1 Affine Varieties Affine varieties and their basic properties 2 Projective Varieties Projective varieties and homogeneous coordinates 3 Quasi-Projective Varieties Quasi-projective varieties and regular maps 4 Rational Maps Rational maps and birational equivalence 5 Dimension Equivalent definitions of dimension for algebraic varieties 6 Tangent Spaces and Smoothness Tangent spaces and smoothness of algebraic varieties 7 Grassmann Varieties Grassmannians as parameter spaces of linear subspaces 8 Algebraic Groups Algebraic group action 9 Divisors Weil divisors, Cartier divisors, and divisor class groups 10 Line Bundles and Vector Bundles Line bundles, invertible sheaves, and the Picard group 11 Linear Systems Complete linear systems, base loci, and ampleness 12 Canonical Line Bundle Canonical bundle and canonical divisor 13 Sheaf Cohomology Sheaf cohomology and its applications 14 Cohomology of Projective Space Bott’s formula and the cohomology of line bundles on projective space 15 Serre Duality Serre duality theorem and its applications 16 The Riemann–Roch Theorem for Curves The Riemann–Roch theorem for curves 17 The Riemann–Roch Theorem for Surfaces Intersection theory on surfaces and its applications 18 Kodaira Vanishing Theorem The Kodaira vanishing theorem and its applications 19 Chow Groups Chow groups and the cycle class map 20 Intersection Product The intersection product on Chow groups