Affine Varieties
Affine varieties and their basic properties
Algebraic Varieties
Projective Varieties
Projective varieties and homogeneous coordinates
Quasi-Projective Varieties
Quasi-projective varieties and regular maps
Rational Maps
Rational maps and birational equivalence
Dimension
Dimension of algebraic varieties
Tangent Spaces and Smoothness
Tangent spaces and smoothness of algebraic varieties
Grassmannians
Grassmannians as parameter spaces of linear subspaces
Divisors
Weil divisors, Cartier divisors, and divisor class groups
Line Bundles and Vector Bundles
Line bundles, invertible sheaves, and the Picard group
Linear Systems
Complete linear systems, base loci, and ampleness
Canonical Bundle
Canonical Bundle and Canonical Divisor
Sheaf Cohomology
Sheaf cohomology and its applications
Cohomology of Projective Space
Bott’s formula and the cohomology of line bundles on projective space
Serre Duality
Serre duality theorem and its applications
The Riemann–Roch Theorem for Curves
The Riemann–Roch theorem for curves
The Riemann–Roch Theorem for Surfaces
Intersection theory on surfaces and its applications
Kodaira Vanishing Theorem
The Kodaira vanishing theorem and its applications
Chow Groups
Chow groups and the cycle class map
Intersection Product
The intersection product on Chow groups
Bézout’s Theorem
Bézout’s theorem and its applications