These are my lecture notes from a graduate topics class I taught at Harvard in Spring 2017. I haven't edited them very much, and there are probably (mostly minor) errors. Sections 3-5 and 7-9 mostly follow Lazarsfeld's excellent "Lectures on Linear Series." Sections 10-20 follow Lazarsfeld's "Positivity in Algebraic Geometry" books I and II.

Section 1: Linear systems

Section 2: Basics of intersection numbers and Chern class calculations

Section 3: Extensions of line bundles

Section 4: The Serre construction

Section 5: Elementary transformations

Section 6: Tautological bundles on Hilbert schemes

Section 7: Bogomolov Instability

Section 8: Reider's Theorem

Section 9: Gonality of complete intersection curves

Section 10: Q-divisors and R-divisors

Section 11: Cones I -- ample, nef, and curves

Section 12: Iitaka dimension and Kodaira dimension

Section 13: Semiample line bundles

Section 14: Iitaka fibrations

Section 15: Big line bundles

Section 16: Cones II -- big and pseudoeffective

Section 17: Volumes of line bundles

Section 18: Fujita's approximation theorem

Section 19: Section rings and Zariski's construction

Section 20: Cutkosky's construction and lots of examples