# Path connectedness of varieties

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 non-singular curves.

• If $X$ is quasi-projective and of dim $\ge 2$, you can use Bertini's theorem on a sufficiently general hyperplane section through P and Q. Apr 24, 2011 at 14:36
• The version in Hartshorne requires $X$ has at most a finite number of singular points and that $X$ projective (or equivalently, projective with a finite number of points removed). Do you have a more general form in mind? Also, your answer leads to another question (probably a dumb one that I cannot think of): curves are parametrizable, i.e. any segment on a curve is an image of a non-singular curve? Apr 24, 2011 at 14:46
• Brian, J.C. Ottem is right. You can just use Bertini. To your question of whether every curve is the image of a non-singular one, the answer is yes, just take the normalization of the curve (see the section on curves in the first chapter of Hartshorne). I don't know what you mean by segment on a curve though. Apr 24, 2011 at 15:24
• Dear Karl Schwede: Thanks a lot for your answer. My question about the curve is indeed a very dumb one. Apr 24, 2011 at 15:29
• We have some more than adequate answers given in the comments. Would one of the commenters be willing to step up and actually answer the question in the formal MO sense? Apr 24, 2011 at 22:29

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