 
Closed points are images of geometric points, but it is not true that they are always in 1-1 correspondence. Consider $\mathfrak{p}=(x^2+1)\in\operatorname{Spec}(\mathbb{R}[x])$: it's a closed $\mathbb{C}$-point, but there are two distinct maps corresponding to $x\mapsto\pm i$, so there are two geometric points lying above it.
For the second question, my interpretation would be that $\cup Y_i$ as a closed subscheme of $X$ has exactly one closed point which is singular, and furthermore, the point has a unique structure as a geometric point. (If you were to think about more than one singular geometric point, you'd have to change this - I'm using here that exactly one geometric point means exactly one closed point. As soon as you have more than one geometric point, your situation could be like the example in the second paragraph, multiple instances of the example in this paragraph, or some mix of them both.)
