This question is motivated by the question <a href="https://mathoverflow.net/questions/62843/path-connectedness-of-varieties">Path Connectedness of Varieties</a> 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.