Projective Algebraic Sets: Definitions and Properties

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Projective Varieties: An Algebraic Study

Covers the study of projective varieties and their relation to compact manifolds.

Complex Manifolds: GAGA Principle

Covers the GAGA principle, stating that any morphism on projective varieties is constant.

Exponential Maps: Properties and Applications in Lie Groups

Covers the properties of the exponential map in Lie groups and their algebras, including smoothness and the relationship between subgroups and algebras.

Field of Rational Functions

Explores the construction of the field of rational functions on a variety and its connection with local rings and dimensions.

Projective Space and Algebraic Sets

Introduces projective space and algebraic sets, discussing homogeneous coordinates, ideals, and generators.

Primary Decomposition: Behaviour and Localisation

Explores primary decomposition, recovering rings from spectra, and studying regular functions on spectra.

Weil Conjectures: Rationality and Function Equation

Covers the Weil conjectures on rationality, functional equation, and the Riemann hypothesis, exploring properties of varieties in algebraic geometry.

Tangent Spaces in Algebraic Geometry

Explores tangent spaces in algebraic geometry, defining them as finite-dimensional subspaces of derivations on varieties.

Morphisms between Affine Varieties

Covers defining morphisms between affine algebraic varieties and constructing morphisms using algebraic homomorphisms.

Albanese Morphism: Kodaira Dimension Zero

Explores the Albanese morphism of varieties with Kodaira dimension zero in positive characteristic.