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 Dimensions of algebraic varieties 6 Tangent Spaces and Smoothness Tangent spaces and smoothness of algebraic varieties 7 Grassmannians Grassmannians as parameter spaces of linear subspaces 8 Divisors Weil divisors, Cartier divisors, and divisor class groups 9 Line Bundles and Vector Bundles Line bundles, invertible sheaves, and the Picard group 10 Linear Systems Complete linear systems, base loci, and ampleness 11 Canonical Line Bundle Canonical bundle and canonical divisor 12 Sheaf Cohomology Sheaf cohomology and its applications 13 Cohomology of Projective Space Bott’s formula and the cohomology of line bundles on projective space 14 Serre Duality Serre duality theorem and its applications 15 The Riemann–Roch Theorem for Curves The Riemann–Roch theorem for curves 16 The Riemann–Roch Theorem for Surfaces Intersection theory on surfaces and its applications 17 Kodaira Vanishing Theorem The Kodaira vanishing theorem and its applications 18 Chow Groups Chow groups and the cycle class map 19 Intersection Product The intersection product on Chow groups 20 Bézout’s Theorem Bézout’s theorem and its applications