Let $X$ be a projective variety over an algebraically closed field $k$. Let $x,y \in X(k)$, call $x \sim y$ iff there exists a map $f : \mathbb{P}^1_k \rightarrow X$ such that $x,y \in f(\mathbb{P}^1_k)$.
Is the above relation an equivalence relation on $X(k)$? If not what would be a counterexample?
I am just posting Mohan's comment here so as to close this question.
"I think in general, transitivity can fail. Think of blowing up points on, say an abelian surface, then you can have two rational curves $L,M$ meeting at a point, say $P$. Then points on $L$ are related to $P$ and same for $M$, but points other than $P$ one on $L$, another on $M$ are not related.That is why the relation `linearly connected', allowing several rational curves in X is better." – Mohan