Projective Algebraic Sets: Definitions and Properties

This lecture covers the definitions and properties of projective algebraic sets, including the review of concepts from the 'Algebraic curves' course. It discusses homogeneous ideals, projective algebraic sets, quasi-projective algebraic sets, standard open charts, dehomogenization, homogenization, and morphisms for quasi-projective varieties. The lecture also explores the field of rational functions, local rings, and localization in algebraic geometry, emphasizing the importance of local structure and the role of affine varieties. Key theorems such as Bezout's Theorem are presented to illustrate the application of algebraic concepts in geometry.