Let $X$ be a variety. Then, is $X$ path connected? And by path connected, I mean any two closed points $P, Q$ on the variety can be connected by the image of a finite number of nonsingular curves.

Given any two points on a projective variety, blow them up and re embed the blownup variety in P^N. Then by Bertini, any general linear section of the right codimension will meet the variety in an irreducible curve which also meets both exceptional divisors. Then blowing back down gives an irreducible curve connecting the original two points. Normalizing that curve gives a map from just one smooth connected curve that connects your two points. (I learned this trick from David Mumford.) – roy smith 11 hours ago 


In the affine setting over $\mathbb{C}$, an algebraic set is pathconnected in the analytic topology if it is irreducible (in fact, its smooth locus is pathconnected too). Conversely, it is irreducible if and only if it contains a dense open pathconnected subset of smooth points. See the appendix here. 

