Another remark on local reducibilitya nd irreducibility of varieties for the interested. Neither local nor global reducibility imply each other, but a connected variety which is globally reduciblwe is also locally so at any point common to two or more components.
This is used to prove that such a variety is singular (not a manifold) at such points as follows:
for affine varieties, and all varieties are locally affine, irreducibilioty corresponds precisely to the ring of functions, or local ring of functions being a domain, i.e. to the ideal of functions defining the bariety being prime.
then there is a big theorem that at all smooth (non singular, manifold) points, the local is a regular local ring, and also that all such rings are domains, u.f.d.'s in fact (after Auslander and Buchsbaum in general).
so every non singular variety is everywhere locally irreducible. Hence the union of two varieties is locally reducible at any intersection point, hence also singular.
this interplay between zero divisors and components is just one aspect of the beautiful relationship between algebra and geometry revealed in modern algebraic geometry.