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I mean by an algebraic variety a locally closed(in Zarisky topolgy) subset of a projective space over an algebraically closed field. If this is not the case, I would like to know counter-examples in dimension both one and two over the field of complex numbers.

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A non-singular variety $V$ is alway locally integral, meaning that the local rings $O_{V,x}$ are integral. This is because by hypothesis, $O_{V,x}$ is a regular local ring, and regular local rings are integral domains (even UFD). This implies that two distinct irreducible components can never intersect (the local ring at an intersection point is not integral: it has as many minimal prime ideals as the nubmer of irreducible components passing through this point).

Now if $V$ is connected, it must be irreducible (otherwise there will always be some intersection points by connectedness).

Note that this holds over any field and even any noetherian regular schemes.

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