In Hartshorne's proof of a result of Igusa (see III, 9.13 of Hartshorne) he claims without proof that any two closed points on a variety can be connected by the image of a nonsingular curve, or by a finite number of such curves. I've seen something like this come up in other places as well, and I don't know why such a fact should be true or so obvious as not to explain. Can anyone explain it to me? Thanks

Mumford's book "Abelian Varieties" contains a proof of the following statement: given two points $x$ and $y$ on a variety, there is an irreducible curve containing both (Lemma on p. 56 in the section on the Theorem of the Cube). The normalization of the curve is nonsingular, so this yields the desired result. The proof is roughly what Artie said. Arguing by dimension, it is enough to produce an irreducible codimension $1$ subvariety containing the points. Pick a blowup $f: X' \to X$ such that $X'$ is projective and the fibers $f^{1}(x)$, $f^{1}(y)$ are positive dimensional. Now fix a projective embedding of $X'$ and take a general hyperplane section $H$. This section is irreducible (Bertini) and meets the fibers $f^{1}(x)$ and $f^{1}(y)$ (for dimensional reasons). The image of $H$ under $f$ is the desired subvariety. Question for the experts: What's an example where it is impossible to take the curve to be smooth? 

