I am wondering what is the difference between algebraic sets and algebraic varieties in complex projective space.

It seems that both are zero sets of polynomials, so what is the difference?


According to Wikipedia, some sources require that a variety be irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. In this case, a non-irreducible algebraic variety is called an algebraic set. In other sources, this convention is not followed and both cases are simply called variety.


If we give $\mathbb A^n$ (resp. $ \mathbb P^n$) their Zariski topology, an algebraic set is just a closed subset $V\subset \mathbb A^n$ (resp. $V\subset \mathbb P^n$).
An algebraic variety is a vastly more general general concept, the basic object in classical algebraic geometry (even if since Grothendieck we have the even more general notion of scheme).
Every algebraic set, which a priori is a topological subspace, can be endowed with the structure of algebraic variety: the supplementary datum consists of decreeing which functions on open subsets $U\subset V$ are considered acceptable, thus obtaining the ring $\mathcal O_V(U)$ of "regular" functions on $U$.
By using this procedure we get the affine (resp. projective varieties).
However there are many algebraic varieties which are neither affine nor projective, and thus are completely different from algebraic subsets: the simplest example is the plane with the origin removed, $V=\mathbb A^2\setminus \{(0,0)\}$.


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