## 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.

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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. – J.C. Ottem 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? – Brian 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. – Karl Schwede 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. – Brian 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? – Pete L. Clark Apr 24 2011 at 22:29
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Thanks, Roy.  – Pete L. Clark Apr 25 2011 at 4:10