This course on basic algebraic geometry aims to teach the importance of local rings in detecting smoothness or non-singularity in algebraic geometry. Students will learn about local ring isomorphism, function field isomorphism, and the geometric meaning of isomorphism of local rings. The course covers topics such as varieties, smooth hypersurfaces, smooth manifolds, rational functions, and function fields of affine and projective varieties. The teaching method involves explaining the concepts and theories related to local rings and their applications in algebraic geometry. This course is intended for learners interested in gaining a foundational understanding of algebraic geometry and its practical implications.
Overview
Syllabus
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.
Taught by
Ch 30 NIOS: Gyanamrit
