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Let $X$ be an algebraic variety over, say, some algebraically closed field. Consider a point in the intersection of two irreducible components. Is this point always singular? If yes, can you prove me why it is?

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Please accept your older questions. – Listing Nov 26 '11 at 14:59
Which older questions? – user17090 Nov 26 '11 at 16:01
up vote 4 down vote accepted

Yes, a point is nonsingular if and only if the local ring at that point is a regular local ring, and at the intersection of two irreducible components, the local ring won't be an integral domain even.

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