This question is motivated by the question Path Connectedness of Varieties and some funny little theorem I was trying to prove. Let $X$ be a (quasiprojective smooth connected) algebraic variety over an algebraically closed field of an arbitrary characteristic. We know from the answer above that any two points of $X$ can be connected by a curve. Can we control its genus?

More precisely, we say that the two points $x$ and $y$ are $n$-path-connected, if there is a sequence of smooth curves from $x$ to $y$ such that every curve has genus less or equal to $n$. It is an equivalence relations and I am interested in its equivalence classes that are reasonably to be called $n$-path-connected components. Let me ask three precise questions.

1. Is it true that there exists $n$ such that $X$ is $n$-path-connected?
2. How do you find such minimal $n$?
3. Is there an algorithm/method for computing/describing $n$-path-connected components of $X$?

PS A curve of genus $g$ is an instructive example. It is $g$-path-connected but its $(g-1)$-path-connected components are points.