Algebraic Geometry
Definition at a Point
Local Ring of Affine Variety
Local Ring of Projective Variety
Properties and Examples
Derivations and Tangent Vectors
Cotangent Space
Definition via Maximal Ideal
Dimension Properties
Statement and Applications
Definition and Examples
Types of Singularities
Definition and Properties
Definition and Characterizations
Connection to Smoothness
Local Dimension
Relationship to Global Dimension
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6. Rational Maps and Birational Geometry
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8. Introduction to Schemes