Is a union of varieties singular in the intersection of its irreducible components? Consider a separated, reduced scheme $X$ of finite type over some algebraically closed field $\Bbbk$. Let $X=X_1\cup\cdots X_r$  be its irreducible components, each of which is then a variety. Assume that $P\in X_i \cap X_j$  for $i\ne j$. I then wonder when $X$ is singular in $P$ - is this always the case or do I need some additional assumption? Also, can I weaken the assumptions on $X$?

For any Noetherian scheme $X$, if $P$ is a nonsingular point of $X$ then the local ring at $P$ is UFD.  In particular it's a domain (which is not entirely trivial to prove!).  But where two irreducible components cross the local ring will not be a domain, as you will have more than one minimal prime ideal.  
Maybe you don't even need Noetherian: I guess if the local ring at $P$ is not Noetherian then by definition the point cannot be nonsingular.  But for some reason I am uncomfortable calling it a "singular point": it just seems out of bounds of the singular / nonsingular dichotomy.
