Notes from Math 252 -- Linear systems and positivity of vector bundles

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