**Course description**: This is the second of a two-quarter introduction to algebraic geometry.

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, at least. 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 II, sections 6-8 (divisors and line bundles, projective embeddings, tangent bundle). 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.

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.