Consider the affine space $\mathbb C^n$ and then, because of reasons, compactify it to obtain the projective space $\mathbb P^n$. One of the most basic axioms or propositions of geometry is that through two distinct points in $\mathbb P^n$ there passes a single projective line.
Somehow this seems to be very much a fact of projective space. Thus it seems natural to ask how this fact changes when we consider a different ambient space.
So, for simplicity, let's consider a compact complex surface $X$. Again, for simplicity, assume that it admits some rational curves
, of degree 1 even. How many of those pass through two (or more?) points of $X$? (We do not assume that a line passes through any two points. We just want to know that given two points that lie on a line, could another line pass through those points?)
From what I understand, basic intersection theory was developed to answer exactly these problems, so one should be able to obtain the answer by intersecting relevant Chern classes on the right vector bundles. Any reasonable attempt at this kind of answer leads to the badly understood problem of calculating the Chow groups and intersection products of surfaces or manifolds. Thus we arrive at a perhaps answerable
Question: Is there an example of a surface (or manifold) $X$ on which we know that more than one projective line passes through two given points? In that case, do we know how many points are necessary to determine a line?