This quarter will focus on coherent sheaf cohomology, the main technical tool in algebraic geometry. It gives invariants for algebraic varieties that are closely related to geometric problems. One application is the Riemann-Roch theorem for algebraic curves. Using that, we will classify algebraic curves of genus 0, 1, 2, 3. We begin the birational classification of algebraic varieties of any dimension.
In terms of Hartshorne, here are the topics I plan to cover. The course can only include part of the material in each section. Chapter I, sections 5-7 (smooth varieties, smooth curves, intersections in projective space). Chapter III, sections 1-5 (coherent sheaf cohomology) and some topics from the later sections (Serre duality, flat morphisms, smooth morphisms). Chapter IV, sections 1-5 (algebraic curves: Riemann-Roch theorem, embeddings in projective space, elliptic curves, canonical embedding). Introduction to birational geometry in higher dimensions.
Instructor: Burt Totaro.
Complex algebraic geometry (using differential geometry and complex analysis) is not the main focus of the class, but it is an important alternative point of view. The standard reference on complex algebraic geometry is Griffiths and Harris's Principles of Algebraic Geometry (Wiley); Huybrechts's Complex Geometry (Springer) is a shorter introduction.