Basic Algebraic Geometry
Explore the fundamental concepts of algebraic geometry in this comprehensive 9-hour course. Delve into the significance of local rings and their role in detecting smoothness and non-singularity. Examine how local ring isomorphisms relate to function field isomorphisms and discover the geometric implications of these relationships. Investigate the concept of varieties as smooth hypersurfaces and manifolds, and learn how local rings provide a framework for calculus without limits in algebraic geometry. Analyze the global properties of rational functions in local rings and understand the non-intrinsic nature of homogeneous coordinate rings through the D-uple embedding. Study the fields of rational functions and function fields of affine and projective varieties to gain a deeper understanding of algebraic geometric structures.