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