Projective Space and Algebraic Sets

This lecture covers the concept of projective space, where affine varieties are extended to avoid complications at infinity. It introduces projective n-space as a set of points with homogeneous coordinates, discussing the equivalence class and hyperplane. The lecture also explores projective algebraic sets, focusing on homogeneous polynomials, algebraic subsets, and the ideal generated by them. It explains the notion of a homogeneous ideal and the conditions for a set of generators to be homogeneous. Additionally, it delves into the concept of the irrelevant ideal and provides examples to illustrate the theory.