I want to prove the following statement:

> For any two points $x$ and $y$ in an irreducible variety $X$, there is a one-dimensional, irreducible subvariety $C\subseteq X$ containing $x$ and $y$.

Both [here](http://mathoverflow.net/questions/62843/path-connectedness-of-varieties/62883#62883) and [here](http://mathoverflow.net/questions/75320/connecting-points-on-a-variety-by-the-image-of-a-nonsingular-curve/75327#75327), the same argument is outlined to prove this statement. In both cases, Bertini's theorem is applied to the exceptional divisors of the blow-up $\tilde X$ of $X$ in $x$ and $y$. The curve joining the exceptional divisors in $\tilde X$ is then mapped to a curve connecting $x$ and $y$ in $X$. 

**Question:** I do not see why Bertini can be applied in the case where $x$ or $y$ are singular points: The exceptional divisor will not be smooth in general. Can you tell me why this works?

Another (but far less important) problem I have with the proof is the application of Chow's lemma - the variety is not required to be complete. This is irrelevant to me because I can assume $X$ to be quasi-projective. I was still wondering.