a projective variety

Two parallel lines never meet in the ordinary plane, but if you stand on a railway track they appear to converge at a point on the horizon. Projective space is the rigorous version of adding those horizon points: it completes affine space with points at infinity so that, for instance, any two distinct lines meet in exactly one point. A projective variety is the zero locus of polynomials living in this completed, more symmetric space — and that completion is what makes intersection theory clean (Bézout's theorem counts every intersection, including those at infinity).

Projective n-space P^n over k is the set of lines through the origin in k^{n+1}: a point is an equivalence class [a_0 : a_1 : ... : a_n] of nonzero (n+1)-tuples, where two tuples are identified if one is a nonzero scalar multiple of the other. An ordinary polynomial f does not give a well-defined function here, since scaling all coordinates by lambda scales f unpredictably. The fix is to use homogeneous polynomials, in which every term has the same total degree d: then f(lambda a) = lambda^d f(a), so the condition f = 0 is independent of the chosen representative. A projective variety is the common zero locus V(S) of a set S of homogeneous polynomials; it is irreducible if it is not a union of two strictly smaller such loci.

Why bother? Compactness and completeness. Over C a projective variety is compact in the usual topology, so functions and cohomology behave well, and the loss of solutions to infinity that plagues affine geometry disappears. The standard cover of P^n by n+1 affine charts (set one coordinate to 1) shows that projective varieties are locally affine, so all affine machinery transfers. A caveat: the only global regular functions on an irreducible projective variety are the constants — the rich structure lives in homogeneous coordinate rings, rational functions, and line bundles, not in honest functions.