On Wikipedia, an Algebraic Variety is defined (I think) as either an Affine Variety or a Projective Variety. From my understanding, an affine variety is the set of common zeroes of a collection of polynomials. And from the wiki, a projective variety is as well, but is also a closed subvariety of a projective space, where closed refers to the Zariski topology. Finally, a quasi-projective variety is a locally closed subset of a projective variety. The quasi-* is used to define projective variety, but is defined in terms of it so it's a bit cyclical.
More specifically (as specific as I could find), given a set $S \subset F[x_1, \dotsc ,x_n]$ of polynomials, the affine variety defined by $S$ is the set
$$V(S) := \{a \in A^n \mid f(a) = 0\ \ \ \forall f \in S\}$$
This is an affine subvariety of $A^n$ or simply a variety.
That's as specific as I found definitions for affine variety and projective variety.
I am wondering if one could:
- Provide a formal definition of algebraic varieties (independent of sheaves/schemes).
- And if the definition includes projective/affine varieties, to also define those. Trying to get an intuition of a formal definition of these 3 (affine, projective, and algebraic varieties).
Thank you for your help.