Projective variety

Summary

A projective variety is an irreducible subset of projective space $\mathbb{P}^n$ over an algebraically closed field cut out by homogeneous polynomial equations. Projective varieties are the algebraic analogue of compact spaces. Over $\C$ they are compact in the usual topology, and images of projective varieties under morphisms are closed. Most classical algebraic geometry, such as plane curves, Bézout's theorem and intersection theory, takes place on projective varieties.