TITLE: $n$-path-connected components of a variety
QUESTION [6 upvotes]: 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.

Is it true that there exists $n$ such that $X$ is $n$-path-connected?
How do you find such minimal $n$?
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.

REPLY [2 votes]: Here is a variation on Tom's argument (for the 1st question) that works: I will also work over ${\mathbb C}$, although, I do not think it is important. I will be proving that $n$ can be bounded depending only on dimension of ambient space and degree $d$ of the subvariety $V\subset P^N$. The proof is by induction on dimension $N$ of the ambient projective space. 
Everything is clear, if $N=1$ or if $dim(V)=1$. Suppose that we have a bound $n(d,N-1)$;  consider a smooth subvariety $V\subset P^N$. Now, given any pair of points $p, q\in V$, I can find a projective hyperplane $H\subset P^N$ intersecting $V$ transversally (algebraic geometers would call it Bertini's transversality, topologists would call it Sard's theorem). By Lefschetz, if $dim(V)\ge 2$, then $W=V\cap H$ is connected. For two generic projective hyperplanes $H_p, H_q$ through $p, q$, intersections $W_p=H_p\cap W, W_q=H_q\cap W$ are nonempty, smooth and connected. Now, we are done by induction applied to $W, W_p, W_q$ (connect $p$ to $x\in W\cap W_p$, then connect $x$ to $y\in W\cap W_q$, etc.).