Algebraic Geometry

7.

Local Properties and Singularities

7.1.

Local Rings of Varieties

7.1.1.

Definition at a Point

7.1.2.

Local Ring of Affine Variety

7.1.3.

Local Ring of Projective Variety

7.1.4.

Properties and Examples

7.2.

7.2.1.

Geometric Definition for Hypersurfaces

7.2.2.

Intrinsic Definition via Derivations

7.2.2.1.

Derivations and Tangent Vectors

7.2.2.2.

Cotangent Space

7.2.3.

Zariski Tangent Space

7.2.3.1.

Definition via Maximal Ideal

7.2.3.2.

Dimension Properties

7.3.

Smooth and Singular Points

7.3.1.

Definition of Smooth Points

7.3.2.

Jacobian Criterion

7.3.2.1.

Statement and Applications

7.3.3.

Singular Points

7.3.3.1.

Definition and Examples

7.3.3.2.

Types of Singularities

7.3.4.

7.3.4.1.

Definition and Properties

7.4.

Regular Local Rings

7.4.1.

Definition and Characterizations

7.4.2.

Connection to Smoothness

7.5.

Dimension at a Point

7.5.1.

Local Dimension

7.5.2.

Relationship to Global Dimension

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8. Introduction to Schemes