These are my lecture notes from an undergraduate algebraic geometry class (math 137) I taught at Harvard in 2018, 2019, and 2020. They loosely follow Fulton's book on algebraic curves, and they are heavily influenced by an algebraic geometry course I took with Fulton in Fall 2010 at the University of Michigan.
Section 1: What is algebraic geometry?
Section 2: Algebraic sets
Section 3: The ideal of a subset of affine space
Section 4: Irreducibility and the Hilbert Basis Theorem
Section 5: Hilbert's Nullstellensatz
Section 6: Algebra detour
Section 7: Affine varieties and coordinate rings
Section 8: Regular maps
Section 9: Rational functions and local rings
Section 10: Affine plane curves
Section 11: Discrete valuation rings and multiplicities
Section 12: Intersection numbers
Section 13: Projective space
Section 14: Projective algebraic sets
Section 15: Homogeneous coordinate rings and rational functions
Section 16: Affine and projective varieties
Section 17: Morphism of projective varieties
Section 18: Projective plane curves
Section 19: Linear systems of curves
Section 20: Bézout's Theorem
Section 21: Abstract varieties
Section 22: Rational maps and dimension
Section 23: Rational maps of curves
Section 24: Blowing up a point in the plane