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.

The Importance of Local rings - A morphism is an isomorphism if it is a homeomorphis.
Any variety is a smooth hypersurface on an open dense subset.
Any Variety is a smooth manifold with or without Non-smooth boundary.
Local Ring isomorphism,Equals Function Field Isomorphism.
How local rings detect smoothness or non-singularity in algaebraic geometry.
Why Local rings provide calculus without limits for Algaebraic geometric pun intended?.
Geometric meaning of Isomorphism of Local Rings - Local rings are almost global.
The Importance of Local rings - A Rational functional in Every local ring is globally regular.
The D-uple embedding and the non-intrinsic nature of the homogeneous coordinate ring.
Fields of Rational Functions or Function fields of Affine and Projective varieties.